Calculus/Points, paths, surfaces, and volumes

This chapter will provide an intuitive interpretation of vector calculus using simple concepts such as multi-points, multi-paths, multi-surfaces, and multi-volumes. Scalar fields will not be simply treated as a function $$f : \R^3 \to \R$$ that returns a number given an input point, and vector fields will not be simply treated as a function $$\mathbf{F} : \R^3 \to \R^3$$ that returns a vector given an input point.

Basic structures
The basic structures are multi-points, multi-paths, multi-surfaces, and multi-volumes.

Multi-points
A point $$\mathbf{q}_0$$ is an arbitrary position. A "multi-point" is a set of point/weight pairs: $$\mathbf{Q} = \{(\mathbf{q}_1,w_1), (\mathbf{q}_2,w_2), ..., (\mathbf{q}_k,w_k)\}$$ where $$w_i$$ is the "weight" that is assigned to point $$\mathbf{q}_i$$. Given two point/weight pairs $$(\mathbf{q},w_1)$$ and $$(\mathbf{q},w_2)$$ that cover the same point $$\mathbf{q}$$, the weights add up to get $$(\mathbf{q},w_1+w_2)$$ which replaces $$(\mathbf{q},w_1)$$ and $$(\mathbf{q},w_2)$$. Any pair $$(\mathbf{q},0)$$ is removed. $$\mathbf{Q}$$ can consist of infinitely many points, and each point may have an infinitesimal weight.

An arbitrary point $$\mathbf{q}_0$$ can be described by the scalar field $$\delta_0(\mathbf{q};\mathbf{q}_0) = \left\{\begin{array}{cc} +\infty^3 & (\mathbf{q} = \mathbf{q}_0) \\ 0 & (\mathbf{q} \neq \mathbf{q}_0) \end{array}\right.$$. This is the "Dirac delta function" centered on point $$\mathbf{q}_0$$. The $$+\infty^3$$ is the inverse of an infinitely small volume that wraps point $$\mathbf{q}_0$$. To further explain this, let $$\omega_0(\mathbf{q}_0,v)$$ be a tiny volume with volume $$v$$ that wraps point $$\mathbf{q}_0$$. $$\delta_0(\mathbf{q};\mathbf{q}_0)$$ can be approximated by $$\Delta_0(\mathbf{q};\mathbf{q}_0,v) = \left\{\begin{array}{cc} 1/v & (\mathbf{q} \in \omega_0(\mathbf{q}_0,v)) \\ 0 & (\mathbf{q} \notin \omega_0(\mathbf{q}_0,v)) \end{array}\right.$$. A mass of 1 is being crammed into $$\omega_0(\mathbf{q}_0,v)$$ yielding an infinitely high density. Since $$\delta_0(\mathbf{q};\mathbf{q}_0)$$ is essentially a density function, it brings with it the units $$[\text{length}^{-3}]$$.

Multi-point $$\mathbf{Q} = \{(\mathbf{q}_1,w_1), (\mathbf{q}_2,w_2), ..., (\mathbf{q}_k,w_k)\}$$ can be described by the scalar field $$\delta_0(\mathbf{q};\mathbf{Q}) = \sum_{i=1}^k w_i\delta_0(\mathbf{q};\mathbf{q}_i)$$. If $$\mathbf{Q}$$ consists of infinitely many points with each point having infinitesimal weight, then $$\delta_0(\mathbf{q};\mathbf{Q})$$ is a density function.

In the image below, the multi-point in the left panel is converted to the scalar field in the center panel by averaging the point weight over each cell. The volume of each cell should be infinitesimal. The multi-point in the right panel corresponds to the same scalar field, and is in a more canonical form where oppositely weighted points have cancelled out.



The image below shows how a continuous scalar field $$\rho : \R^3 \to \R$$ can be generated as a collection of points. Consider position $$\mathbf{q}_0$$ and the infinitesimal volume $$\omega_0(\mathbf{q}_0,v)$$ with volume $$v$$. The total point weight contained by $$\omega_0(\mathbf{q}_0,v)$$ is $$\iiint_{\mathbf{q} \in \omega_0(\mathbf{q}_0,v)} \rho(\mathbf{q})dV \approx v \cdot \rho(\mathbf{q}_0) $$. This weight of $$v \cdot \rho(\mathbf{q}_0)$$ is then split up over an arbitrarily large number of points that are scattered over the volume $$\omega_0(\mathbf{q}_0,v)$$.



In summary, a multi-point is denoted by a scalar field that quantifies the density at each point, and any scalar field that quantifies density at each point is best interpreted as a multi-point.

Multi-paths
A simple path (also called a simple curve) $$C$$ is an oriented continuous curve that extends from a starting point $$C(0)$$ to an ending point $$C(1)$$. Intermediate points are indexed by $$t \in [0,1]$$ and are denoted by $$C(t)$$. A simple path should be continuous (no breaks), and may intersect or retrace itself. A "multi-path" is a set of simple-path/weight pairs: $$\mathbf{C} = \{(C_1,w_1), (C_2,w_2), ..., (C_k,w_k)\}$$ where $$w_i$$ is the weight that is assigned to path $$C_i$$. Given two path/weight pairs $$(C,w_1)$$ and $$(C,w_2)$$ that cover the same path $$C$$, the weights add up to get $$(C,w_1+w_2)$$ which replaces $$(C,w_1)$$ and $$(C,w_2)$$. Any pair $$(C,0)$$ is removed. In addition given two path/weight pairs $$(C_1,w)$$ and $$(C_2,w)$$ with the same weight $$w$$ and $$C_1(1) = C_2(0)$$, then $$C_1$$ and $$C_2$$ can be linked end-to-end to get the pair $$(C_1+C_2,w)$$ which replaces $$(C_1,w)$$ and $$(C_2,w)$$. Assigning a path a negative weight effectively reverses its orientation: if $$-C$$ denotes path $$C$$ with the opposite orientation, then $$(C,-w)$$ is equivalent to $$(-C,w)$$. $$\mathbf{C}$$ can consist of infinitely many paths, and each path may have an infinitesimal weight.



An arbitrary curve $$C$$ can be described by the vector field $$\delta_1(\mathbf{q};C) = \left\{\begin{array}{cc} (+\infty^2)\hat{\mathbf{n}}(\mathbf{q};C) & (\mathbf{q} \in C) \\ \mathbf{0} & (\mathbf{q} \notin C) \end{array}\right.$$. This is the "Dirac delta function" for the curve $$C$$. $$\hat{\mathbf{n}}(\mathbf{q};C)$$ is the unit length tangent vector to path $$C$$ at point $$\mathbf{q} \in C$$. $$\hat{\mathbf{n}}(\mathbf{q};C) = \mathbf{0}$$ if $$\mathbf{q} \notin C$$. If there are multiple tangent vectors due to $$C$$ intersecting itself, then $$\hat{\mathbf{n}}(\mathbf{q};C)$$ is the sum of these tangent vectors. The $$+\infty^2$$ is the inverse of the cross-sectional area of an infinitely thin tube that encloses $$C$$. To further explain this, let $$\omega_1(C,a)$$ be a thin tube with cross-sectional area $$a$$ that encloses $$C$$. $$\delta_1(\mathbf{q};C)$$ can be approximated by $$\Delta_1(\mathbf{q};C,a) = \left\{\begin{array}{cc} (1/a)\hat{\mathbf{n}}_*(\mathbf{q};C,a) & (\mathbf{q} \in \omega_1(C,a)) \\ \mathbf{0} & (\mathbf{q} \notin \omega_1(C,a)) \end{array}\right.$$. $$\hat{\mathbf{n}}_*(\mathbf{q};C,a)$$ is the generalization of $$\hat{\mathbf{n}}(\mathbf{q};C)$$ to the tube $$\omega_1(C,a)$$. A path weight of 1 is being crammed into the cross-sectional area of $$\omega_1(C,a)$$ yielding an infinitely high path density. Since $$\delta_1(\mathbf{q};C)$$ is essentially a density over area, it brings with it the units $$[\text{length}^{-2}]$$.

The image to the right gives a depiction of the Dirac delta function for a simple curve. The vector field $$\delta_1(\mathbf{q};C)$$ is $$\mathbf{0}$$ everywhere outside of an infinitely thin tube that encloses the path. Inside the tube, the vectors are parallel to the path, and have a magnitude equal to the inverse of the cross-sectional area. The Dirac delta function is the limit as the tube becomes infinitely thin.

Multi-path $$\mathbf{C} = \{(C_1,w_1), (C_2,w_2), ..., (C_k,w_k)\}$$ can be described by the vector field $$\delta_1(\mathbf{q};\mathbf{C}) = \sum_{i=1}^k w_i\delta_1(\mathbf{q};C_i)$$. If $$\mathbf{C}$$ consists of infinitely many paths with each path having infinitesimal weight, then $$\delta_1(\mathbf{q};\mathbf{C})$$ is a flow density function.

In the image below, the multi-path in the left panel is converted to the vector field in the center panel by computing the total displacement in each cell and averaging over the volume. The volume of each cell should be infinitesimal. The multi-path in the right panel corresponds to the same vector field, and is in a more canonical form where the individual paths do not cross each other.



In summary, a multi-path is denoted by a vector field that quantifies the path/flow density at each point, and any vector field that quantifies a flow density at each point (such as current density) is best interpreted as a multi-path. (Flow density is a vector that points in the net direction of a flow, and has a length equal to the flow rate per unit area through a surface that is perpendicular to the net flow.)

Multi-surfaces
A simple surface $$\sigma$$ is an oriented continuous surface. A simple surface may intersect or fold back on itself. A "multi-surface" is a set of simple-surface/weight pairs: $$\mathbf{S} = \{(\sigma_1,w_1), (\sigma_2,w_2), ..., (\sigma_k,w_k)\}$$ where $$w_i$$ is the weight that is assigned to surface $$\sigma_i$$. Given two surface/weight pairs $$(\sigma,w_1)$$ and $$(\sigma,w_2)$$ that cover the same surface $$\sigma$$, the weights add up to get $$(\sigma,w_1+w_2)$$ which replaces $$(\sigma,w_1)$$ and $$(\sigma,w_2)$$. Any pair $$(\sigma,0)$$ is removed. In addition given two surface/weight pairs $$(\sigma_1,w)$$ and $$(\sigma_2,w)$$ with the same weight $$w$$, then $$\sigma_1$$ and $$\sigma_2$$ can be combined to get the pair $$(\sigma_1+\sigma_2,w)$$ which replaces $$(\sigma_1,w)$$ and $$(\sigma_2,w)$$. Assigning a surface a negative weight effectively reverses its orientation: if $$-\sigma$$ denotes surface $$\sigma$$ with the opposite orientation, then $$(\sigma,-w)$$ is equivalent to $$(-\sigma,w)$$. $$\mathbf{S}$$ can consist of infinitely many surfaces, and each surface may have an infinitesimal weight.

An arbitrary surface $$\sigma$$ can be described by the vector field $$\delta_2(\mathbf{q};\sigma) = \left\{\begin{array}{cc} (+\infty)\hat{\mathbf{n}}(\mathbf{q};\sigma) & (\mathbf{q} \in \sigma) \\ \mathbf{0} & (\mathbf{q} \notin \sigma) \end{array}\right.$$. This is the "Dirac delta function" for the surface $$\sigma$$. $$\hat{\mathbf{n}}(\mathbf{q};\sigma)$$ is the unit length normal vector to surface $$\sigma$$ at point $$\mathbf{q} \in \sigma$$. $$\hat{\mathbf{n}}(\mathbf{q};\sigma) = \mathbf{0}$$ if $$\mathbf{q} \notin \sigma$$. If there are multiple normal vectors due to $$\sigma$$ intersecting itself, then $$\hat{\mathbf{n}}(\mathbf{q};\sigma)$$ is the sum of these normal vectors. The $$+\infty$$ is the inverse of the thickness of an infinitely thin membrane that encloses $$\sigma$$. To further explain this, let $$\omega_2(\sigma,t)$$ be a thin membrane with thickness $$t$$ that encloses $$\sigma$$. $$\delta_2(\mathbf{q};\sigma)$$ can be approximated by $$\Delta_2(\mathbf{q};\sigma,t) = \left\{\begin{array}{cc} (1/t)\hat{\mathbf{n}}_*(\mathbf{q};\sigma,t) & (\mathbf{q} \in \omega_2(\sigma,t)) \\ \mathbf{0} & (\mathbf{q} \notin \omega_2(\sigma,t)) \end{array}\right.$$. $$\hat{\mathbf{n}}_*(\mathbf{q};\sigma,t)$$ is the generalization of $$\hat{\mathbf{n}}(\mathbf{q};\sigma)$$ to the membrane $$\omega_2(\sigma,t)$$. A surface weight of 1 is being sandwiched into the thickness of $$\omega_2(\sigma,t)$$ yielding an infinitely high surface density. Since $$\delta_2(\mathbf{q};\sigma)$$ is essentially a density over length, it brings with it the units $$[\text{length}^{-1}]$$.

Multi-surface $$\mathbf{S} = \{(\sigma_1,w_1), (\sigma_2,w_2), ..., (\sigma_k,w_k)\}$$ can be described by the vector field $$\delta_2(\mathbf{q};\mathbf{S}) = \sum_{i=1}^k w_i\delta_2(\mathbf{q};\sigma_i)$$. If $$\mathbf{S}$$ consists of infinitely many surfaces with each surface having infinitesimal weight, then $$\delta_2(\mathbf{q};\mathbf{S})$$ is a rate-of-gain function.

In the image below, the multi-surface in the left panel is converted to the vector field in the center panel by computing the total surface in each cell and averaging over the volume. The volume of each cell should be infinitesimal. The multi-surface in the right panel corresponds to the same vector field, and is in a more canonical form where the individual surfaces do not intersect each other.



In summary, a multi-surface is denoted by a vector field that quantifies the rate of gain at each point. To describe the rate-of-gain, imagine that passing through a surface in the preferred direction gives "energy". The rate of gain is a vector that points in the direction that yields the greatest rate of energy increase per unit length, and has a length equal to the maximum rate of energy increase per unit length. Any vector field that quantifies a rate of gain at each point (such as a force field) is best interpreted as a multi-surface.

Multi-volumes
A volume $$\Omega$$ is an arbitrary region of space. A "multi-volume" is a set of volume/weight pairs: $$\mathbf{U} = \{(\Omega_1,w_1), (\Omega_2,w_2), ..., (\Omega_k,w_k)\}$$ where $$w_i$$ is the "weight" that is assigned to volume $$\Omega_i$$. Given two volume/weight pairs $$(\Omega,w_1)$$ and $$(\Omega,w_2)$$ that cover the same volume $$\Omega$$, the weights add up to get $$(\Omega,w_1+w_2)$$ which replaces $$(\Omega,w_1)$$ and $$(\Omega,w_2)$$. Any pair $$(\Omega,0)$$ is removed. In addition given two volume/weight pairs $$(\Omega_1,w)$$ and $$(\Omega_2,w)$$ with the same weight $$w$$ and $$\Omega_1 \cap \Omega_2 = \emptyset$$, then $$\Omega_1$$ and $$\Omega_2$$ can be combined to get the pair $$(\Omega_1 \cup \Omega_2, w)$$ which replaces $$(\Omega_1,w)$$ and $$(\Omega_2,w)$$. $$\mathbf{U}$$ can consist of infinitely many volumes, and each volume may have an infinitesimal weight.

An arbitrary volume $$\Omega$$ can be described by the scalar field $$\delta_3(\mathbf{q};\Omega) = \left\{\begin{array}{cc} 1 & (\mathbf{q} \in \Omega) \\ 0 & (\mathbf{q} \notin \Omega) \end{array}\right.$$. This is the "Dirac delta function" analog for volumes, and is essentially an indicator function that indicates whether or not a position is contained by $$\Omega$$ or not, 1 being yes and 0 being no. Since $$\delta_3(\mathbf{q};\Omega)$$ is simply an indicator function, it brings with it no units (it is dimensionless).

