Talk:Calculus/Introduction to multivariable calculus

Green's theorem
Yo, I added in Green's Theorem and order of integration into Multivariable Calculus, and Projection/derivatives/limits/integral/vector-valued functions/position, velocity, acceleration, tangent, normal, binormal, vectors/curvature into Vector Calculus. However, I'm not sure how much of this scope from Vector Calculus spills into this Multivariable Calculus, what else should be added (as I don't really want to edit for grammatical errors and ordering nicety, just adding articles, since I suck at ordering and English), and where I can put independence of path or if you have that in here, at one point it looks like you do, but to be honest I can't decipher what you're trying to teach; but then again I just randomly put stuff in here also. Anyways, what's going on/where can I read about, thanks, feedback? Fephisto 05:52, 14 January 2006 (UTC)

Comment
Starting with the topology of Rn feels like an higher level approach than the the other chapters in this section of this book - more suited for the fully formal third section.

I put the div, grad, hyperbolic, parabolic, etc subheadings in to indicate the expected scope of this chapter, and to allow people to fill it in from the middle. Was there any particular objection to doing this?

Does anyone have any comments on the scope of this chapter, and the rest of this book? As I understand it, it's to cover calculus from 16-19, A' Level and 1st year undergrad, mostly without full rigor. At that level, I'd expect a course on multivariable calculus to cover most or all the headings I listed, and nothing beyond. Carandol 04:48, 13 Apr 2004 (UTC)


 * You're on the wrong page. Try Talk:Calculus:Multivariable calculus. The reason why I'm doing it this way is that I'm trying to be complete, but not especially dryly rigorous. It's not that bad, is it? Anyway, I'll be getting to divs and grads etc later. Dysprosia 04:52, 13 Apr 2004 (UTC)

I hadn't noticed the redirect, left from moving the simple vector stuff to an earlier chapter.

I'd have just relied on intuitive definitions at this level, and maybe a couple of illustrations. Everyone knows what the edge of a surface or volume is, even if that intuitive notion is incomplete. That was the general approach followed by the textbooks I read at this level.

And why not leave the div, grad, etc headings in? Then people can drop stuff in the middle of the chapter, if that happens to be what they feel like writing, and it gives a better idea of the general level expected. Carandol 05:09, 13 Apr 2004 (UTC)


 * I'll put in illustrations later. I want to get most of the text out of the way first.
 * I also think certain pages should go a bit more "above and beyond the call of duty". My aesthetic for these sorts of pages is to be complete, rigorous but not formal, or to the point of excessive dryness, and to have gentler language. Maybe it's a bit too much now, so I wouldn't know where else to take it. It's not formal enough to deserve being under the "formal" section on the Calculus intro page.
 * As to the headers, I probably won't be using them exactly, so I don't know how useful it would be. Are you thinking of contributing too? Maybe we can work something out, so you can pitch in at the middle if you so choose. Dysprosia 05:29, 13 Apr 2004 (UTC)

You could start with the intuitive notion of boundaries first, then say this is how we talk about them rigorously.

I am thinking of contributing. That's another reason I put the headers in, as markers of what I intended when I was ready. I've put a section on grad in now, which belongs perhaps a third of the way down the chapter, which should also indicate the lines I was thinking along. Carandol 06:11, 13 Apr 2004 (UTC)

Your last edit is better for boundaries, Dysprosia. The problem wasn't precisely excessive rigor, more lack of motivation. We should try and say why we're making these definitions, make it clear we're going somewhere with them and where we;ve plucked them from, or some readers will get lost at this point.

Along the same lines, perhaps you could say something about why the difference between open and closed sets matters in calculus, or why it's interesting in its own right: you, because I'm reluctant to edit a subsection you're currently active in. Carandol 07:32, 13 Apr 2004 (UTC)


 * Okay, I've tried to add more motivation to these points, but feel free to pitch in at any time, however. I'll probably continue to add more later on tonight, so you can add some stuff if you want :) Dysprosia 08:04, 13 Apr 2004 (UTC)

How common is the notation &part;x h? I've seen extensive use of hx, but not the former... Dysprosia 08:22, 13 Apr 2004 (UTC)

