Talk:Calculus/Discrete vector calculus

The purpose of this chapter is primarily to simplify the concept of vector calculus by using a graph as a discrete model of space. Operators such as the gradient, divergence, and curl, and their associated theorems can be expressed in a form that is devoid of any explicit calculus which will make these concepts intuitive and easy to grasp. Another benefit to using graphs as a lumped model of space is that the space being modeled does not have to be Euclidean. The lumped model provides the groundwork for generalizing vector calculus to non-Euclidean geometries.

The notation has changed, see the bottom of this section for the currently used notation.

The prime notation that will be used in this chapter is the following:


 * $$\mathcal{G}$$ denotes the subject directed graph.
 * $$\mathcal{N}$$ denotes the set of nodes.
 * $$\mathcal{E}$$ denotes the set of edges.
 * $$t_\bullet(e)$$ denotes the start of edge $$e$$.
 * $$t^\bullet(e)$$ denotes the end of edge $$e$$.
 * $$V(n)$$ denotes the volume of node $$n$$.
 * $$l(e)$$ denotes the length of edge $$e$$.
 * $$A(e)$$ denotes the area/thickness of edge $$e$$.
 * $$\mathbf{q}(n)$$ denotes the point in space associated with node $$n$$.
 * $$\Omega(n)$$ denotes the volume/region of space associated with node $$n$$.
 * $$C(e)$$ denotes the oriented path associated with edge $$e$$.
 * $$\sigma(e)$$ denotes the oriented surface associated with edge $$e$$.

Care should be taken to avoid any overlaps with the above notation.

Math buff (discuss • contribs) 02:25, 28 April 2018 (UTC)

To cleanup the notation and to release letters and symbols that are of too great a risk of causing notation overlap, the following notation changes will be made:


 * The length of edge $$e$$ will be denoted by $$\tau_1(e)$$ instead of $$l(e)$$.
 * The area/thickness of edge $$e$$ will be denoted by $$\tau_2(e)$$ instead of $$A(e)$$.
 * The volume of node $$n$$ will be denoted by $$\tau_3(n)$$ instead of $$V(n)$$.
 * The point in space associated with node $$n$$ will be denoted by $$\omega_0(n)$$ instead of $$\mathbf{q}(n)$$.
 * The oriented path associated with edge $$e$$ will be denoted by $$\omega_1(e)$$ instead of $$C(e)$$.
 * The oriented surface associated with edge $$e$$ will be denoted by $$\omega_2(e)$$ instead of $$\sigma(e)$$.
 * The volume/region of space associated with node $$n$$ will be denoted by $$\omega_3(n)$$ instead of $$\Omega(n)$$.

The new notation will be:
 * $$\mathcal{G}$$ denotes the subject directed graph.
 * $$\mathcal{N}$$ denotes the set of nodes.
 * $$\mathcal{E}$$ denotes the set of edges.
 * $$t_\bullet(e)$$ denotes the start of edge $$e$$.
 * $$t^\bullet(e)$$ denotes the end of edge $$e$$.
 * $$\tau_1(e)$$ denotes the length of edge $$e$$.
 * $$\tau_2(e)$$ denotes the area/thickness of edge $$e$$.
 * $$\tau_3(n)$$ denotes the volume of node $$n$$.
 * $$\omega_0(n)$$ denotes the point in space associated with node $$n$$.
 * $$\omega_1(e)$$ denotes the oriented path associated with edge $$e$$.
 * $$\omega_2(e)$$ denotes the oriented surface associated with edge $$e$$.
 * $$\omega_3(n)$$ denotes the volume/region of space associated with node $$n$$.
 * $$\delta_0(n;n_0)$$ denotes the node based function that denotes point $$n_0$$.
 * $$\delta_1(e;P)$$ denotes the edge based function that denotes path $$P$$.
 * $$\delta_2(e;S)$$ denotes the edge based function that denotes surface $$S$$.
 * $$\delta_3(n;U)$$ denotes the node based function the denotes volume $$U$$.
 * $$\Sigma_{\mathcal{N}}(f,g)$$ denotes the net intersection of multi-point $$f$$ with multi-volume $$g$$.
 * $$\Sigma_{\mathcal{E}}(f,g)$$ denotes the net intersection of multi-path $$f$$ with multi-surface $$g$$.

Math buff (discuss • contribs) 23:30, 19 July 2018 (UTC)