Multi-volume $$\mathbf{U} = \{(\Omega_1,w_1), (\Omega_2,w_2), ..., (\Omega_k,w_k)\}$$ can be described by the scalar field $$\delta_3(\mathbf{q};\mathbf{U}) = \sum_{i=1}^k w_i\delta_3(\mathbf{q};\Omega_i)$$. If $$\mathbf{U}$$ consists of infinitely many volumes with each volume having infinitesimal weight, then $$\delta_3(\mathbf{q};\mathbf{U})$$ is a potential function.

In the image below, the multi-volume in the left panel is converted to the scalar field in the center panel by averaging the volume weight in each cell. The volume of each cell should be infinitesimal. The multi-volume in the right panel corresponds to the same scalar field, and is in a more canonical form where oppositely weighted volumes have cancelled out, and the remaining volume has diffused to fill each cell.



In summary, a multi-volume is denoted by a scalar field that quantifies a potential at each point, and any scalar field that quantifies a potential at each point is best interpreted as a multi-volume.

At infinity
An important requirement is that all multi-points, multi-paths, multi-surfaces, and multi-volumes not extend to infinity. All structures can extend to an arbitrarily large range, as long as this range is not unbounded. Allowing the structures to extend to infinity will cause problems in the later discussions.

Totals
These sections will describe the total weight of multi-points, the total displacement of multi-paths, the total surface of multi-surfaces, and the total volumes of multi-volumes.

Total point weight
Given a multi-point $$\mathbf{Q} = \{(\mathbf{q}_1,w_1), (\mathbf{q}_2,w_2), ..., (\mathbf{q}_k,w_k)\}$$, the total point weight is clearly $$\sum_{i=1}^k w_i$$. Given a scalar field $$\rho$$ that denotes a multi-point, the total weight of $$\rho$$ is $$\iiint_{\mathbf{q} \in \R^3} \rho(\mathbf{q})dV$$. Given a simple point $$\mathbf{q}_0$$, the total weight is 1 so $$\iiint_{\mathbf{q} \in \R^3} \delta_0(\mathbf{q};\mathbf{q}_0)dV = 1$$.

Total displacement


Given a simple path $$C$$ that starts at point $$C(0)$$ and ends at point $$C(1)$$, the total displacement generated by $$C$$ is $$\int_{\mathbf{q} \in C} d\mathbf{q} = C(1)-C(0)$$. This displacement is solely dependent on the endpoints as indicated by the top image to the right.

The displacement generated by a closed loop is $$\mathbf{0}$$.

Given a multi-path $$\mathbf{C} = \{(C_1,w_1), (C_2,w_2), ..., (C_k,w_k)\}$$, the total displacement generated by $$\mathbf{C}$$ is $$\sum_{i=1}^k w_i\int_{\mathbf{q} \in C_i} d\mathbf{q} = \sum_{i=1}^k w_i(C_i(1)-C_i(0))$$.

Given a vector field $$\mathbf{J}$$ that denotes a multi-path, the total displacement generated by $$\mathbf{J}$$ is $$\iiint_{\mathbf{q} \in \R^3} \mathbf{J}(\mathbf{q})dV$$. Since the displacement generated by a simple path $$C$$ is $$\int_{\mathbf{q} \in C} d\mathbf{q} = C(1)-C(0)$$, it is the case that $$\iiint_{\mathbf{q} \in \R^3} \delta_1(\mathbf{q};C)dV = \int_{\mathbf{q} \in C} d\mathbf{q} = C(1)-C(0)$$.



One important observation from $$\int_{\mathbf{q} \in C} d\mathbf{q} = \iiint_{\mathbf{q} \in \R^3} \delta_1(\mathbf{q};C)dV$$ is that given a path integral over path $$C$$, the differential $$d\mathbf{q}$$ is equal to $$\delta_1(\mathbf{q};C)dV$$ in a volume integral: $$\int_{\mathbf{q} \in C} f(\mathbf{q},d\mathbf{q}) = \iiint_{\mathbf{q} \in \R^3} f(\mathbf{q},\delta_1(\mathbf{q};C)dV)$$ provided that function $$f$$ is linear in the second parameter. In the lower image to the right, the displacement differential $$d\mathbf{q} = \hat{\mathbf{n}} \cdot \Delta l$$ is equated to the volume differential $$\left(\frac{\hat{\mathbf{n}}}{\Delta A}\right)dV = \delta_1(\mathbf{q};C)dV$$ by diffusing the path over an infintely thin cross-sectional area $$\Delta A$$. The integrand at points outside of the infinitely thin tube is 0: for all points $$\mathbf{q} \notin C$$, $$f(\mathbf{q},\delta_1(\mathbf{q};C)dV) = f(\mathbf{q},\mathbf{0}) = 0$$.

Total surface vector


Given an arbitrary oriented surface $$\sigma$$, its "counter-clockwise boundary", denoted by $$\partial\sigma$$, is the boundary of $$\sigma$$ whose orientation is determined in the following manner: Looking at $$\sigma$$ so that the preferred direction through $$\sigma$$ is oriented towards the viewer, the boundary $$\partial\sigma$$ wraps $$\sigma$$ in a counter-clockwise direction.

Given a flat surface as shown in the image to the right, this surface can be quantified by the "surface vector" which is a vector that is perpendicular (normal) to the surface in the preferred orientation, and has a length equal to the area of the surface. In the image to the right, a flat surface has an area of $$A$$ and is oriented to be perpendicular to unit-length normal vector $$\hat{\mathbf{n}}$$. The "surface vector" of this surface is $$A \cdot \hat{\mathbf{n}}$$.

Given a non-flat surface $$\sigma$$, the total surface vector of $$\sigma$$ is computed by summing the surface vectors of each infinitesimal portion of $$\sigma$$. The total surface vector is $$\mathbf{S} = \iint_{\mathbf{q} \in \sigma} d\mathbf{S}$$.

In a manner similar to how the total displacement of a path is solely a function of the endpoints, the total surface vector of a surface is solely a function of its counter-clockwise boundary. This is not intuitive, and will be explained in greater detail below using two approaches:



Generalizing from surfaces in 2D space
Below are shown two images related to surface vectors in 2D space. The image to the left shows surface vectors in 2D space. In 2 dimensions, surfaces are called 1D surfaces and are similar to paths. The boundary of a 1D surface consists of 2 points. The surface vector of a 1D surface segment is a 90 degree rotation of the segment and is oriented in the direction of the surface's orientation. The total surface vector of a 1D surface is the sum of all of the surface vectors of the individual components. For each component of the surface, the surface vector is a 90 degree rotation of the displacement that traverses the component, so the total surface vector is a 90 degree rotation of the displacement between the points that form the boundary of the surface. This proves that in two dimensions, the total surface vector depends only on the boundary of the 1D surface.

The image to the right extrudes the 1D surfaces in two dimensional space into 2D "ribbons" in 3 dimensional space. At the top a closed "ribbon" is shown. This "ribbon" is a surface that is always parallel to the vertical dimension, and whose boundary forms two identical loops that are vertically displaced from each other. The boundary loops are also perpendicular to the vertical dimension. The ribbon itself is partitioned into tiny rectangles whose height is equal to that of the ribbon. To the bottom left, a view of the same ribbon from the top down is shown. It can be seen that the length of the each surface vector is proportional to the length of the corresponding rectangular segment, since the heights are all uniform. To the bottom right, by rotating the surface vectors 90 degrees around the vertical dimension, the surface vectors now sum to $$\mathbf{0}$$, so the sum of the unrotated surface vectors is also $$\mathbf{0}$$.

The fact that the total surface vector of a closed ribbon is $$\mathbf{0}$$ means that if relief is added to a surface without changing its boundaries, the total surface vector is conserved. The two left images below give examples of distorting the interior of a surface by hammering in relief. The vertical surfaces introduced by the relief are ribbons which contribute $$\mathbf{0}$$ to the total surface vector, while the horizontal surfaces are simply displaced vertically be the relief. The rightmost image below shows how the total surface vector is preserved if the "texture" of the surface at infinitesimal scales is converted from "steps" (a union of horizontal and vertical surfaces) to "smooth slopes" and vice versa. The surface formed from the red and green planes is a step, while the surface formed from the blue plane is a slope. These two surfaces can be seen to have equal total surface vectors from the right-angled triangle at the right side of the image.

Generalizing from displacement vectors
The total displacement along a simple oriented curve can be used to compute the net displacement in a specific direction. Given a simple oriented curve $$C$$ and an oriented straight line with the direction indicated by normal vector $$\hat{\mathbf{n}}$$, the total displacement $$\Delta\mathbf{q}$$ along $$C$$ can be used to compute the net displacement in the direction indicated by the line. This displacement is $$\hat{\mathbf{n}} \cdot \Delta\mathbf{q}$$, and depends only on the endpoints of the curve.

In a direct analogy, given a simple oriented surface $$\sigma$$ with counter-clockwise boundary $$\partial\sigma$$, and an oriented flat plane whose surface normal is $$\hat{\mathbf{n}}$$, a quantity of interest is the total signed area of $$\sigma$$ perpendicularly projected onto the plane. The signed area that is projected by a flat infinitesimal portion of $$\sigma$$ with surface vector $$d\mathbf{S}$$ is $$\hat{\mathbf{n}} \cdot d\mathbf{S}$$, and the total signed area is $$\iint_{\mathbf{q} \in \sigma} \hat{\mathbf{n}} \cdot d\mathbf{S} = \hat{\mathbf{n}} \cdot \iint_{\mathbf{q} \in \sigma} d\mathbf{S} = \hat{\mathbf{n}} \cdot \mathbf{S}$$ where $$\mathbf{S}$$ is the total surface vector of $$\sigma$$.

The total signed projected area $$\hat{\mathbf{n}} \cdot \mathbf{S}$$ onto the plane is purely a function of the boundary $$\partial\sigma$$, and does not depend on how $$\sigma$$ fills its boundary $$\partial\sigma$$. This is much more obvious and clearer than the claim that the total surface vector $$\mathbf{S}$$ is only a function of $$\partial\sigma$$: the area enclosed by a boundary in 2D space is purely a function of that boundary. Since the projected area is signed, "upside down" surfaces project negative area, and folds and overhangs cancel each other out.

Since $$\hat{\mathbf{n}} \cdot \mathbf{S}$$ is purely a function of $$\partial\sigma$$ for all choices of plane orientation $$\hat{\mathbf{n}}$$, then the total surface vector $$\mathbf{S}$$ is purely a function of $$\partial\sigma$$.

Summary
The total surface vector generated by a closed surface is $$\mathbf{0}$$.

Given a multi-surface $$\mathbf{S} = \{(\sigma_1,w_1), (\sigma_2,w_2), ..., (\sigma_k,w_k)\}$$ the total surface vector generated by $$\mathbf{S}$$ is $$\sum_{i=1}^k w_i\iint_{\mathbf{q} \in \sigma_i} d\mathbf{S}$$.

Given a vector field $$\mathbf{F}$$ that denotes a multi-surface, the total surface vector generated by $$\mathbf{F}$$ is $$\iiint_{\mathbf{q} \in \R^3} \mathbf{F}(\mathbf{q})dV$$. Since the surface vector generated by simple surface $$\sigma$$ is $$\iint_{\mathbf{q} \in \sigma} d\mathbf{S}$$, it is the case that $$\iiint_{\mathbf{q} \in \R^3} \delta_2(\mathbf{q};\sigma)dV = \iint_{\mathbf{q} \in \sigma} d\mathbf{S}$$. One important observation is that given a surface integral over $$\sigma$$, the differential $$d\mathbf{S}$$ is equal to $$\delta_2(\mathbf{q};\sigma)dV$$ in a volume integral: $$\iint_{\mathbf{q} \in \sigma} f(\mathbf{q},d\mathbf{S}) = \iiint_{\mathbf{q} \in \R^3} f(\mathbf{q},\delta_2(\mathbf{q};\sigma)dV)$$ provided that function $$f$$ is a linear in the second parameter.

Total volume
Consider a multi-volume $$\mathbf{U} = \{(\Omega_1,w_1), (\Omega_2,w_2), ..., (\Omega_k,w_k)\}$$, where the volumes of $$\Omega_1, \Omega_2, ..., \Omega_k$$ are respectively $$V_1, V_2, ..., V_k$$, then the total volume of $$\mathbf{U}$$ is $$\sum_{i=1}^k w_iV_i$$. Each volume $$V_i$$ can be computed by $$V_i = \iiint_{\mathbf{q} \in \Omega_i} dV = \iiint_{\mathbf{q} \in \R^3} \delta_3(\mathbf{q};\Omega_i)dV$$. The total volume of $$\mathbf{U}$$ is $$V = \sum_{i=1}^k w_iV_i = \sum_{i=1}^k w_i\iiint_{\mathbf{q} \in \Omega_i} dV = \sum_{i=1}^k w_i\iiint_{\mathbf{q} \in \R^3} \delta_3(\mathbf{q};\Omega_i)dV $$ $$ = \iiint_{\mathbf{q} \in \R^3} \left(\sum_{i=1}^k w_i\delta_3(\mathbf{q};\Omega_i)\right)dV = \iiint_{\mathbf{q} \in \R^3} \delta_3(\mathbf{q};\mathbf{U})dV$$.

If a multi-volume $$\mathbf{U}$$ can be denoted by scalar field $$U$$, then the volume of $$\mathbf{U}$$ is $$\iiint_{\mathbf{q} \in \R^3} U(\mathbf{q})dV$$.

Given an arbitrary volume $$\Omega$$, a volume integral over $$\Omega$$ can be converted to a volume integral over $$\R^3$$ by replacing the differential $$dV$$ with $$\delta_3(\mathbf{q};\Omega)dV$$:

$$\iiint_{\mathbf{q} \in \Omega} f(\mathbf{q},dV) = \iiint_{\mathbf{q} \in \R^3} f(\mathbf{q},\delta_3(\mathbf{q};\Omega)dV)$$ provided that $$f$$ is linear in the second parameter.

Intersections
The union of two multi-points denoted by scalar fields $$\rho_1$$ and $$\rho_2$$ is simply $$\rho_1+\rho_2$$, and the same is true for the union of two multi-paths, the union of two multi-surfaces, and the union of two multi-volumes. The union of two structures with different types, such as a multi-point with a multi-path, is forbidden however.

The intersection on the other hand, is less trivial and can occur between structures of different types.

Point-Volume intersections
When a point $$\mathbf{q}$$ with weight $$w_1$$ intersects a volume $$\Omega$$ with weight $$w_2$$, then the intersection is point $$\mathbf{q}$$ with weight $$w_1w_2$$. Given a multi-point and a multi-volume, the intersection is the sum of the pair-wise intersections of each simple point with each simple volume. The image below gives an example of the intersection of a multi-point with a multi-volume.



Given a multi-point with scalar field $$\rho$$, and a multi-volume with scalar field $$U$$, then the intersection is a multi-point with scalar field $$\rho U$$.

The total intersection between a multi-point $$\rho$$ and a multi-volume $$U$$ is $$\iiint_{\mathbf{q} \in \R^3} \rho(\mathbf{q})U(\mathbf{q})dV$$.

If $$\rho$$ denotes a simple point $$\mathbf{q}_0$$, then the total intersection is $$\iiint_{\mathbf{q} \in \R^3} \delta_0(\mathbf{q};\mathbf{q}_0)U(\mathbf{q})dV = U(\mathbf{q}_0)$$.

If $$U$$ denotes a simple volume $$\Omega$$, then the total intersection is $$\iiint_{\mathbf{q} \in \R^3} \rho(\mathbf{q})\delta_3(\mathbf{q};\Omega)dV = \iiint_{\mathbf{q} \in \Omega} \rho(\mathbf{q})dV$$.