I've seen &part;x used, (e.g PDE: Erich Zauderer ex 2.1.2), but it isn't that common. Using numbers as subscripts is more common, especially with tensors. The letters are only seem to be used that way where space is short, the names of the axes matter, and hx wouldn't work, e.g (&part;x-c&part;t)(&part;x+c&part;t)h=0, which compactly exhibits the factorisation. Carandol 12:56, 13 Apr 2004 (UTC)~


 * That reminds me of the operator form... If so, maybe use the Heaviside "big-D"? I've been using that form in other pages... what do you think? Dysprosia 13:56, 13 Apr 2004 (UTC)

That's more common than &part;x, and needs less typing. I'd use it preferentially, except in fluid dynamics (where the total derivative D/Dt is too similiar). The meaning of &part;x is obvious, there's just no call for it. Carandol 14:21, 13 Apr 2004 (UTC)

How does a set being open/closed matter for doing multi-variable calculus at this level? Which theorems will we be quoting that require the distinction? Similarly for the formal defintions of neighbourhoods etc. On reflection, I feel these details are too much of a digression from the main topic of this chapter. They might fit better in the vector chapter, which is more about the structure of Rn. Either that, or a chapter specifically about such topics.

In 1-D, direction of tangent is arctan(f'), the extension to df(t)/dt is not immediately obvious - principally because the 1-D csae isn't being thought of as parametric.

Lower case L looks too similar to 1, especially when editing, best avoided Better to premultiply &nabla; by constants, not postmultiply, to avoid ambiguity

&nabla; as vector should be mentioned at first opportunity Carandol 20:39, 19 Apr 2004 (UTC)


 * There was something that uses open/closed sets, but I can't quite remember it now...I might remember it later, though ;)...Oh, remember now. It may be useful to mention neighbourhoods in discussing the implicit function theorem later, but certainly isn't necessary to.
 * The change in l/L is fine. There is that funky loopy l but I don't know how to generate that in TeX...Dysprosia 22:26, 19 Apr 2004 (UTC)

I'd then mention these things just before using them, putting them in the right context. It may slightly out of place, logically, but it should improve readability.Carandol 23:17, 19 Apr 2004 (UTC)

layout
I'm assuming that this wikibook isn't very active? If I'm mistaken, I would like to propose this general layout, if agreed by other editors:

Ch. 1:

Ch. 2:
 * Mostly general vectors. Discuss addition, subtraction, dot product and applications, cross product and applications.

Discuss Vector Valued functions in planes first (xy plane to be specific). Introduce projectile motion perhaps. Then move on to 3d VVF's. Here we can discuss Unit Tangent, Unit Normals, curvature etc.
 * Vector Valued Functions.

Ch. 3:
 * Partial Derivatives and Applications.

What partial derivatives are, what they're used for. Fairly self-explanatory. Then of course we need to discuss optimization and LaGrange Multipliers.

Ch. 4:
 * Integration with multiple variables.
 * Double/Triple Integrals; area/volume.

Ch. 5: damneinstien 00:32, 17 Dec 2005 (UTC)
 * Vector Fields.

Ch. 6:
 * line and surface integrals
 * Green's

Ch. 7: --Cronholm144 09:05, 18 June 2007 (UTC)
 * Curl and Div
 * Stoke's and Div theorem

Topology
Is topology really the best way to start a chapter on multi-variable? (assuming that this is supposed to be equivalent to a CAL III course) (answering my own question) I would say no, the more traditional place to start is partial derivation and the implications thereof. --Cronholm144 12:10, 19 June 2007 (UTC)

Boundary points
As someone who recently understood what a boundary point is (I amn studying calculus on my own as prepartion for next year math) I confess that definition you guys used is very hard to understand. Consider definition from the following site, both precise and not:

http://www.economics.utoronto.ca/osborne/MathTutorial/CLNF.HTM

Definitions from this site made me understand what a boundary points are much easier, even easier than it is written in the book I am using (Spivak's Calculus on manifolds). Tank you.