Path-Surface intersections
When a path $$C$$ with weight $$w_1$$ intersects a surface $$\sigma$$ with weight $$w_2$$ at point $$\mathbf{q}$$, then the intersection is point $$\mathbf{q}$$ with weight $$\pm w_1w_2$$. The weight is $$+w_1w_2$$ if $$C$$ passes through $$\sigma$$ in the direction in which $$\sigma$$ is oriented. The weight is $$-w_1w_2$$ if $$C$$ passes through $$\sigma$$ opposite the direction in which $$\sigma$$ is oriented. Given a multi-path and a multi-surface, the intersection is the sum of the pair-wise intersections of each simple path with each simple surface. The images below give examples of the intersections of a multi-path with a multi-surface.

In the image above to the far right, the multi-path is denoted by a vector field which has the value $$\mathbf{F}$$ inside the blue tube, and is $$\mathbf{0}$$ everywhere else. The multi-surface is denoted by a vector field which has the value $$\mathbf{G}$$ among the red sheets, and is $$\mathbf{0}$$ everywhere else. The total path weight in the blue tube is $$|\mathbf{F}|\Delta A$$. The total surface weight in the red sheets is $$|\mathbf{G}|\Delta t$$. The total weight of all the intersection points is $$(|\mathbf{F}|\Delta A)(|\mathbf{G}|\Delta t) = |\mathbf{F}||\mathbf{G}|\Delta A\Delta t$$. The volume that the intersection points are evenly spread out in is $$\Delta A\Delta t/\cos\theta$$. The intersection point density is $$\frac{|\mathbf{F}||\mathbf{G}|\Delta A\Delta t}{\Delta A\Delta t/\cos\theta} = |\mathbf{F}||\mathbf{G}|\cos\theta = \mathbf{F} \cdot \mathbf{G}$$.

Given a multi-path with vector field $$\mathbf{J}$$, and a multi-surface with vector field $$\mathbf{F}$$, then the intersection is a multi-point with scalar field $$\mathbf{J} \cdot \mathbf{F}$$.

The total intersection between a multi-path $$\mathbf{J}$$ and a multi-surface $$\mathbf{F}$$ is $$\iiint_{\mathbf{q} \in \R^3} (\mathbf{J}(\mathbf{q}) \cdot \mathbf{F}(\mathbf{q}))dV$$.

If $$\mathbf{J}$$ is a simple path $$C$$, then the total intersection is $$\iiint_{\mathbf{q} \in \R^3} (\delta_1(\mathbf{q};C) \cdot \mathbf{F}(\mathbf{q}))dV = \int_{\mathbf{q} \in C} \mathbf{F}(\mathbf{q}) \cdot d\mathbf{q}$$.

If $$\mathbf{F}$$ is a simple surface $$\sigma$$, then the total intersection is $$\iiint_{\mathbf{q} \in \R^3} (\mathbf{J}(\mathbf{q}) \cdot \delta_2(\mathbf{q};\sigma))dV = \iint_{\mathbf{q} \in \sigma} \mathbf{J}(\mathbf{q}) \cdot d\mathbf{S}$$.

Path-Volume intersections
When a path $$C$$ with weight $$w_1$$ intersects a volume $$\Omega$$ with weight $$w_2$$, then the intersection is path $$C \cap \Omega$$ with weight $$w_1w_2$$. Given a multi-path and a multi-volume, the intersection is the sum of the pair-wise intersections of each simple path with each simple volume. The image below gives an example of the intersection of a multi-path with a multi-volume.



Given a multi-path with vector field $$\mathbf{J}$$, and a multi-volume with scalar field $$U$$, then the intersection is a multi-path with vector field $$\mathbf{J}U$$.

The total intersection between a multi-path $$\mathbf{J}$$ and a multi-volume $$U$$ is $$\iiint_{\mathbf{q} \in \R^3} \mathbf{J}(\mathbf{q})U(\mathbf{q})dV$$.

If $$\mathbf{J}$$ denotes a simple path $$C$$, then the total intersection is $$\iiint_{\mathbf{q} \in \R^3} \delta_1(\mathbf{q};C)U(\mathbf{q})dV = \int_{\mathbf{q} \in C} U(\mathbf{q})d\mathbf{q}$$.

If $$U$$ denotes a simple volume $$\Omega$$, then the total intersection is $$\iiint_{\mathbf{q} \in \R^3} \mathbf{J}(\mathbf{q})\delta_3(\mathbf{q};\Omega)dV = \iiint_{\mathbf{q} \in \Omega} \mathbf{J}(\mathbf{q})dV$$.

Surface-Surface intersections
When a surface $$\sigma_1$$ with weight $$w_1$$ intersects a surface $$\sigma_2$$ with weight $$w_2$$, then the intersection is the path $$\sigma_1 \cap \sigma_2$$ with weight $$w_1w_2$$. The orientation given to path $$\sigma_1 \cap \sigma_2$$ is defined as follows: viewing the intersection where the surface normal vectors of $$\sigma_1$$ and $$\sigma_2$$ are oriented towards the viewer, the intersection path has $$\sigma_1$$ to its right, and $$\sigma_2$$ to its left. Put another way, the intersection path is oriented according to the "right-hand rule" where the surface normals of $$\sigma_1$$ are the "x" direction, and the surface normals of $$\sigma_2$$ are the "y" direction. The images below give examples of the intersections of a multi-surface with a multi-surface.

In the image above to the right, the first multi-surface is denoted by a vector field that has the value $$\mathbf{F}$$ among the blue sheets, and is $$\mathbf{0}$$ everywhere else. The second multi-surface is denoted by a vector field that has the value $$\mathbf{G}$$ among the red sheets, and is $$\mathbf{0}$$ everywhere else. The total surface weight in the blue sheets is $$|\mathbf{F}|\Delta t_1$$, and the total surface weight in the red sheets is $$|\mathbf{G}|\Delta t_2$$. The total weight of all the intersection paths is $$(|\mathbf{F}|\Delta t_1)(|\mathbf{G}|\Delta t_2) = |\mathbf{F}||\mathbf{G}|\Delta t_1\Delta t_2$$. The cross-sectional area that the intersection paths are evenly spread out over is $$\Delta t_1\Delta t_2/\sin\theta$$. The intersection path density is $$\frac{|\mathbf{F}||\mathbf{G}|\Delta t_1\Delta t_2}{\Delta t_1\Delta t_2/\sin\theta} = |\mathbf{F}||\mathbf{G}|\sin\theta = |\mathbf{F} \times \mathbf{G}|$$. Lastly, it should be noted that the intersection paths are oriented out of the screen as per the right-hand rule.

Given a multi-surface with vector field $$\mathbf{F}_1$$, and a multi-surface with vector field $$\mathbf{F}_2$$, then the intersection is the multi-path with vector field $$\mathbf{F}_1 \times \mathbf{F}_2$$.

The total intersection between multi-surface $$\mathbf{F}_1$$ and multi-surface $$\mathbf{F}_2$$ is $$\iiint_{\mathbf{q} \in \R^3} (\mathbf{F}_1(\mathbf{q}) \times \mathbf{F}_2(\mathbf{q}))dV$$.

If $$\mathbf{F}_2$$ denotes a simple surface $$\sigma$$, then the total intersection is $$\iiint_{\mathbf{q} \in \R^3} (\mathbf{F}_1(\mathbf{q}) \times \delta_2(\mathbf{q};\sigma))dV = \iint_{\mathbf{q} \in \sigma} \mathbf{F}_1(\mathbf{q}) \times d\mathbf{S}$$.

Surface-Volume intersections
When a surface $$\sigma$$ with weight $$w_1$$ intersects a volume $$\Omega$$ with weight $$w_2$$, then the intersection is surface $$\sigma \cap \Omega$$ with weight $$w_1w_2$$. Given a multi-surface and a multi-volume, the intersection is the sum of the pair-wise intersections of each simple surface with each simple volume. The image below gives an example of the intersection of a multi-surface with a multi-volume.



Given a multi-surface with vector field $$\mathbf{F}$$, and a multi-volume with scalar field $$U$$, then the intersection is a multi-surface with vector field $$\mathbf{F}U$$.

The total intersection between a multi-surface $$\mathbf{F}$$ and a multi-volume $$U$$ is $$\iiint_{\mathbf{q} \in \R^3} \mathbf{F}(\mathbf{q})U(\mathbf{q})dV$$.

If $$\mathbf{F}$$ denotes a simple surface $$\sigma$$, then the total intersection is $$\iiint_{\mathbf{q} \in \R^3} \delta_2(\mathbf{q};\sigma)U(\mathbf{q})dV = \iint_{\mathbf{q} \in \sigma} U(\mathbf{q})d\mathbf{S}$$.

If $$U$$ denotes a simple volume $$\Omega$$, then the total intersection is $$\iiint_{\mathbf{q} \in \R^3} \mathbf{F}(\mathbf{q})\delta_3(\mathbf{q};\Omega)dV = \iiint_{\mathbf{q} \in \Omega} \mathbf{F}(\mathbf{q})dV$$.

Volume-Volume intersections
When a volume $$\Omega_1$$ with weight $$w_1$$ intersects a volume $$\Omega_2$$ with weight $$w_2$$, then the intersection is the volume $$\Omega_1 \cap \Omega_2$$ with weight $$w_1w_2$$. Given two multi-volumes, the intersection is the sum of the pair-wise intersections of each simple volume from the first multi-volume with each simple volume from the second multi-volume. The image below gives an example of the intersection between two multi-volumes.



Given a multi-volume with scalar field $$U_1$$, and a multi-volume with scalar field $$U_2$$, then the intersection is a multi-volume with scalar field $$U_1U_2$$.

The total intersection between multi-volume $$U_1$$ and multi-volume $$U_2$$ is $$\iiint_{\mathbf{q} \in \R^3} U_1(\mathbf{q})U_2(\mathbf{q})dV$$.

If $$U_2$$ denotes a simple volume $$\Omega$$, then the total intersection is $$\iiint_{\mathbf{q} \in \R^3} U_1(\mathbf{q})\delta_3(\mathbf{q};\Omega)dV = \iiint_{\mathbf{q} \in \Omega} U_1(\mathbf{q})dV$$.

Other intersections
Other types of intersections, such as Point-Point intersections, Point-Path intersections, Point-Surface intersections, and Path-Path intersections, are not considered since these kinds of intersections can occur only by design. For example, the probability that two randomly chosen points will intersect each other is 0, but if a point and a volume are randomly chosen, then the probability of the point landing in the volume is nonzero. Given two unrelated points, these two points will never land on each other, since a prior relationship has to exist for the points to coincide. Below is summarized the various types of intersections:

The endpoints of paths
Given a simple path $$C$$ that starts at point $$C(0)$$ and ends at point $$C(1)$$, the "endpoints" of $$C$$ is the multi-point $$\{(C(0),+1),(C(1),-1)\}$$ that consists of the starting point with a weight of +1, and the final point with a weight of -1. While $$C$$ is denoted by the vector field $$\delta_1(\mathbf{q};C)$$, the endpoints are denoted by the scalar field $$\delta_0(\mathbf{q};C(0))-\delta_0(\mathbf{q};C(1))$$. The image below gives several examples of simple paths and their associated endpoints.



Given a multi-path $$\mathbf{C} = \{(C_1,w_1),(C_2,w_2),...,(C_k,w_k)\}$$, the endpoints of $$\mathbf{C}$$ is the multi-point $$\{(C_1(0),+1),(C_1(1),-1),(C_2(0),+1),(C_2(1),-1),...,(C_k(0),+1),(C_k(1),-1)\}$$.

Given a vector field $$\mathbf{J}$$ that denotes a multi-path, the multi-point that denotes the endpoints of $$\mathbf{J}$$ is denoted by scalar field $$\nabla\cdot\mathbf{J}$$. The scalar field $$\nabla\cdot\mathbf{J}$$ evaluated at point $$\mathbf{q}$$ is denoted by $$\nabla\cdot\mathbf{J}(\mathbf{q})$$, $$(\nabla\cdot\mathbf{J})(\mathbf{q})$$ or $$\nabla\cdot\mathbf{J}|_{\mathbf{q}}$$.

The requirement that no path extends to infinity means that every starting point is paired with a final point, and therefore the total weight of all of the endpoints together is 0: $$\iiint_{\mathbf{q} \in \R^3} (\nabla\cdot\mathbf{J}(\mathbf{q}))dV = 0$$.



The similarity of the notation $$\nabla\cdot\mathbf{J}$$ to the intersection of multi-path $$\mathbf{J}$$ with multi-surface $$\mathbf{F}$$, denoted by $$\mathbf{F}\cdot\mathbf{J}$$, makes sense if $$\nabla$$ is interpreted as the "surface of reality". A starting point forms when a path pokes into reality, and a final point forms when a path pokes out of reality.

In the image to the right, a depiction of the "surface of reality" interpretation of $$\nabla$$ is shown. On the right is a simple path $$\mathbf{F}$$, along with its endpoints $$\nabla\cdot\mathbf{F}$$. On the left $$\mathbf{F}_\text{ext}$$ is an extension of $$\mathbf{F}$$ that is behind the "veil" of surface $$\mathbf{G}_\nabla$$. $$\mathbf{F}_\text{ext}$$ pokes out of and into $$\mathbf{G}_\nabla$$ at points consistent with the endpoints of $$\mathbf{F}$$: i.e. $$\mathbf{G}_\nabla \cdot \mathbf{F}_\text{ext} = \nabla\cdot\mathbf{F}$$.

The counter-clockwise oriented boundaries of surfaces
Given an oriented surface $$\sigma$$, the "counter-clockwise oriented boundary" of $$\sigma$$ is a path $$\partial\sigma$$ that traces the boundary of $$\sigma$$ in a counter-clockwise direction. The counter-clockwise direction is better described as follows: While located on the boundary, the counter-clockwise direction is the "forwards" direction when the surface normal vectors point "up" and the surface itself is on the "left". The image below gives several examples of oriented surfaces and their counter-clockwise boundaries. Note in particular the 4th panel that shows that the orientation around a hole in the surface appears to be clockwise.



Given a multi-surface $$\mathbf{S} = \{(\sigma_1,w_1),(\sigma_2,w_2),...,(\sigma_k,w_k)\}$$, the counter-clockwise boundary of $$\mathbf{S}$$ is the multi-path $$\{(\partial\sigma_1,w_1),(\partial\sigma_2,w_2),...,(\partial\sigma_k,w_k)\}$$.

Given a vector field $$\mathbf{F}$$ that denotes a multi-surface, the multi-path that denotes the counter-clockwise boundary of $$\mathbf{F}$$ is denoted by vector field $$\nabla\times\mathbf{F}$$. The vector field $$\nabla\times\mathbf{F}$$ evaluated at point $$\mathbf{q}$$ is denoted by $$\nabla\times\mathbf{F}(\mathbf{q})$$, $$(\nabla\times\mathbf{F})(\mathbf{q})$$, or $$\nabla\times\mathbf{F}|_{\mathbf{q}}$$.

The requirement that no surface weight extends to infinity means that all counter-clockwise boundaries form closed loops, and therefore the total displacement of the total counter-clockwise boundary is $$\mathbf{0}$$: $$\iiint_{\mathbf{q} \in \R^3} (\nabla\times\mathbf{F}(\mathbf{q}))dV = \mathbf{0}$$.

It is also important to note that the counter-clockwise boundary has no endpoints: $$\nabla \cdot (\nabla \times \mathbf{F}) = 0$$.



The similarity of the notation $$\nabla\times\mathbf{F}_2$$ to the intersection of multi-surface $$\mathbf{F}_1$$ with multi-surface $$\mathbf{F}_2$$, denoted by $$\mathbf{F}_1 \times \mathbf{F}_2$$, again makes sense if $$\nabla$$ is interpreted as the "surface of reality". An edge is formed when a surface "slices" into reality.

In the image to the right, a depiction of the "surface of reality" interpretation of $$\nabla$$ is shown. On the right is a simple surface $$\mathbf{F}$$, along with its counter-clockwise boundary $$\nabla\times\mathbf{F}$$. On the left $$\mathbf{F}_\text{ext}$$ is an extension of $$\mathbf{F}$$ that is behind the "veil" of surface $$\mathbf{G}_\nabla$$. $$\mathbf{F}_\text{ext}$$ slices into $$\mathbf{G}_\nabla$$ at curves consistent with the boundary of $$\mathbf{F}$$: i.e. $$\mathbf{G}_\nabla \times \mathbf{F}_\text{ext} = \nabla \times \mathbf{F}$$.

The inwards-oriented surfaces of volumes
Given a volume $$\Omega$$, the "inwards oriented surface" of $$\Omega$$ is a surface $$\partial\Omega$$ that wraps the volume $$\Omega$$ with the surface normals pointing inwards. The image below gives several examples of volumes and their inwards oriented surfaces.



Given a multi-volume $$\mathbf{U} = \{(\Omega_1,w_1),(\Omega_2,w_2),...,(\Omega_k,w_k)\}$$, the inwards oriented surface of $$\mathbf{U}$$ is the multi-surface $$\{(\partial\Omega_1,w_1),(\partial\Omega_2,w_2),...,(\partial\Omega_k,w_k)\}$$.

Given a scalar field $$U$$ that denotes a multi-volume, the multi-surface that denotes the inwards oriented surface of $$U$$ is denoted by vector field $$\nabla U$$. The vector field $$\nabla U$$ evaluated at point $$\mathbf{q}$$ is denoted by $$\nabla U(\mathbf{q})$$, $$(\nabla U)(\mathbf{q})$$, or $$\nabla U|_{\mathbf{q}}$$.

The requirement that no volume weight extends to infinity means that all inwards oriented surfaces form closed surfaces, and therefore the total surface vector of the total inwards oriented surface is $$\mathbf{0}$$: $$\iiint_{\mathbf{q} \in \R^3} (\nabla U(\mathbf{q}))dV = \mathbf{0}$$.

It is also important to note that the inwards oriented surface has no boundary: $$\nabla \times (\nabla U) = \mathbf{0}$$.



The similarity of the notation $$\nabla U$$ to the intersection of multi-surface $$\mathbf{F}$$ with multi-volume $$U$$, denoted by $$\mathbf{F}U$$, again makes sense if $$\nabla$$ is interpreted as the "surface of reality". A surface is formed from the surface of reality when the volume "pushes" into reality.

In the image to the right, a depiction of the "surface of reality" interpretation of $$\nabla$$ is shown. The image is a 2D cross-section for simplicity. On the right is a simple volume $$U$$, along with its inwards oriented surface $$\nabla U$$. On the left $$U_\text{ext}$$ is an extension of $$U$$ that is behind the "veil" of surface $$\mathbf{G}_\nabla$$. $$U_\text{ext}$$ pushes through $$\mathbf{G}_\nabla$$ at surfaces consistent with the surface of $$U$$: i.e. $$\mathbf{G}_\nabla U_\text{ext} = \nabla U$$.

Closed loops and closed surfaces
A simple path is "closed" or a "loop" if its starting and final points are the same, so the total endpoints is 0 since the weights of the starting and final points cancel out. More generally, a multi-path $$\mathbf{J}$$ is "closed" or a "multi-loop" if $$\nabla \cdot \mathbf{J} = 0$$. As previously noted, the counter-clockwise boundary of a surface is closed.

A simple surface is "closed" or a "bubble" if it has no boundary. More generally, a multi-surface $$\mathbf{F}$$ is "closed" or a "multi-bubble" if $$\nabla \times \mathbf{F} = \mathbf{0}$$. As previously noted, the inwards oriented surface of a volume is closed.

It is clear that the total displacement present in a closed multi-path is $$\mathbf{0}$$: $$\nabla \cdot \mathbf{J} = 0 \implies \iiint_{\mathbf{q} \in \R^3} \mathbf{J}dV = \mathbf{0}$$, and it is also clear that the total surface vector of a closed multi-surface is also $$\mathbf{0}$$: $$\nabla \times \mathbf{F} = \mathbf{0} \implies \iiint_{\mathbf{q} \in \R^3} \mathbf{F}dV = \mathbf{0}$$.



Given both a simple loop and a simple bubble, the total point weight of all intersection points is 0: every time the loop enters the bubble, it must also leave the bubble, and the weights of these two intersection points cancel out. More generally, given a closed multi-path $$\mathbf{J}$$ and a closed multi-surface $$\mathbf{F}$$, then the total intersection point weight is 0: $$(\nabla \cdot \mathbf{J} = 0 \;\text{and}\; \nabla \times \mathbf{F} = \mathbf{0}) \implies \iiint_{\mathbf{q} \in \R^3} (\mathbf{J} \cdot \mathbf{F})dV = 0$$.

The above identity gives rise to the following observations:
 * The total intersection point weight of a multi-loop and a multi-surface is purely a function of the multi-loop and the multi-surface's counter-clockwise boundary: the interior of the multi-surface does not matter. If $$\nabla \cdot \mathbf{J} = 0$$ and $$\nabla \times \mathbf{F}_1 = \nabla \times \mathbf{F}_2$$, then $$\iiint_{\mathbf{q} \in \R^3} (\mathbf{J} \cdot \mathbf{F}_1)dV = \iiint_{\mathbf{q} \in \R^3} (\mathbf{J} \cdot \mathbf{F}_2)dV$$.
 * The total intersection point weight of a multi-path and a multi-bubble is purely a function of the multi-bubble and the multi-path's endpoints: the interior of the multi-path does not matter. If $$\nabla \times \mathbf{F} = \mathbf{0}$$ and $$\nabla \cdot \mathbf{J}_1 = \nabla \cdot \mathbf{J}_2$$, then $$\iiint_{\mathbf{q} \in \R^3} (\mathbf{J}_1 \cdot \mathbf{F})dV = \iiint_{\mathbf{q} \in \R^3} (\mathbf{J}_2 \cdot \mathbf{F})dV$$.

The inwards oriented surface of a volume is closed. Conversely, given a closed surface, there exists a volume that "fills" the surface. More generally, given a multi-bubble $$\mathbf{F}$$, there exists a multi-volume $$U$$ for which $$\mathbf{F}$$ is the inwards oriented multi-surface of $$U$$: $$\nabla \times \mathbf{F} = \mathbf{0} \implies \exists U : \nabla U = \mathbf{F}$$. This multi-volume is referred to as the "scalar potential" of $$\mathbf{F}$$. The requirement that volumes cannot extend to infinity means that $$U$$ is unique.

The counter-clockwise oriented boundary of a surface is closed. Conversely, given a loop, there exists a surface that "fills" the loop. More generally, given a multi-loop $$\mathbf{J}$$, there exists a multi-surface $$\mathbf{F}$$ for which $$\mathbf{J}$$ is the counter-clockwise boundary of $$\mathbf{F}$$: $$\nabla \cdot \mathbf{J} = 0 \implies \exists \mathbf{F} : \nabla \times \mathbf{F} = \mathbf{J}$$. This multi-surface is referred to as the "vector potential" of $$\mathbf{J}$$. Even with the requirement that surfaces cannot extend to infinity, $$\mathbf{F}$$ is not unique without additional restrictions.

Coordinate Systems


This section will describe how to compute various quantities such as intersections, endpoints, boundaries, and surfaces given a curvilinear coordinate system.

Let the curvilinear coordinate system be arbitrary. Let the 3 coordinates that index all points be $$c_1, c_2, c_3$$. Coordinates will be denoted by the triple $$(c_1,c_2,c_3)$$.

The following notation will be used in the following discussions:


 * Given an arbitrary expression $$f: \{1,2,3\} \to \R$$ that assigns a real number to each index $$i = 1,2,3$$, then $$(i;f(i))$$ will denote the triple $$(f(1),f(2),f(3))$$.


 * Given index variables $$i,j \in \{1,2,3\}$$, the expression $$\mathbf{1}(i = j)$$ equals 1 if $$i = j$$ and 0 if otherwise.


 * Given an arbitrary expression $$f: \{1,2,3\} \to \R$$ that assigns a real number to each index $$i = 1,2,3$$, then $$\sum_i f(i)$$ will denote the sum $$f(1)+f(2)+f(3)$$.


 * Given an index variable $$i \in \{1,2,3\}$$, $$i+1$$ will rotate $$i$$ forwards by 1, and $$i+2$$ will rotate $$i$$ forwards by 2. In essence, $$i+1 = \left\{\begin{array}{cc} i+1 & (i=1,2) \\ 1 & (i=3) \end{array}\right.$$ and $$i+2 = \left\{\begin{array}{cc} 3 & (i=1) \\ i-1 & (i=2,3) \end{array}\right.$$.

Start with an arbitrary coordinate $$(c'_1,c'_2,c'_3) = (j;c'_j)$$ and infinitesimal differences $$\Delta c_1$$, $$\Delta c_2$$, and $$\Delta c_3$$. The following 3 paths, 3 surfaces, and volume will be associated with point $$(j;c'_j)$$:


 * For each $$i \in \{1,2,3\}$$ there exists an infinitely short path $$C_i((j;c'_j))$$ starting from point $$(j;c'_j)$$ and ending on point $$(j;c'_j + \Delta c_i \mathbf{1}(j = i))$$ along the curve defined by $$c'_i \leq c_i < c'_i + \Delta c_i$$, $$c_{i+1} = c'_{i+1}$$ and $$c_{i+2} = c'_{i+2}$$. The displacement covered by $$C_i((j;c'_j))$$ is approximately $$\Delta c_i \cdot l_i((j;c'_j)) \cdot \hat{\mathbf{a}}_i((j;c'_j))$$ where $$\hat{\mathbf{a}}_i((j;c'_j))$$ is a unit length vector that is parallel to the displacement between points $$(j;c'_j)$$ and $$(j;c'_j + \Delta c_i \mathbf{1}(j = i))$$, and $$\Delta c_i \cdot l_i((j;c'_j))$$ is the length of the displacement. Note that the length of the displacement is proportional to $$\Delta c_i$$, with $$l_i((j;c'_j))$$ being the constant of proportionality. The set of vectors $$\{\hat{\mathbf{a}}_1((j;c'_j)),\hat{\mathbf{a}}_2((j;c'_j)),\hat{\mathbf{a}}_3((j;c'_j))\}$$ is the set of displacement basis vectors.


 * For each $$i \in \{1,2,3\}$$ there exists an infinitely small surface $$\sigma_i((j;c'_j))$$ that is defined by the following: $$c_i = c'_i$$, $$c'_{i+1} \leq c_{i+1} < c'_{i+1} + \Delta c_{i+1}$$, and $$c'_{i+2} \leq c_{i+2} < c'_{i+2} + \Delta c_{i+2}$$. The orientation of $$\sigma_i((j;c'_j))$$ is in the direction of increasing $$c_i$$. The surface vector of $$\sigma_i((j;c'_j))$$ is approximately $$\Delta c_{i+1} \Delta c_{i+2} \cdot A_i((j;c'_j)) \cdot \hat{\mathbf{a}}^i((j;c'_j))$$ where $$\hat{\mathbf{a}}^i((j;c'_j))$$ is a unit length vector that is perpendicular to $$\sigma_i((j;c'_j))$$, and $$\Delta c_{i+1} \Delta c_{i+2} \cdot A_i((j;c'_j))$$ is the area of $$\sigma_i((j;c'_j))$$. Note that the area of $$\sigma_i((j;c'_j))$$ is proportional to $$\Delta c_{i+1} \Delta c_{i+2}$$, with $$A_i((j;c'_j))$$ being the constant of proportionality. The set of vectors $$\{\hat{\mathbf{a}}^1((j;c'_j)),\hat{\mathbf{a}}^2((j;c'_j)),\hat{\mathbf{a}}^3((j;c'_j))\}$$ is the set of surface basis vectors.


 * There is an infinitely small volume $$\Omega((j;c'_j))$$ defined by $$c'_1 \leq c_1 < c'_1+\Delta c_1$$, $$c'_2 \leq c_2 < c'_2+\Delta c_2$$, and $$c'_3 \leq c_3 < c'_3+\Delta c_3$$. $$\Omega((j;c'_j))$$ has a shape that is approximately that of a parallelepiped. The volume of $$\Omega((j;c'_j))$$ is approximately $$\Delta c_1 \Delta c_2 \Delta c_3 \cdot V((j;c'_j))$$. Note that the volume of $$\Omega((j;c'_j))$$ is proportional to $$\Delta c_1 \Delta c_2 \Delta c_3$$, with $$V((j;c'_j))$$ being the constant of proportionality.

It is important to note that:
 * $$(i;c_i) \in \Omega((j;c'_j))$$ if and only if $$c'_1 \leq c_1 < c'_1 + \Delta c_1$$, $$c'_2 \leq c_2 < c'_2 + \Delta c_2$$, and $$c'_3 \leq c_3 < c'_3 + \Delta c_3$$ (note the strictness of the upper bounds).
 * For all $$i \in \{1,2,3\}$$, $$C_i((j;c_j)) \subseteq \Omega((j;c'_j))$$ if and only if $$c_i = c'_i$$, $$c'_{i+1} \leq c_{i+1} < c'_{i+1} + \Delta c_{i+1}$$, and $$c'_{i+2} \leq c_{i+2} < c'_{i+2} + \Delta c_{i+2}$$ (note the strictness of the upper bounds).
 * For all $$i \in \{1,2,3\}$$, $$\sigma_i((j;c_j)) \subseteq \Omega((j;c'_j))$$ if and only if $$c'_i \leq c_i < c'_i + \Delta c_i$$ (note the strictness of the upper bound), $$c_{i+1} = c'_{i+1}$$, and $$c_{i+2} = c'_{i+2}$$.

Converting between multi-points, multi-paths, multi-surfaces, and multi-volumes and their respective scalar fields and vector fields proceeds as follows:

This conversion is performed by subdividing space into discrete volumes or cells. Infinitesimal differences $$\Delta c_1$$, $$\Delta c_2$$, and $$\Delta c_3$$ are chosen, and a lattice consisting of the points $$(j;k_j\Delta c_j)$$ where $$(j;k_j)$$ is an arbitrary triple of integers is generated. The cell indexed by $$(j;k_j)$$ consists of the point $$(j;k_j\Delta c_j)$$, the paths $$C_i((j;k_j\Delta c_j))$$ for each $$i \in \{1,2,3\}$$, the surfaces $$\sigma_i((j;k_j\Delta c_j))$$ for each $$i \in \{1,2,3\}$$, and the volume $$\Omega((j;k_j\Delta c_j))$$. All points $$(j;c_j)$$ where $$k_i\Delta c_i \leq c_i < (k_i+1)\Delta c_i$$ for all $$i \in \{1,2,3\}$$ "belong" to the cell indexed by $$(j;k_j)$$ (note that the upper bounds are excluded). Given an arbitrary point $$(j;c_j)$$, the cell that contains $$(j;c_j)$$ is indexed by $$(j;k_j) = \left(j;\left\lfloor\frac{c_j}{\Delta c_j}\right\rfloor\right)$$. The point $$(j;c'_j) = (j;k_j\Delta c_j)$$ is the vertex that the cell is associated with.

A multi-point, multi-path, multi-surface, or multi-volume is converted to a scalar field or vector field by computing the total point weight, displacement, surface vector, or volume contained by each cell and then averaging over the cell's volume.

A scalar-field $$\rho$$ is converted to a multi-point by doing the following for each cell $$(j;k_j)$$. First compute the total point weight contained inside the cell: $$\iiint_{\mathbf{q} \in \Omega((j;k_j\Delta c_j))} \rho(\mathbf{q})dV \approx \rho((j;k_j\Delta c_j)) V((j;k_j\Delta c_j)) \Delta c_1 \Delta c_2 \Delta c_3$$. Next assign this weight to the point $$(j;k_j\Delta c_j)$$.