Errata in the Jacobian Matrix section
if you define $$\left(J_\mathbf{p} \mathbf{f}\right)_{ij} = \left.{\partial f_i \over \partial x_j}\right|_\mathbf{p}$$ then $$J \in R^{nxm}$$, i think this should be corrected. I will do this if no there aren't any objections. --Drunken sapo (talk) 14:24, 29 September 2009 (UTC)

Approach using vectors
I don't like how this page approaches it initially dealing with vectors rather than multi-variable scalar functions. It also fails to discuss how limits and continuity work in multivariate functions.--Jasper Deng (talk|meta) 02:41, 19 April 2013 (UTC)

Inverting differentials
With respect to the inverting differentials subsection of the section "Line integrals", the latter two assertions


 * If div u = V
 * $$\mathbf{u}(\mathbf{p}) =\int_{\mathbf{p}_0}^{\mathbf{p}} V \, d\mathbf{r} + \mathbf{w}(\mathbf{p})$$
 * where w is any function of zero divergence.


 * If curl u = v
 * $$\mathbf{u}(\mathbf{p}) =

\frac{1}{2}\int_{\mathbf{p}_0}^{\mathbf{p}} \mathbf{v} \times d\mathbf{r} + \mathbf{w}(\mathbf{p})$$
 * where w is any function of zero curl.

I do not believe are accurate. Let me start with the proposed formula for inverting the divergence:

It should be noted that vector field $$\mathbf{u}$$ exists no matter the scalar field $$V$$. The path integral $$\int_{\mathbf{p}_0}^{\mathbf{p}} V \, d\mathbf{r}$$ is dependent on the specific path used, which is not specified in this integral.

Now consider the spherically symmetric vector field: $$\mathbf{u}(\mathbf{p}) = |\mathbf{p}|^\alpha\frac{\mathbf{p}}{|\mathbf{p}|}$$ where $$\alpha \geq 1$$ is arbitrary. The divergence of this vector field is: $$V(\mathbf{p}) = (\nabla \bullet \mathbf{u})(\mathbf{p}) = (\alpha+2)|\mathbf{p}|^{\alpha-1}$$.

A line integral that starts at the origin gives $$\int_{0}^{\mathbf{p}} V(\mathbf{r}) \, d\mathbf{r} = \frac{\alpha+2}{\alpha}|\mathbf{p}|^\alpha\frac{\mathbf{p}}{|\mathbf{p}|} = \frac{\alpha+2}{\alpha}\mathbf{u}(\mathbf{p}) \neq \mathbf{u}(\mathbf{p})$$. The difference is not a vector field with 0 divergence. This contradicts the proposed formula.

I am certain that a similar argument can be made for the proposed formula for inverting the curl.

Math buff (discuss • contribs) 20:30, 9 February 2017 (UTC)

With respect to the proposed formula for inverting the curl, consider the cylindrical symmetric vector field: $$\mathbf{u}(\mathbf{p}) = (\rho(\mathbf{p}))^\alpha\hat{\phi}(\mathbf{p})$$ where $$\rho(\mathbf{p}) = |\mathbf{p} - (\mathbf{p} \cdot \mathbf{k})\mathbf{k}| = |\mathbf{k} \times \mathbf{p}|$$ is the distance of position $$\mathbf{p}$$ to the z-axis; $$\hat{\phi}(\mathbf{p}) = \frac{\mathbf{k} \times \mathbf{p}}{|\mathbf{k} \times \mathbf{p}|}$$ is the unit vector at position $$\mathbf{p}$$ that points in a counterclockwise direction around the z-axis; and $$\alpha \geq 1$$ is an arbitrary constant.

The curl of this vector field is $$\mathbf{v}(\mathbf{p}) = (\nabla \times \mathbf{u})(\mathbf{p}) = (\alpha + 1)(\rho(\mathbf{p}))^{\alpha-1}\mathbf{k}$$.