A vector-field $$\mathbf{J} = \sum_i J_i\hat{\mathbf{a}}_i$$ is converted to a multi-path by doing the following for each cell $$(j;k_j)$$. First compute the total displacement contained inside the cell: $$\iiint_{\mathbf{q} \in \Omega((j;k_j\Delta c_j))} \mathbf{J}(\mathbf{q})dV \approx \left(\sum_i J_i((j;k_j\Delta c_j))\hat{\mathbf{a}}_i((j;k_j\Delta c_j))\right) V((j;k_j\Delta c_j)) \Delta c_1 \Delta c_2 \Delta c_3$$. Next separate this total displacement into components according to the basis $$\hat{\mathbf{a}}_1$$, $$\hat{\mathbf{a}}_2$$, and $$\hat{\mathbf{a}}_3$$: for each $$i \in \{1,2,3\}$$ the coefficient of $$\hat{\mathbf{a}}_i$$ is $$\iiint_{\mathbf{q} \in \Omega((j;k_j\Delta c_j))} J_i(\mathbf{q})dV \approx J_i((j;k_j\Delta c_j)) V((j;k_j\Delta c_j)) \Delta c_1 \Delta c_2 \Delta c_3$$. Next for each $$i \in \{1,2,3\}$$, divide the coefficient of $$\hat{\mathbf{a}}_i$$ by the length of $$C_i((j;k_j\Delta c_j))$$, which results in approximately $$J_i((j;k_j\Delta c_j)) \frac{V((j;k_j\Delta c_j))}{l_i((j;k_j\Delta c_j))} \Delta c_{i+1} \Delta c_{i+2}$$, and assign this weight to $$C_i((j;k_j\Delta c_j))$$.

A vector-field $$\mathbf{F} = \sum_i F_i\hat{\mathbf{a}}^i$$ is converted to a multi-surface by doing the following for each cell $$(j;k_j)$$. First compute the total surface vector contained inside the cell: $$\iiint_{\mathbf{q} \in \Omega((j;k_j\Delta c_j))} \mathbf{F}(\mathbf{q})dV \approx \left(\sum_i F_i((j;k_j\Delta c_j))\hat{\mathbf{a}}^i((j;k_j\Delta c_j))\right) V((j;k_j\Delta c_j)) \Delta c_1 \Delta c_2 \Delta c_3$$. Next separate this total surface vector into components according to the basis $$\hat{\mathbf{a}}^1$$, $$\hat{\mathbf{a}}^2$$, and $$\hat{\mathbf{a}}^3$$: for each $$i \in \{1,2,3\}$$ the coefficient of $$\hat{\mathbf{a}}^i$$ is $$\iiint_{\mathbf{q} \in \Omega((j;k_j\Delta c_j))} F_i(\mathbf{q})dV \approx F_i((j;k_j\Delta c_j)) V((j;k_j\Delta c_j)) \Delta c_1 \Delta c_2 \Delta c_3$$. Next for each $$i \in \{1,2,3\}$$, divide the coefficient of $$\hat{\mathbf{a}}^i$$ by the area of $$\sigma_i((j;k_j\Delta c_j))$$, which results in approximately $$F_i((j;k_j\Delta c_j)) \frac{V((j;k_j\Delta c_j))}{A_i((j;k_j\Delta c_j))} \Delta c_i$$, and assign this weight to $$\sigma_i((j;k_j\Delta c_j))$$.

A scalar-field $$U$$ is converted to a multi-volume by doing the following for each cell $$(j;k_j)$$. First compute the total volume contained inside the cell: $$\iiint_{\mathbf{q} \in \Omega((j;k_j\Delta c_j))} U(\mathbf{q})dV \approx U((j;k_j\Delta c_j)) V((j;k_j\Delta c_j)) \Delta c_1 \Delta c_2 \Delta c_3$$. Next divide this weight by the volume of $$\Omega((j;k_j\Delta c_j))$$, which results in approximately $$U((j;k_j\Delta c_j))$$, and assign this weight to $$\Omega((j;k_j\Delta c_j))$$.

Computing various intersections
Computing the intersection of any structure with a multi-volume is trivial matter: Simply multiply the scalar of vector field by the scalar field that denotes the multi-volume. When both structures are denoted by vector fields however, computing the intersection is far less trivial.

Computing path-surface intersections
To save space, the notation $$(j;c_j)$$ and $$(j;k_j\Delta c_j)$$ will be omitted from the various terms.

Given a multi-path $$\mathbf{C}$$ denoted by vector field $$\mathbf{J} = \sum_i J_i\hat{\mathbf{a}}_i$$, and a multi-surface $$\mathbf{S}$$ denoted by vector field $$\mathbf{F} = \sum_i F_i\hat{\mathbf{a}}^i$$, the scalar field that denotes the intersection can be computed as follows:

The following computations applies to each cell:

For each $$i \in \{1,2,3\}$$, the weight assigned to $$C_i$$ by $$\mathbf{C}$$ is computed as follows: $$\Delta c_1 \Delta c_2 \Delta c_3 \cdot V \cdot J_i$$ is the $$\hat{\mathbf{a}}_i$$ component of the total displacement contained by the current cell. Computing the weight assigned to $$C_i$$ requires that this displacement be spread over the length of $$C_i$$: $$\frac{\Delta c_1 \Delta c_2 \Delta c_3 \cdot V \cdot J_i}{\Delta c_i \cdot l_i} = \frac{V}{l_i} \cdot \Delta c_{i+1} \Delta c_{i+2} \cdot J_i$$.

For each $$i \in \{1,2,3\}$$, the weight assigned to $$\sigma_i$$ by $$\mathbf{S}$$ is computed as follows: $$\Delta c_1 \Delta c_2 \Delta c_3 \cdot V \cdot F_i$$ is the $$\hat{\mathbf{a}}^i$$ component of the total surface vector contained by the current cell. Computing the weight assigned to $$\sigma_i$$ requires that this surface vector be spread over the area of $$\sigma_i$$: $$\frac{\Delta c_1 \Delta c_2 \Delta c_3 \cdot V \cdot F_i}{\Delta c_{i+1} \Delta c_{i+2} \cdot A_i} = \frac{V}{A_i} \cdot \Delta c_i \cdot F_i$$.

The intersection between $$C_i$$ and $$\sigma_i$$ is the current lattice point with weight $$\left(\frac{V}{l_i} \cdot \Delta c_{i+1} \Delta c_{i+2} \cdot J_i\right)\left(\frac{V}{A_i} \cdot \Delta c_i \cdot F_i\right)$$$$ = \frac{V^2}{l_i A_i} \cdot \Delta c_1 \Delta c_2 \Delta c_3 \cdot J_i F_i$$.

Aside from the intersections between $$C_i$$ and $$\sigma_i$$ for each cell and $$i \in \{1,2,3\}$$, no other intersections occur. The total weight of the intersection at the vertex of the current cell is $$\sum_i \frac{V^2}{l_i A_i} \cdot \Delta c_1 \Delta c_2 \Delta c_3 \cdot J_i F_i$$$$ = V^2 \cdot \Delta c_1 \Delta c_2 \Delta c_3 \sum_i \frac{1}{l_i A_i} \cdot J_i F_i$$.

The value of $$\mathbf{J} \cdot \mathbf{F}$$ at the current cell is approximately $$\frac{1}{\Delta c_1 \Delta c_2 \Delta c_3 \cdot V} \cdot V^2 \cdot \Delta c_1 \Delta c_2 \Delta c_3 \sum_i \frac{1}{l_i A_i} \cdot J_i F_i$$$$ = V \sum_i \frac{1}{l_i A_i} \cdot J_i F_i$$. The coefficient of $$\frac{1}{\Delta c_1 \Delta c_2 \Delta c_3 \cdot V}$$ exists to spread the point weight over the current cell.

Therefore $$\mathbf{J} \cdot \mathbf{F} = V \sum_i \frac{1}{l_i A_i} \cdot J_i F_i$$. Note that $$\mathbf{J}$$ is expressed using the displacement basis vectors, while $$\mathbf{F}$$ is expressed using the surface basis vectors.

Computing surface-surface intersections
To save space, the notation $$(j;c_j)$$ and $$(j;k_j\Delta c_j)$$ will be omitted from the various terms.

Given a multi-surface $$\mathbf{S}_1$$ denoted by vector field $$\mathbf{F} = \sum_i F_i\hat{\mathbf{a}}^i$$, and a multi-surface $$\mathbf{S}_2$$ denoted by vector field $$\mathbf{G} = \sum_i G_i\hat{\mathbf{a}}^i$$, the vector field that denotes the intersection can be computed as follows:

The following computations applies to each cell:

For each $$i \in \{1,2,3\}$$, the weight assigned to $$\sigma_i$$ by $$\mathbf{S}_1$$ is computed as follows: $$\Delta c_1 \Delta c_2 \Delta c_3 \cdot V \cdot F_i$$ is the $$\hat{\mathbf{a}}^i$$ component of the total surface vector contained by the current cell. Computing the weight assigned to $$\sigma_i$$ requires that this surface vector be spread over the area of $$\sigma_i$$: $$\frac{\Delta c_1 \Delta c_2 \Delta c_3 \cdot V \cdot F_i}{\Delta c_{i+1} \Delta c_{i+2} \cdot A_i} = \frac{V}{A_i} \cdot \Delta c_i \cdot F_i$$. Similarly, the weight assigned to $$\sigma_i$$ by $$\mathbf{S}_2$$ is $$\frac{V}{A_i} \cdot \Delta c_i \cdot G_i$$.

The intersection between $$\sigma_{i+1}$$ and $$\sigma_{i+2}$$ is path $$C_i$$ with weight $$\left(\frac{V}{A_{i+1}} \cdot \Delta c_{i+1} \cdot F_{i+1}\right)\left(\frac{V}{A_{i+2}} \cdot \Delta c_{i+2} \cdot G_{i+2}\right)$$$$ = \frac{V^2}{A_{i+1} A_{i+2}} \cdot \Delta c_{i+1} \Delta c_{i+2} \cdot F_{i+1} G_{i+2}$$. Conversely, the intersection between $$\sigma_{i+2}$$ and $$\sigma_{i+1}$$ is path $$C_i$$ with weight $$-\frac{V^2}{A_{i+2} A_{i+1}} \cdot \Delta c_{i+2} \Delta c_{i+1} \cdot F_{i+2} G_{i+1}$$.

Aside from the intersections between $$\sigma_{i+1}$$ and $$\sigma_{i+2}$$, and the intersections between $$\sigma_{i+2}$$ and $$\sigma_{i+1}$$, for each cell and $$i \in \{1,2,3\}$$, no other intersections occur. For each $$i \in \{1,2,3\}$$, the total weight assigned to $$C_i$$ is $$\frac{V^2}{A_{i+1} A_{i+2}} \cdot \Delta c_{i+1} \Delta c_{i+2} \cdot F_{i+1} G_{i+2} - \frac{V^2}{A_{i+2} A_{i+1}} \cdot \Delta c_{i+2} \Delta c_{i+1} \cdot F_{i+2} G_{i+1}$$ $$ = \frac{V^2}{A_{i+1} A_{i+2}} \cdot \Delta c_{i+1} \Delta c_{i+2} \cdot (F_{i+1} G_{i+2} - F_{i+2} G_{i+1})$$.

The value of $$\mathbf{F} \times \mathbf{G}$$ at the current cell is approximately $$\sum_i \frac{l_i \cdot \Delta c_i \cdot \hat{\mathbf{a}}_i}{V \cdot \Delta c_1 \Delta c_2 \Delta c_3} \cdot \frac{V^2}{A_{i+1} A_{i+2}} \cdot \Delta c_{i+1} \Delta c_{i+2} \cdot (F_{i+1} G_{i+2} - F_{i+2} G_{i+1})$$ $$ = \sum_i \frac{V l_i}{A_{i+1} A_{i+2}} \cdot (F_{i+1} G_{i+2} - F_{i+2} G_{i+1})\hat{\mathbf{a}}_i$$. The coefficient of $$\frac{l_i \cdot \Delta c_i \cdot \hat{\mathbf{a}}_i}{V \cdot \Delta c_1 \Delta c_2 \Delta c_3}$$ exists to spread the displacement of each path over the current cell.

Therefore $$\mathbf{F} \times \mathbf{G} = \sum_i \frac{V l_i}{A_{i+1} A_{i+2}} \cdot (F_{i+1} G_{i+2} - F_{i+2} G_{i+1})\hat{\mathbf{a}}_i$$. Note that both $$\mathbf{F}$$ and $$\mathbf{G}$$ are both expressed using the surface basis vectors, but $$\mathbf{F} \times \mathbf{G}$$ is using the displacement basis vectors.

Computing the endpoints of paths
To save space, the notation $$(j;k_j)$$, $$(j;c_j)$$, and $$(j;k_j \Delta c_j)$$ will be omitted from the various terms. However, given a quantity $$Q$$ and an arbitrary $$i \in \{1,2,3\}$$, the notation $$[Q]_{-i}$$ will denote the quantity in the adjacent cell by moving back one step along the dimension indexed by $$i$$. This cell will be referred to as the $$-i$$ neighbor of the current cell.

Given a multi-path $$\mathbf{C}$$ denoted by vector field $$\mathbf{J} = \sum_i J_i\hat{\mathbf{a}}_i$$, the scalar field that denotes the endpoints can be computed as follows:

The following computations apply to each cell:

For each $$i \in \{1,2,3\}$$, the weight assigned to $$C_i$$ by $$\mathbf{C}$$ is computed as follows: $$\Delta c_1 \Delta c_2 \Delta c_3 \cdot V \cdot J_i$$ is the $$\hat{\mathbf{a}}_i$$ component of the total displacement contained by the current cell. Computing the weight assigned to $$C_i$$ requires that this displacement be spread over the length of $$C_i$$: $$\frac{\Delta c_1 \Delta c_2 \Delta c_3 \cdot V \cdot J_i}{\Delta c_i \cdot l_i} = \frac{V}{l_i} \cdot \Delta c_{i+1} \Delta c_{i+2} \cdot J_i$$.

For each $$i \in \{1,2,3\}$$, path $$C_i$$ contributes a weight of $$+\frac{V}{l_i} \cdot \Delta c_{i+1} \Delta c_{i+2} \cdot J_i$$ to the lattice point of the current cell, and path $$[C_i]_{-i}$$ contributes a weight of $$-\left[\frac{V}{l_i} \cdot \Delta c_{i+1} \Delta c_{i+2} \cdot J_i\right]_{-i}$$ to the lattice point of the current cell.

The total weight assigned to the lattice point of the current cell is $$\sum_i \left(+\frac{V}{l_i} \cdot \Delta c_{i+1} \Delta c_{i+2} \cdot J_i - \left[\frac{V}{l_i} \cdot \Delta c_{i+1} \Delta c_{i+2} \cdot J_i\right]_{-i}\right)$$ $$\approx \sum_i \Delta c_i \frac{\partial}{\partial c_i}\left(\frac{V}{l_i} \cdot \Delta c_{i+1} \Delta c_{i+2} \cdot J_i\right)$$ $$= \Delta c_1 \Delta c_2 \Delta c_3 \sum_i \frac{\partial}{\partial c_i}\left(\frac{V}{l_i} \cdot J_i\right)$$.

Spreading the weight assigned to the current lattice point over the volume of the current cell gives: $$\nabla \cdot \mathbf{J} = \frac{1}{\Delta c_1 \Delta c_2 \Delta c_3 \cdot V} \cdot \Delta c_1 \Delta c_2 \Delta c_3 \sum_i \frac{\partial}{\partial c_i}\left(\frac{V}{l_i} \cdot J_i\right)$$ $$= \frac{1}{V} \sum_i \frac{\partial}{\partial c_i}\left(\frac{V}{l_i} \cdot J_i\right)$$.

Therefore: $$\nabla \cdot \mathbf{J} = \frac{1}{V} \sum_i \frac{\partial}{\partial c_i}\left(\frac{V}{l_i} \cdot J_i\right)$$. Note that $$\mathbf{J}$$ is expressed using the displacement basis vectors.

Computing the counter-clockwise boundaries of surfaces
To save space, the notation $$(j;k_j)$$, $$(j;c_j)$$, and $$(j;k_j \Delta c_j)$$ will be omitted from the various terms. However, given a quantity $$Q$$ and an arbitrary $$i \in \{1,2,3\}$$, the notation $$[Q]_{-i}$$ will denote the quantity in the adjacent cell by moving back one step along the dimension indexed by $$i$$. This cell will be referred to as the $$-i$$ neighbor of the current cell.

Given a multi-surface $$\mathbf{S}$$ denoted by vector field $$\mathbf{F} = \sum_i F_i\hat{\mathbf{a}}^i$$, the vector field that denotes the counter-clockwise boundary can be computed as follows:

The following computations apply to each cell:

For each $$i \in \{1,2,3\}$$, the weight assigned to $$\sigma_i$$ by $$\mathbf{S}$$ is computed as follows: $$\Delta c_1 \Delta c_2 \Delta c_3 \cdot V \cdot F_i$$ is the $$\hat{\mathbf{a}}^i$$ component of the total surface vector contained by the current cell. Computing the weight assigned to $$\sigma_i$$ requires that this surface vector be spread over the area of $$\sigma_i$$: $$\frac{\Delta c_1 \Delta c_2 \Delta c_3 \cdot V \cdot F_i}{\Delta c_{i+1} \Delta c_{i+2} \cdot A_i} = \frac{V}{A_i} \cdot \Delta c_i \cdot F_i$$.

For each $$i \in \{1,2,3\}$$, surfaces that contain path $$C_i$$ as part of their boundary include $$\sigma_{i+1}$$, $$[\sigma_{i+1}]_{-(i+2)}$$, $$\sigma_{i+2}$$, and $$[\sigma_{i+2}]_{-(i+1)}$$. $$C_i$$ receives a mass of $$-\frac{V}{A_{i+1}} \cdot \Delta c_{i+1} \cdot F_{i+1}$$ from $$\sigma_{i+1}$$; a mass of $$+\left[\frac{V}{A_{i+1}} \cdot \Delta c_{i+1} \cdot F_{i+1}\right]_{-(i+2)}$$ from $$[\sigma_{i+1}]_{-(i+2)}$$; a mass of $$+\frac{V}{A_{i+2}} \cdot \Delta c_{i+2} \cdot F_{i+2}$$ from $$\sigma_{i+2}$$; and a mass of $$-\left[\frac{V}{A_{i+2}} \cdot \Delta c_{i+2} \cdot F_{i+2}\right]_{-(i+1)}$$ from $$[\sigma_{i+2}]_{-(i+1)}$$. The total mass assigned to $$C_i$$ is $$-\frac{V}{A_{i+1}} \cdot \Delta c_{i+1} \cdot F_{i+1} + \left[\frac{V}{A_{i+1}} \cdot \Delta c_{i+1} \cdot F_{i+1}\right]_{-(i+2)} + \frac{V}{A_{i+2}} \cdot \Delta c_{i+2} \cdot F_{i+2} - \left[\frac{V}{A_{i+2}} \cdot \Delta c_{i+2} \cdot F_{i+2}\right]_{-(i+1)}$$ $$\approx -\Delta c_{i+2} \frac{\partial}{\partial c_{i+2}}\left(\frac{V}{A_{i+1}} \cdot \Delta c_{i+1} \cdot F_{i+1}\right) + \Delta c_{i+1} \cdot \frac{\partial}{\partial c_{i+1}}\left(\frac{V}{A_{i+2}} \cdot \Delta c_{i+2} \cdot F_{i+2}\right)$$ $$= \Delta c_{i+1} \Delta c_{i+2} \left(\frac{\partial}{\partial c_{i+1}}\left(\frac{V}{A_{i+2}} \cdot F_{i+2}\right) - \frac{\partial}{\partial c_{i+2}}\left(\frac{V}{A_{i+1}} \cdot F_{i+1}\right)\right)$$.

Spreading the displacement generated by each $$C_i$$ over the volume of the current cell gives: $$\nabla \times \mathbf{F} = \sum_i \frac{\Delta c_i \cdot l_i \cdot \hat{\mathbf{a}}_i}{\Delta c_1 \Delta c_2 \Delta c_3 \cdot V} \cdot \Delta c_{i+1} \Delta c_{i+2} \left(\frac{\partial}{\partial c_{i+1}}\left(\frac{V}{A_{i+2}} \cdot F_{i+2}\right) - \frac{\partial}{\partial c_{i+2}}\left(\frac{V}{A_{i+1}} \cdot F_{i+1}\right)\right)$$ $$ = \sum_i \frac{l_i}{V}\left(\frac{\partial}{\partial c_{i+1}}\left(\frac{V}{A_{i+2}} \cdot F_{i+2}\right) - \frac{\partial}{\partial c_{i+2}}\left(\frac{V}{A_{i+1}} \cdot F_{i+1}\right)\right)\hat{\mathbf{a}}_i$$.

Therefore: $$\nabla \times \mathbf{F} = \sum_i \frac{l_i}{V}\left(\frac{\partial}{\partial c_{i+1}}\left(\frac{V}{A_{i+2}} \cdot F_{i+2}\right) - \frac{\partial}{\partial c_{i+2}}\left(\frac{V}{A_{i+1}} \cdot F_{i+1}\right)\right)\hat{\mathbf{a}}_i$$. Note that $$\mathbf{F}$$ is expressed using the surface basis vectors, but $$\nabla \times \mathbf{F}$$ is using the displacement basis vectors.

Computing the inwards-oriented surfaces of volumes
To save space, the notation $$(j;k_j)$$, $$(j;c_j)$$, and $$(j;k_j \Delta c_j)$$ will be omitted from the various terms. However, given a quantity $$Q$$ and an arbitrary $$i \in \{1,2,3\}$$, the notation $$[Q]_{-i}$$ will denote the quantity in the adjacent cell by moving back one step along the dimension indexed by $$i$$. This cell will be referred to as the $$-i$$ neighbor of the current cell.

Given a multi-volume $$\mathbf{U}$$ denoted by scalar field $$U$$, the vector field that denotes the inwards-oriented surface can be computed as follows:

The following computations apply to each cell:

The cell's volume $$\Omega$$ has the weight $$U$$.

For each $$i \in \{1,2,3\}$$, surface $$\sigma_i$$ receives a weight of $$U$$ from the current cell, and a weight of $$-[U]_{-i}$$ from the $$-i$$ neighbor of the current cell. The total weight is simply $$U - [U]_{-i} \approx \Delta c_i \frac{\partial U}{\partial c_i}$$. Spreading the surface vector generated by each $$\sigma_i$$ over the volume of the current cell gives: $$\nabla U = \sum_i \frac{\Delta c_{i+1} \Delta c_{i+2} \cdot A_i \cdot \hat{\mathbf{a}}^i}{\Delta c_1 \Delta c_2 \Delta c_3 \cdot V} \cdot \Delta c_i \frac{\partial U}{\partial c_i}$$ $$= \sum_i \frac{A_i}{V}\frac{\partial U}{\partial c_i}\hat{\mathbf{a}}^i$$.

Therefore: $$\nabla U = \sum_i \frac{A_i}{V}\frac{\partial U}{\partial c_i}\hat{\mathbf{a}}^i$$. Note that $$\nabla U$$ uses the surface basis vectors.

Summary

 * Given multi-path $$\mathbf{J} = \sum_i J_i\hat{\mathbf{a}}_i$$ and multi-surface $$\mathbf{F} = \sum_i F_i\hat{\mathbf{a}}^i$$, the intersection of $$\mathbf{J}$$ with $$\mathbf{F}$$ is multi-point $$\mathbf{J} \cdot \mathbf{F} = \sum_i \frac{V}{l_i A_i} J_i F_i$$.
 * Given multi-surfaces $$\mathbf{F} = \sum_i F_i\hat{\mathbf{a}}^i$$ and $$\mathbf{G} = \sum_i G_i\hat{\mathbf{a}}^i$$, the intersection of $$\mathbf{F}$$ with $$\mathbf{G}$$ is multi-path $$\mathbf{F} \times \mathbf{G} = \sum_i \frac{l_i V}{A_{i+1} A_{i+2}}(F_{i+1}G_{i+2} - F_{i+2}G_{i+1})\hat{\mathbf{a}}_i$$.
 * Given multi-path $$\mathbf{J} = \sum_i J_i\hat{\mathbf{a}}_i$$, the endpoints of $$\mathbf{J}$$ is multi-point $$\nabla \cdot \mathbf{J} = \sum_i \frac{1}{V} \frac{\partial}{\partial c_i}\left(\frac{V}{l_i}J_i\right)$$.
 * Given multi-surface $$\mathbf{F} = \sum_i F_i\hat{\mathbf{a}}^i$$, the counter-clockwise boundary of $$\mathbf{F}$$ is multi-path $$\nabla \times \mathbf{F} = \sum_i \frac{l_i}{V}\left(\frac{\partial}{\partial c_{i+1}}\left(\frac{V}{A_{i+2}}F_{i+2}\right) - \frac{\partial}{\partial c_{i+2}}\left(\frac{V}{A_{i+1}}F_{i+1}\right)\right)\hat{\mathbf{a}}_i$$.
 * Given multi-volume $$U$$, the inwards-oriented surface of $$U$$ is multi-surface $$\nabla U = \sum_i \frac{A_i}{V}\frac{\partial U}{\partial c_i}\hat{\mathbf{a}}^i$$.

Orthogonal coordinate systems
In the special case where the displacement basis vectors $$\{\hat{\mathbf{a}}_1, \hat{\mathbf{a}}_2, \hat{\mathbf{a}}_3\}$$ are all mutually orthogonal (perpendicular), then:
 * The surface basis vectors are identical to the displacement basis vectors: $$\forall i \in \{1,2,3\} : \hat{\mathbf{a}}^i = \hat{\mathbf{a}}_i$$.
 * For each $$i \in \{1,2,3\}$$, $$A_i = l_{i+1}l_{i+2}$$.
 * $$V = l_1 l_2 l_3$$.

The above formulas simplify to:
 * $$\left(\sum_i J_i\hat{\mathbf{a}}_i\right) \cdot \left(\sum_i F_i\hat{\mathbf{a}}^i\right) = \sum_i J_i F_i$$.
 * $$\left(\sum_i F_i\hat{\mathbf{a}}^i\right) \times \left(\sum_i G_i\hat{\mathbf{a}}^i\right) = \sum_i (F_{i+1}G_{i+2} - F_{i+2}G_{i+1})\hat{\mathbf{a}}_i$$.
 * $$\nabla \cdot \left(\sum_i J_i\hat{\mathbf{a}}_i\right) = \sum_i \frac{1}{l_1 l_2 l_3} \frac{\partial}{\partial c_i}\left(l_{i+1} l_{i+2} J_i\right)$$.
 * $$\nabla \times \left(\sum_i F_i\hat{\mathbf{a}}^i\right) = \sum_i \frac{1}{l_{i+1} l_{i+2}}\left(\frac{\partial}{\partial c_{i+1}}(l_{i+2} F_{i+2}) - \frac{\partial}{\partial c_{i+2}}\left(l_{i+1} F_{i+1}\right)\right)\hat{\mathbf{a}}_i$$.
 * $$\nabla U = \sum_i \frac{1}{l_i}\frac{\partial U}{\partial c_i}\hat{\mathbf{a}}^i$$.

For Cartesian coordinates, $$c_1 = x$$, $$c_2 = y$$, $$c_3 = z$$, and $$\hat{\mathbf{a}}^1 = \hat{\mathbf{a}}_1 = \hat{\mathbf{x}}$$, $$\hat{\mathbf{a}}^2 = \hat{\mathbf{a}}_2 = \hat{\mathbf{y}}$$, $$\hat{\mathbf{a}}^3 = \hat{\mathbf{a}}_3 = \hat{\mathbf{z}}$$, and $$l_1 = 1$$, $$l_2 = 1$$, $$l_3 = 1$$. Therefore:
 * $$(J_x\hat{\mathbf{x}} + J_y\hat{\mathbf{y}} + J_z\hat{\mathbf{z}}) \cdot (F_x\hat{\mathbf{x}} + F_y\hat{\mathbf{y}} + F_z\hat{\mathbf{z}}) = J_x F_x + J_y F_y + J_z F_z$$.
 * $$(F_x\hat{\mathbf{x}} + F_y\hat{\mathbf{y}} + F_z\hat{\mathbf{z}}) \times (G_x\hat{\mathbf{x}} + G_y\hat{\mathbf{y}} + G_z\hat{\mathbf{z}}) = (F_y G_z - F_z G_y)\hat{\mathbf{x}} + (F_z G_x - F_x G_z)\hat{\mathbf{y}} + (F_x G_y - F_y G_x)\hat{\mathbf{z}}$$.
 * $$\nabla \cdot (J_x\hat{\mathbf{x}} + J_y\hat{\mathbf{y}} + J_z\hat{\mathbf{z}}) = \frac{\partial J_x}{\partial x} + \frac{\partial J_y}{\partial y} + \frac{\partial J_z}{\partial z}$$.
 * $$\nabla \times (F_x\hat{\mathbf{x}} + F_y\hat{\mathbf{y}} + F_z\hat{\mathbf{z}}) = \left(\frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}\right)\hat{\mathbf{x}} + \left(\frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x}\right)\hat{\mathbf{y}} + \left(\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}\right)\hat{\mathbf{z}}$$.
 * $$\nabla U = \frac{\partial U}{\partial x}\hat{\mathbf{x}} + \frac{\partial U}{\partial y}\hat{\mathbf{y}} + \frac{\partial U}{\partial z}\hat{\mathbf{z}}$$.

For cylindrical coordinates, $$c_1 = \rho$$, $$c_2 = \phi$$, $$c_3 = z$$, and $$\hat{\mathbf{a}}^1 = \hat{\mathbf{a}}_1 = \hat{\mathbf{\rho}}$$, $$\hat{\mathbf{a}}^2 = \hat{\mathbf{a}}_2 = \hat{\mathbf{\phi}}$$, $$\hat{\mathbf{a}}^3 = \hat{\mathbf{a}}_3 = \hat{\mathbf{z}}$$, and $$l_1 = 1$$, $$l_2 = \rho$$, $$l_3 = 1$$. Therefore:
 * $$(J_\rho\hat{\mathbf{\rho}} + J_\phi\hat{\mathbf{\phi}} + J_z\hat{\mathbf{z}}) \cdot (F_\rho\hat{\mathbf{\rho}} + F_\phi\hat{\mathbf{\phi}} + F_z\hat{\mathbf{z}}) = J_\rho F_\rho + J_\phi F_\phi + J_z F_z$$.
 * $$(F_\rho\hat{\mathbf{\rho}} + F_\phi\hat{\mathbf{\phi}} + F_z\hat{\mathbf{z}}) \times (F_\rho\hat{\mathbf{\rho}} + F_\phi\hat{\mathbf{\phi}} + F_z\hat{\mathbf{z}}) = (F_\phi G_z - F_z G_\phi)\hat{\mathbf{\rho}} + (F_z G_\rho - F_\rho G_z)\hat{\mathbf{\phi}} + (F_\rho G_\phi - F_\phi G_\rho)\hat{\mathbf{z}}$$.
 * $$\nabla \cdot (J_\rho\hat{\mathbf{\rho}} + J_\phi\hat{\mathbf{\phi}} + J_z\hat{\mathbf{z}}) = \frac{1}{\rho}\left(\frac{\partial}{\partial \rho}(\rho J_\rho) + \frac{\partial F_\phi}{\partial \phi} + \frac{\partial}{\partial z}(\rho F_z)\right)$$.
 * $$\nabla \times (F_\rho\hat{\mathbf{\rho}} + F_\phi\hat{\mathbf{\phi}} + F_z\hat{\mathbf{z}}) = \frac{1}{\rho}\left(\frac{\partial F_z}{\partial \phi} - \frac{\partial}{\partial z}(\rho F_\phi)\right)\hat{\mathbf{\rho}} + \left(\frac{\partial F_\rho}{\partial z} - \frac{\partial F_z}{\partial \rho}\right)\hat{\mathbf{\phi}} + \frac{1}{\rho}\left(\frac{\partial}{\partial \rho}(\rho F_\phi) - \frac{\partial F_\rho}{\partial \phi}\right)\hat{\mathbf{z}}$$.
 * $$\nabla U = \frac{\partial U}{\partial \rho}\hat{\mathbf{\rho}} + \frac{1}{\rho}\frac{\partial U}{\partial \phi}\hat{\mathbf{\phi}} + \frac{\partial U}{\partial z}\hat{\mathbf{z}}$$.