Let $$\mathbf{p}$$ be arbitrary. Integrating along a line from the origin to $$\mathbf{p}$$ gives:

$$\frac{1}{2}\int_{\mathbf{r} = \mathbf{0}}^{\mathbf{p}} \mathbf{v}(\mathbf{r}) \times d\mathbf{r} = \frac{1}{2}\int_{\rho = 0}^{\rho(\mathbf{p})} (\alpha + 1)\rho^{\alpha-1}\hat{\phi}(\mathbf{p})d\rho = \frac{\alpha+1}{2\alpha}(\rho(\mathbf{p}))^\alpha\hat{\phi}(\mathbf{p}) = \frac{\alpha+1}{2\alpha}\mathbf{u}(\mathbf{p})$$

For $$\alpha > 1$$, the formula $$\frac{1}{2}\int_{\mathbf{r} = \mathbf{0}}^{\mathbf{p}} \mathbf{v}(\mathbf{r}) \times d\mathbf{r}$$ fails to return $$\mathbf{u}(\mathbf{p})$$, and the difference is not a vector field with 0 curl. This contradicts the proposed formula.

Math buff (discuss • contribs) 21:30, 31 March 2017 (UTC)

I've now removed the inaccurate formulas. Math buff (discuss • contribs) 00:13, 1 April 2017 (UTC)

Splitting the page
I am currently working on this book and I stumbled upon this page that a Wikibookian suggested splitting it into multiple smaller pages. I think it contains too much information in an over-condensed way. I believe I do not have the ability to split the page on my own, so the following is my thoughts on how the page should be split if technically possible and someone actually can help split the page. I will try to complete each separated page with more content and simpler explanations so that other readers can understand multivariable calculus easier in the following year. Sorry if I caused too much trouble. I will probably ask for help in the reading room if other editors also agree with this page being too long or this discussion is left unattended for a period of time.CalciumTetraoxide (discuss • contribs) 03:30, 27 January 2021 (UTC)
 * Make Section 2 Curves and parameterizations into a separate page and rename it into "Calculus/Vector Functions"
 * Make Section 3 Limits and continuity into a separate page and name it "Calculus/Limits and Continuity"
 * Make Section 4.1 Properties into a separate page and rename it into "Calculus/Partial Derivatives"
 * Make Section 4.2 Rules of taking Jacobians into a separate page and rename it into "Calculus/Multivariable Chain Rule"
 * Make Section 4.4 Directional derivatives and 4.5 Gradient vectors into a separate page and rename it into "Calculus/Directional Derivatives and the Gradient Vector"
 * Make Sections 5.1 Riemann sums and 5.2 Iterated integrals into a separate page and rename it into "Calculus/Riemann sums and Iterated Integrals"
 * Delete Sections 4.3 Alternate notations, 4.6 Divergence, 4.7 Curl, 4.9 Second order differentials, 5.4 Parametric integrals, 5.5 Line integrals, 5.6 Surface and Volume Integrals, 5.7 Gauss's divergence theorem, and 5.8 Stokes' curl theorem because I found another page dedicated to or containing the same information.
 * Create pages with names "Calculus/Double Integrals", "Calculus/Triple Integrals", and "Calculus/Applications of Multiple Integrals".
 * The remaining content in the original page can be deleted.
 * I don't see why you can't split the page on your own. You can just copy-paste and create pages as appropriate. And for the record, I agree with you. However, all of these topics fall under multivariate calculus, so my suggestion would be to put them all under the Multivariate Calculus group. So, instead of "Calculus/Partial Derivatives", we'll have "Calculus/Multivariable calculus/Partial Derivatives". Leaderboard (discuss • contribs) 09:27, 27 January 2021 (UTC)
 * I thought that there is a special "split" action only accessible to administrators because I thought that if I just copy, paste, and delete huge chunks of content within a page in a short time, I might get misidentified as a troll. I'll start splitting the page. Thanks! CalciumTetraoxide (discuss • contribs) 17:06, 27 January 2021 (UTC)
 * I've already split the pages into several smaller pages as listed above. I intend to just delete this page because most of its content is preserved and will be expanded significantly to completion in the smaller pages. And after the splitting, the remaining content in this page cannot stand alone as a page. However, I am afraid that after deleting the page, the discussion page will be deleted as well. Is there any way to relocate the information in this discussion page to one of the smaller pages created or a simple copy and paste will suffice? CalciumTetraoxide (discuss • contribs) 18:07, 27 January 2021 (UTC)
 * I personally do not think the core Calculus/Multivariable calculus page should be deleted, because it can serve as the landing pages to the rest of the content (which are just subpages). We can just link the other pages that you've just split to instead. Leaderboard (discuss • contribs) 19:18, 27 January 2021 (UTC)