For spherical coordinates, $$c_1 = r$$, $$c_2 = \theta$$, $$c_3 = \phi$$, and $$\hat{\mathbf{a}}^1 = \hat{\mathbf{a}}_1 = \hat{\mathbf{r}}$$, $$\hat{\mathbf{a}}^2 = \hat{\mathbf{a}}_2 = \hat{\mathbf{\theta}}$$, $$\hat{\mathbf{a}}^3 = \hat{\mathbf{a}}_3 = \hat{\mathbf{\phi}}$$, and $$l_1 = 1$$, $$l_2 = r$$, $$l_3 = r\sin\theta$$. Therefore:
 * $$(J_r\hat{\mathbf{r}} + J_\theta\hat{\mathbf{\theta}} + J_\phi\hat{\mathbf{\phi}}) \cdot (F_r\hat{\mathbf{r}} + F_\theta\hat{\mathbf{\theta}} + F_\phi\hat{\mathbf{\phi}}) = J_r F_r + J_\theta F_\theta + J_\phi F_\phi$$.
 * $$(F_r\hat{\mathbf{r}} + F_\theta\hat{\mathbf{\theta}} + F_\phi\hat{\mathbf{\phi}}) \times (G_r\hat{\mathbf{r}} + G_\theta\hat{\mathbf{\theta}} + G_\phi\hat{\mathbf{\phi}}) = (F_\theta G_\phi - F_\phi G_\theta)\hat{\mathbf{r}} + (F_\phi G_r - F_r G_\phi)\hat{\mathbf{\theta}} + (F_r G_\theta - F_\theta G_r)\hat{\mathbf{\phi}}$$.
 * $$\nabla \cdot (J_r\hat{\mathbf{r}} + J_\theta\hat{\mathbf{\theta}} + J_\phi\hat{\mathbf{\phi}}) = \frac{1}{r^2 \sin\theta}\left(\frac{\partial}{\partial r}(r^2\sin\theta J_r) + \frac{\partial}{\partial \theta}(r\sin\theta J_\theta) + \frac{\partial}{\partial \phi}(r F_\phi)\right)$$.
 * $$\nabla \times (F_r\hat{\mathbf{r}} + F_\theta\hat{\mathbf{\theta}} + F_\phi\hat{\mathbf{\phi}}) = \frac{1}{r^2 \sin\theta}\left(\frac{\partial}{\partial \theta}(r\sin\theta F_\phi) - \frac{\partial}{\partial \phi}(r F_\theta)\right)\hat{\mathbf{r}} + \frac{1}{r\sin\theta}\left(\frac{\partial F_r}{\partial \phi} - \frac{\partial}{\partial r}(r\sin\theta F_\phi)\right)\hat{\mathbf{\theta}} + \frac{1}{r}\left(\frac{\partial}{\partial r}(r F_\theta) - \frac{\partial F_r}{\partial \theta}\right)\hat{\mathbf{\phi}}$$.
 * $$\nabla U = \frac{\partial U}{\partial r}\hat{\mathbf{r}} + \frac{1}{r}\frac{\partial U}{\partial \theta}\hat{\mathbf{\theta}} + \frac{1}{r\sin\theta}\frac{\partial U}{\partial \phi}\hat{\mathbf{\phi}}$$.

The endpoints of intersections
Many identities related to vector calculus can be derived from examining the endpoints of path-volume intersections and surface-surface intersections.

The endpoints of path-volume intersections




Start with a multi-path $$\mathbf{C}$$, denoted by vector field $$\mathbf{J}$$, and a multi-volume $$\mathbf{U}$$, denoted by scalar field $$U$$. The intersection $$\mathbf{C} \cap \mathbf{U}$$ is denoted by vector field $$\mathbf{J}U$$.

Any time a path $$C$$ with weight $$w_1$$ starts in a volume $$\Omega$$ with weight $$w_2$$, the intersection $$\mathbf{C} \cap \mathbf{U}$$ gains an endpoint at the starting point of $$C$$ with weight $$w_1w_2$$. Any time a path $$C$$ with weight $$w_1$$ finishes in a volume $$\Omega$$ with weight $$w_2$$, the intersection $$\mathbf{C} \cap \mathbf{U}$$ gains an endpoint at the finishing point of $$C$$ with weight $$-w_1w_2$$. The endpoints for $$\mathbf{C} \cap \mathbf{U}$$ that are generated when paths from $$\mathbf{C}$$ start or finish in volumes from $$\mathbf{U}$$ is the intersection of the endpoints of $$\mathbf{C}$$ with multi-volume $$\mathbf{U}$$. This contributes the term $$(\nabla \cdot \mathbf{J})U$$ to $$\nabla \cdot (\mathbf{J}U)$$.

Any time a path $$C$$ with weight $$w_1$$ enters a volume $$\Omega$$ with weight $$w_2$$, the intersection $$\mathbf{C} \cap \mathbf{U}$$ gains an endpoint at the point of entry with weight $$w_1w_2$$. Any time a path $$C$$ with weight $$w_1$$ leaves a volume $$\Omega$$ with weight $$w_2$$, the intersection $$\mathbf{C} \cap \mathbf{U}$$ gains an endpoint at the point of exit with weight $$-w_1w_2$$. The endpoints for $$\mathbf{C} \cap \mathbf{U}$$ that are generated when paths from $$\mathbf{C}$$ enter or exit volumes from $$\mathbf{U}$$ is the intersection of multi-path $$\mathbf{C}$$ with the inwards oriented multi-surface of $$\mathbf{U}$$. This contributes the term $$\mathbf{J} \cdot (\nabla U)$$ to $$\nabla \cdot (\mathbf{J}U)$$.

The total endpoints of $$\mathbf{C} \cap \mathbf{U}$$ are: $$\nabla \cdot (\mathbf{J}U) = (\nabla \cdot \mathbf{J})U + \mathbf{J} \cdot (\nabla U)$$. In essence, the endpoints of $$\mathbf{C} \cap \mathbf{U}$$ are the endpoints of $$\mathbf{C}$$ that are contained in $$\mathbf{U}$$, plus the points at which paths from $$\mathbf{C}$$ enter or exit volumes from $$\mathbf{U}$$. This is depicted in the images on the right.

From the identity $$\nabla \cdot (\mathbf{J}U) = (\nabla \cdot \mathbf{J})U + \mathbf{J} \cdot (\nabla U)$$, counting the total point weight gives: $$\iiint_{\mathbf{q} \in \R^3} \nabla \cdot (\mathbf{J}(\mathbf{q})U(\mathbf{q}))dV = \iiint_{\mathbf{q} \in \R^3} (\nabla \cdot \mathbf{J}(\mathbf{q}))U(\mathbf{q})dV + \iiint_{\mathbf{q} \in \R^3} \mathbf{J}(\mathbf{q}) \cdot (\nabla U(\mathbf{q}))dV$$. For the endpoints of a multi-path, every starting point must be paired with a finishing point so the total point weight of the endpoints of a multi-path is 0. $$\iiint_{\mathbf{q} \in \R^3} \nabla \cdot (\mathbf{J}(\mathbf{q})U(\mathbf{q}))dV = 0$$ so hence $$\iiint_{\mathbf{q} \in \R^3} (\nabla \cdot \mathbf{J}(\mathbf{q}))U(\mathbf{q})dV = -\iiint_{\mathbf{q} \in \R^3} \mathbf{J}(\mathbf{q}) \cdot (\nabla U(\mathbf{q}))dV$$. The total intersection between the endpoints of multi-path $$\mathbf{C}$$ and multi-volume $$\mathbf{U}$$ is the negative of the total intersection between $$\mathbf{C}$$ and the inwards oriented surface of $$\mathbf{U}$$.

If $$\mathbf{J}$$ denotes a simple path $$C$$ that starts at point $$\mathbf{q}_0$$ and ends at point $$\mathbf{q}_1$$, then the above integral identity becomes:

$$\iiint_{\mathbf{q} \in \R^3} (\nabla \cdot \delta_1(\mathbf{q};C))U(\mathbf{q})dV = -\iiint_{\mathbf{q} \in \R^3} \delta_1(\mathbf{q};C) \cdot (\nabla U(\mathbf{q}))dV$$ $$ \iff \iiint_{\mathbf{q} \in \R^3} (\delta_0(\mathbf{q};\mathbf{q}_0)-\delta_0(\mathbf{q};\mathbf{q}_1))U(\mathbf{q})dV = -\int_{\mathbf{q} \in C} (\nabla U(\mathbf{q})) \cdot d\mathbf{q}$$ $$ \iff U(\mathbf{q}_0) - U(\mathbf{q}_1) = -\int_{\mathbf{q} \in C} (\nabla U(\mathbf{q})) \cdot d\mathbf{q}$$ $$ \iff \int_{\mathbf{q} \in C} (\nabla U(\mathbf{q})) \cdot d\mathbf{q} = U(\mathbf{q}_1) - U(\mathbf{q}_0)$$ This is known as the gradient theorem.

If $$U$$ denotes a simple volume $$\Omega$$ with a outwards oriented surface $$\sigma$$, then the integral identity becomes:

$$\iiint_{\mathbf{q} \in \R^3} (\nabla \cdot \mathbf{J}(\mathbf{q}))\delta_3(\mathbf{q};\Omega)dV = -\iiint_{\mathbf{q} \in \R^3} \mathbf{J}(\mathbf{q}) \cdot (\nabla \delta_3(\mathbf{q};\Omega))dV$$ $$ \iff \iiint_{\mathbf{q} \in \Omega} (\nabla \cdot \mathbf{J}(\mathbf{q}))dV = -\iiint_{\mathbf{q} \in \R^3} \mathbf{J}(\mathbf{q}) \cdot (-\delta_2(\mathbf{q};\sigma))dV$$ $$ \iff \iiint_{\mathbf{q} \in \Omega} (\nabla \cdot \mathbf{J}(\mathbf{q}))dV = \iint_{\mathbf{q} \in \sigma} \mathbf{J}(\mathbf{q}) \cdot d\mathbf{S}$$ This is known as Gauss's divergence theorem.

In summary:
 * Given a multi-path denoted by vector field $$\mathbf{J}$$, and a multi-volume denoted by scalar field $$U$$, then the endpoints of the intersection are: $$\nabla \cdot (\mathbf{J}U) = (\nabla \cdot \mathbf{J})U + \mathbf{J} \cdot (\nabla U)$$.
 * Given a multi-path denoted by vector field $$\mathbf{J}$$, and a multi-volume denoted by scalar field $$U$$, then $$\iiint_{\mathbf{q} \in \R^3} (\nabla \cdot \mathbf{J}(\mathbf{q}))U(\mathbf{q})dV = -\iiint_{\mathbf{q} \in \R^3} \mathbf{J}(\mathbf{q}) \cdot (\nabla U(\mathbf{q}))dV$$.
 * Given a simple path $$C$$ that starts at point $$\mathbf{q}_0$$ and ends at point $$\mathbf{q}_1$$, and a multi-volume denoted by scalar field $$U$$, then $$\int_{\mathbf{q} \in C} (\nabla U(\mathbf{q})) \cdot d\mathbf{q} = U(\mathbf{q}_1) - U(\mathbf{q}_0)$$. This is the gradient theorem.
 * Given a multi-path denoted by vector field $$\mathbf{J}$$, and a simple volume $$\Omega$$ with outwards oriented surface $$\sigma$$, then $$\iiint_{\mathbf{q} \in \Omega} (\nabla \cdot \mathbf{J}(\mathbf{q}))dV = \iint_{\mathbf{q} \in \sigma} \mathbf{J}(\mathbf{q}) \cdot d\mathbf{S}$$. This is Gauss's divergence theorem.

The endpoints of surface-surface intersections


Start with multi-surface $$\mathbf{S}_1$$, denoted by vector field $$\mathbf{F}_1$$, and a second multi-surface $$\mathbf{S}_2$$, denoted by vector field $$\mathbf{F}_2$$. The intersection $$\mathbf{S}_1 \cap \mathbf{S}_2$$ is denoted by vector field $$\mathbf{F}_1 \times \mathbf{F}_2$$.

Consider a surface $$\sigma_1$$ with weight $$w_1$$ from $$\mathbf{S}_1$$, and a surface $$\sigma_2$$ with weight $$w_2$$ from $$\mathbf{S}_2$$. Let $$\partial\sigma_1$$ denote the counter-clockwise boundary of $$\sigma_1$$, and let $$\partial\sigma_2$$ denote the counter-clockwise boundary of $$\sigma_2$$. There are 4 scenarios regarding the endpoints of $$\sigma_1 \cap \sigma_2$$:
 * When the $$\partial\sigma_1$$ intersects $$\sigma_2$$ in the preferred direction, the intersection point $$\partial\sigma_1 \cap \sigma_2$$ has a weight of $$+w_1w_2$$, and an endpoint with weight $$+w_1w_2$$ (starting point) for $$\sigma_1 \cap \sigma_2$$ forms at $$\partial\sigma_1 \cap \sigma_2$$.
 * When the $$\partial\sigma_1$$ intersects $$\sigma_2$$ in the opposite direction, the intersection point $$\partial\sigma_1 \cap \sigma_2$$ has a weight of $$-w_1w_2$$, and an endpoint with weight $$-w_1w_2$$ (finishing point) for $$\sigma_1 \cap \sigma_2$$ forms at $$\partial\sigma_1 \cap \sigma_2$$.
 * When the $$\partial\sigma_2$$ intersects $$\sigma_1$$ in the preferred direction, the intersection point $$\sigma_1 \cap \partial\sigma_2$$ has a weight of $$+w_1w_2$$, and an endpoint with weight $$-w_1w_2$$ (finishing point) for $$\sigma_1 \cap \sigma_2$$ forms at $$\sigma_1 \cap \partial\sigma_2$$.
 * When the $$\partial\sigma_2$$ intersects $$\sigma_1$$ in the opposite direction, the intersection point $$\sigma_1 \cap \partial\sigma_2$$ has a weight of $$-w_1w_2$$, and an endpoint with weight $$+w_1w_2$$ (starting point) for $$\sigma_1 \cap \sigma_2$$ forms at $$\sigma_1 \cap \partial\sigma_2$$.

It can be seen that the intersection $$\partial\sigma_1 \cap \sigma_2$$ forms endpoints for $$\sigma_1 \cap \sigma_2$$ with the correct polarity, and that the intersection $$\sigma_1 \cap \partial\sigma_2$$ forms endpoints for $$\sigma_1 \cap \sigma_2$$ with the opposite polarity. This can be observed in the image of the right. This implies that the endpoints of $$\mathbf{S}_1 \cap \mathbf{S}_2$$ are: $$\nabla \cdot (\mathbf{F}_1 \times \mathbf{F}_2) = (\nabla \times \mathbf{F}_1) \cdot \mathbf{F}_2 - \mathbf{F}_1 \cdot (\nabla \times \mathbf{F}_2)$$.



From the identity $$\nabla \cdot (\mathbf{F}_1 \times \mathbf{F}_2) = (\nabla \times \mathbf{F}_1) \cdot \mathbf{F}_2 - \mathbf{F}_1 \cdot (\nabla \times \mathbf{F}_2)$$, counting the total point weight gives: $$\iiint_{\mathbf{q} \in \R^3} \nabla \cdot (\mathbf{F}_1(\mathbf{q}) \times \mathbf{F}_2(\mathbf{q}))dV = \iiint_{\mathbf{q} \in \R^3} (\nabla \times \mathbf{F}_1(\mathbf{q})) \cdot \mathbf{F}_2(\mathbf{q})dV - \iiint_{\mathbf{q} \in \R^3} \mathbf{F}_1(\mathbf{q}) \cdot (\nabla \times \mathbf{F}_2(\mathbf{q}))dV$$. For the endpoints of a multi-path, every starting point must be paired with a finishing point so the total point weight of the endpoints of a multi-path is 0. $$\iiint_{\mathbf{q} \in \R^3} \nabla \cdot (\mathbf{F}_1(\mathbf{q}) \times \mathbf{F}_2(\mathbf{q}))dV = 0$$ so hence $$\iiint_{\mathbf{q} \in \R^3} (\nabla \times \mathbf{F}_1(\mathbf{q})) \cdot \mathbf{F}_2(\mathbf{q})dV = \iiint_{\mathbf{q} \in \R^3} \mathbf{F}_1(\mathbf{q}) \cdot (\nabla \times \mathbf{F}_2(\mathbf{q}))dV$$. The total intersection of the counter-clockwise boundary of multi-surface $$\mathbf{S}_1$$ with multi-surface $$\mathbf{S}_2$$ is the total intersection of the counter-clockwise boundary of $$\mathbf{S}_2$$ with $$\mathbf{S}_1$$. This is illustrated by the image on the right.

If $$\mathbf{F}_2$$ denotes a simple surface $$\sigma$$ with a counter-clockwise boundary $$\partial\sigma$$, then the above integral identity becomes:

$$\iiint_{\mathbf{q} \in \R^3} (\nabla \times \mathbf{F}_1(\mathbf{q})) \cdot \delta_2(\mathbf{q};\sigma)dV = \iiint_{\mathbf{q} \in \R^3} \mathbf{F}_1(\mathbf{q}) \cdot (\nabla \times \delta_2(\mathbf{q};\sigma))dV$$ $$ \iff \iint_{\mathbf{q} \in \sigma} (\nabla \times \mathbf{F}_1(\mathbf{q})) \cdot d\mathbf{S} = \iiint_{\mathbf{q} \in \R^3} \mathbf{F}_1(\mathbf{q}) \cdot \delta_1(\mathbf{q};\partial\sigma)dV$$ $$ \iff \iint_{\mathbf{q} \in \sigma} (\nabla \times \mathbf{F}_1(\mathbf{q})) \cdot d\mathbf{S} = \int_{\mathbf{q} \in \partial\sigma} \mathbf{F}_1(\mathbf{q}) \cdot d\mathbf{q}$$

This is known as Stokes' theorem.

In summary:
 * Given two multi-surfaces denoted by vector fields $$\mathbf{F}_1$$ and $$\mathbf{F}_2$$, then the endpoints of the intersection are: $$\nabla \cdot (\mathbf{F}_1 \times \mathbf{F}_2) = (\nabla \times \mathbf{F}_1) \cdot \mathbf{F}_2 - \mathbf{F}_1 \cdot (\nabla \times \mathbf{F}_2)$$.
 * Given two multi-surfaces denoted by vector fields $$\mathbf{F}_1$$ and $$\mathbf{F}_2$$, then $$\iiint_{\mathbf{q} \in \R^3} (\nabla \times \mathbf{F}_1(\mathbf{q})) \cdot \mathbf{F}_2(\mathbf{q})dV = \iiint_{\mathbf{q} \in \R^3} \mathbf{F}_1(\mathbf{q}) \cdot (\nabla \times \mathbf{F}_2(\mathbf{q}))dV$$.
 * Given a multi-surface denoted by vector field $$\mathbf{F}_1$$ and a simple surface $$\sigma$$ with counter-clockwise oriented boundary $$\partial\sigma$$, then $$\iint_{\mathbf{q} \in \sigma} (\nabla \times \mathbf{F}_1(\mathbf{q})) \cdot d\mathbf{S} = \int_{\mathbf{q} \in \partial\sigma} \mathbf{F}_1(\mathbf{q}) \cdot d\mathbf{q}$$. This is Stokes' theorem.

The boundaries of intersections
In addition to the identities derived from examining the endpoints of intersections, some more identities can be derived by examining the counter-clockwise boundaries of surfaces that result from intersections.

The counter-clockwise boundary of surface-volume intersections


Start with a multi-surface $$\mathbf{S}$$, denoted by vector field $$\mathbf{F}$$, and a multi-volume $$\mathbf{U}$$, denoted by scalar field $$U$$. The intersection $$\mathbf{S} \cap U$$ is denoted by vector field $$\mathbf{F}U$$.

Consider a surface $$\sigma$$ with weight $$w_1$$ from $$\mathbf{S}$$, and a volume $$\Omega$$ with weight $$w_2$$ from $$\mathbf{U}$$. Let $$\partial\sigma$$ denote the counter-clockwise boundary of $$\sigma$$, and let $$\partial\Omega$$ denote the inwards oriented surface of $$\Omega$$. There are two sources for the counter-clockwise boundary of $$\sigma \cap \Omega$$. Any time $$\partial\sigma$$ intersects $$\Omega$$, the intersection $$\partial\sigma \cap \Omega$$ contributes to the boundary of $$\sigma \cap \Omega$$. When $$\partial\sigma$$ leaves $$\Omega$$, the boundary of $$\sigma \cap \Omega$$ cannot follow, and instead must trace along the surface of $$\Omega$$ while remaining in the surface $$\sigma$$ as indicated in the image to the right. The boundary of the total intersection $$\mathbf{S} \cap \mathbf{U}$$, denoted by $$\nabla \times (\mathbf{F}U)$$, consists of two parts: the intersection of the boundary of $$\mathbf{S}$$ with $$\mathbf{U}$$, denoted by $$(\nabla \times \mathbf{F})U$$, and the intersection of the inwards-oriented surface of $$\mathbf{U}$$ with $$\mathbf{S}$$, denoted by $$(\nabla U) \times \mathbf{F} = - \mathbf{F} \times (\nabla U)$$. Therefore: $$\nabla \times (\mathbf{F}U) = (\nabla \times \mathbf{F})U + (\nabla U) \times \mathbf{F} = (\nabla \times \mathbf{F})U - \mathbf{F} \times (\nabla U)$$.

From the identity $$\nabla \times (\mathbf{F}U) = (\nabla \times \mathbf{F})U - \mathbf{F} \times (\nabla U)$$, computing the total displacement gives: $$\iiint_{\mathbf{q} \in \R^3} \nabla \times (\mathbf{F}(\mathbf{q})U(\mathbf{q}))dV = \iiint_{\mathbf{q} \in \R^3} (\nabla \times \mathbf{F}(\mathbf{q}))U(\mathbf{q})dV - \iiint_{\mathbf{q} \in \R^3} \mathbf{F}(\mathbf{q}) \times (\nabla U(\mathbf{q}))dV$$. The counter-clockwise boundary of a multi-surface is a closed multi-loop, and the total displacement generated by a loop is $$\mathbf{0}$$. $$\iiint_{\mathbf{q} \in \R^3} \nabla \times (\mathbf{F}(\mathbf{q})U(\mathbf{q}))dV = \mathbf{0}$$ so hence $$\iiint_{\mathbf{q} \in \R^3} (\nabla \times \mathbf{F}(\mathbf{q}))U(\mathbf{q})dV = \iiint_{\mathbf{q} \in \R^3} \mathbf{F}(\mathbf{q}) \times (\nabla U(\mathbf{q}))dV$$. The total intersection of a the boundary of multi-surface $$\mathbf{S}$$ with multi-volume $$\mathbf{U}$$ is the total intersection of $$\mathbf{S}$$ with the surface of $$\mathbf{U}$$.

If $$\mathbf{F}$$ denotes a simple surface $$\sigma$$ with counter-clockwise boundary $$\partial\sigma$$, then the above integral identity becomes:

$$\iiint_{\mathbf{q} \in \R^3} (\nabla \times \delta_2(\mathbf{q};\sigma))U(\mathbf{q})dV = \iiint_{\mathbf{q} \in \R^3} \delta_2(\mathbf{q};\sigma) \times (\nabla U(\mathbf{q}))dV $$ $$ \iff \iiint_{\mathbf{q} \in \R^3} \delta_1(\mathbf{q};\partial\sigma)U(\mathbf{q})dV = \iint_{\mathbf{q} \in \sigma} d\mathbf{S} \times (\nabla U(\mathbf{q})) $$ $$ \iff \int_{\mathbf{q} \in \partial\sigma} U(\mathbf{q})d\mathbf{q} = -\iint_{\mathbf{q} \in \sigma} (\nabla U(\mathbf{q})) \times d\mathbf{S} $$

If $$\Omega$$ denotes a simple volume with outwards oriented surface $$\sigma$$, then the integral identity becomes:

$$\iiint_{\mathbf{q} \in \R^3} (\nabla \times \mathbf{F}(\mathbf{q}))\delta_3(\mathbf{q};\Omega)dV = \iiint_{\mathbf{q} \in \R^3} \mathbf{F}(\mathbf{q}) \times (\nabla \delta_3(\mathbf{q};\Omega))dV$$ $$ \iff \iiint_{\mathbf{q} \in \Omega} (\nabla \times \mathbf{F}(\mathbf{q}))dV = \iiint_{\mathbf{q} \in \R^3} \mathbf{F}(\mathbf{q}) \times (-\delta_2(\mathbf{q};\sigma))dV$$ $$ \iff \iiint_{\mathbf{q} \in \Omega} (\nabla \times \mathbf{F}(\mathbf{q}))dV = -\iint_{\mathbf{q} \in \sigma} \mathbf{F}(\mathbf{q}) \times d\mathbf{S}$$

In summary:
 * Given a multi-surface denoted by vector field $$\mathbf{F}$$, and a multi-volume denoted by scalar field $$U$$, then the counter-clockwise boundary of the intersection is: $$\nabla \times (\mathbf{F}U) = (\nabla \times \mathbf{F})U - \mathbf{F} \times (\nabla U)$$.
 * Given a multi-surface denoted by vector field $$\mathbf{F}$$, and a multi-volume denoted by scalar field $$U$$, then $$\iiint_{\mathbf{q} \in \R^3} (\nabla \times \mathbf{F}(\mathbf{q}))U(\mathbf{q})dV = \iiint_{\mathbf{q} \in \R^3} \mathbf{F}(\mathbf{q}) \times (\nabla U(\mathbf{q}))dV$$
 * Given a simple surface $$\sigma$$ with counter-clockwise boundary $$\partial\sigma$$, and a multi-volume denoted by scalar field $$U$$, then $$\int_{\mathbf{q} \in \partial\sigma} U(\mathbf{q})d\mathbf{q} = -\iint_{\mathbf{q} \in \sigma} (\nabla U(\mathbf{q})) \times d\mathbf{S} $$.
 * Given a multi-surface denoted by vector field $$\mathbf{F}$$, and a simple volume $$\Omega$$ with outwards oriented surface $$\sigma$$, then $$\iiint_{\mathbf{q} \in \Omega} (\nabla \times \mathbf{F}(\mathbf{q}))dV = -\iint_{\mathbf{q} \in \sigma} \mathbf{F}(\mathbf{q}) \times d\mathbf{S}$$.

The surfaces of intersections
Some more identities can be derived by examining the surfaces of volumes that result from intersections.

The inwards-oriented surface of volume-volume intersections


Start with a multi-volume $$\mathbf{U}_1$$, denoted by scalar field $$U_1$$, and a second multi-volume $$\mathbf{U}_2$$, denoted by scalar field $$U_2$$. The intersection $$\mathbf{U}_1 \cap \mathbf{U}_2$$ is denoted by the scalar field $$U_1 U_2$$.

Consider a volume $$\Omega_1$$ with weight $$w_1$$ from $$\mathbf{U}_1$$, and a volume $$\Omega_2$$ with weight $$w_2$$ from $$\mathbf{U}_2$$. Let $$\sigma_1$$ denote the inwards-oriented surface of $$\Omega_1$$, and let $$\sigma_2$$ denote the inwards-oriented surface of $$\Omega_2$$. There are two parts to the inwards-oriented surface of the intersection $$\Omega_1 \cap \Omega_2$$, as shown in the image to the right. Part of the surface of $$\Omega_1 \cap \Omega_2$$ consists of the portion of $$\sigma_2$$ that is contained by $$\Omega_1$$, which contributes the term $$U_1 (\nabla U_2)$$ to $$\nabla (U_1 U_2)$$. The other part of the surface of $$\Omega_1 \cap \Omega_2$$ consists of the portion of $$\sigma_1$$ that is contained by $$\Omega_2$$, which contributes the term $$(\nabla U_1) U_2$$ to $$\nabla (U_1 U_2)$$. Therefore the total surface of $$\mathbf{U}_1 \cap \mathbf{U}_2$$ is $$\nabla (U_1 U_2) = U_1 (\nabla U_2) + (\nabla U_1) U_2$$.

From the identity $$\nabla (U_1 U_2) = U_1 (\nabla U_2) + (\nabla U_1) U_2$$, computing the total surface vector gives: $$\iiint_{\mathbf{q} \in \R^3} \nabla (U_1(\mathbf{q}) U_2(\mathbf{q}))dV = \iiint_{\mathbf{q} \in \R^3} U_1(\mathbf{q}) (\nabla U_2(\mathbf{q}))dV + \iiint_{\mathbf{q} \in \R^3} (\nabla U_1(\mathbf{q})) U_2(\mathbf{q})dV$$. The inwards-oriented surface of a multi-volume is a closed multi-surface, and the total surface vector of a closed surface is $$\mathbf{0}$$. $$\iiint_{\mathbf{q} \in \R^3} \nabla (U_1(\mathbf{q}) U_2(\mathbf{q}))dV = \mathbf{0}$$ so hence $$\iiint_{\mathbf{q} \in \R^3} U_1(\mathbf{q}) (\nabla U_2(\mathbf{q}))dV = -\iiint_{\mathbf{q} \in \R^3} (\nabla U_1(\mathbf{q})) U_2(\mathbf{q})dV$$. The total surface vector of the intersection of multi-volume $$\mathbf{U}_1$$ with the inwards oriented surface of multi-volume $$\mathbf{U}_2$$ is the opposite of the total surface vector of the intersection of the inwards oriented surface of $$\mathbf{U}_1$$ with $$\mathbf{U}_2$$.

If $$U_1$$ denotes a simple volume $$\Omega$$ with outwards oriented surface $$\sigma$$, then the above integral identity becomes: $$\iiint_{\mathbf{q} \in \R^3} \delta_3(\mathbf{q};\Omega) (\nabla U_2(\mathbf{q}))dV = -\iiint_{\mathbf{q} \in \R^3} (\nabla \delta_3(\mathbf{q};\Omega)) U_2(\mathbf{q})dV$$ $$ \iff \iiint_{\mathbf{q} \in \Omega} (\nabla U_2(\mathbf{q}))dV = -\iiint_{\mathbf{q} \in \R^3} (-\delta_2(\mathbf{q};\sigma)) U_2(\mathbf{q})dV$$ $$ \iff \iiint_{\mathbf{q} \in \Omega} (\nabla U_2(\mathbf{q}))dV = \iint_{\mathbf{q} \in \sigma} U_2(\mathbf{q})d\mathbf{S}$$

In summary:
 * Given two multi-volumes denoted by scalar fields $$U_1$$ and $$U_2$$, then the inwards-oriented surface of the intersection is: $$\nabla (U_1 U_2) = U_1 (\nabla U_2) + (\nabla U_1) U_2$$.
 * Given two multi-volumes denoted by scalar fields $$U_1$$ and $$U_2$$, then $$\iiint_{\mathbf{q} \in \R^3} U_1(\mathbf{q}) (\nabla U_2(\mathbf{q}))dV = -\iiint_{\mathbf{q} \in \R^3} (\nabla U_1(\mathbf{q})) U_2(\mathbf{q})dV$$.
 * Given a simple volume $$\Omega$$ with outwards oriented surface $$\sigma$$ and a multi-volume denoted by scalar field $$U_2$$, then $$\iiint_{\mathbf{q} \in \Omega} (\nabla U_2(\mathbf{q}))dV = \iint_{\mathbf{q} \in \sigma} U_2(\mathbf{q})d\mathbf{S}$$.

Summary
The tables below summarizes the results of the previous sections: