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For questions about differential and integral calculus with more than one independent variable.

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Downloadable MCQs on Mathematics

I am looking for multiple choice question (MCQ) based tests on some Mathematics' topics (details below), which could be downloaded in most preferably tex (LaTex) format or doc/docx format. Kindly ...
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3answers
292 views

Justifying the multi-variable chain rule to students

Suppose that $f(x,y,z) = x + 2xy^2 - yz$, and that $\gamma(u,v) = \langle uv, u\sin(v), u\cos(v)\rangle$. Use the chain rule to calculate $\partial(f \circ \gamma)/\partial u$. This is an exercise ...
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Recommend a vector calculus textbook/resource with an algebraic geometry flavor

Is there a resource or textbook that presents the basics of vector calculus, specifically the gradient, directional derivatives, curves and surfaces, and extrema, from a more algebraic geometry ...
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4answers
319 views

Who actually uses $\mathbf i$, $\mathbf j$, $\mathbf k$ for the standard unit vectors?

I am wondering which research communities use the notation $\mathbf i$, $\mathbf j$, $\mathbf k$ for the three-dimensional unit vectors. The calculus textbook I have to use (Stewart) uses that ...
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3answers
158 views

Resources on solving systems of polynomial equations in multivariable calculus setting

Whenever I teach multivariable calculus I find students really struggle with both finding critical points and the method of Lagrange multipliers. I think that the reason is the same: solving systems ...
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3answers
252 views

Polymorphic functions in vector calculus

While teaching multi-variable calculus for the first time in a while, I came across a tricky notational point in our textbook (Thomas' calculus - I'm not sure how widespread this notation is). When $\...
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2answers
296 views

Is “hat notation” for unit vectors commonly used in mathematics?

As an undergraduate, I clearly remember learning and using "hat notation" to describe unit vectors. That is, if $\vec{v}$ is any vector (in 2 or 3 dimensions) then $\hat{v}$ denotes the unit vector ...
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4answers
278 views

A question about Vector Analysis problems

Why is it difficult to find really challenging vector analysis problems (problems about Green's, Stokes' and Gauss' theorems in a Calculus 3 course) in Calculus books? Most of the problems are ...
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4answers
253 views

How can we focus students on the various data types in multivariable calculus?

To try to find out if students knew what the gradient was, after the computational questions, I asked the following question on an exam: Let $f(x, y) = 5 - x - y$. Why doesn't it make sense to find ...
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3answers
287 views

The use of software to formulate problems in multivariable calculus

I know it's common for high school teachers to use software (such as Geogebra) to formulate geometry problems for their students, so I wonder: Do professors of multivariable calculus use softwares (...
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1answer
62 views

Multivariable limit problem [closed]

Im triying to explain this delta-epsilon problem, but I didnt find a way to attack effectively this rigorous demonstration I actually i tried a lot of inequalities (Cauchy-Schwarz etc), but nothing ...
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1answer
134 views

Vector Algebra Text [closed]

Recent developments in Geometric Algebra have extended vector algebra to include the outer product (wedge product) and bivectors. Is there a Vector Algebra text (preferably at the advanced high ...
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3answers
132 views

How to motivate the surface element

$\newcommand{\RR}{\mathbb{R}} \newcommand{\dd}{\mathrm{d}}$ In teaching multivariable integration on sub-manifolds in $\RR^n$, i.e. integrals over $k$-dimensional surfaces $M\subset \RR^n$ you define ...
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3answers
398 views

Differential forms in mechanics?

I teach mechanics (including large deformation and flow of continua) to mechanical engineering students and have a continuing mission to drag the teaching of mechanics into the 20th century (I'll ...
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5answers
1k views

Good examples of Lagrange multiplier problems

I've noticed that most Lagrange multiplier problems I've seen can be solved with other methods. Often the method of Lagrange multipliers takes longer than the other available methods. I don't like ...
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3answers
643 views

What is an efficient way of drawing surfaces in multivariable calculus?

I've noticed that some surfaces are difficult to draw in multivariable calculus. For instance, I always have trouble with hyperbolic paraboloids. What is an efficient way to draw the following ...
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1answer
324 views

Textbook for multivariable calculus with interesting modern applications

A colleague of mine in a math department at another university is looking for a textbook on multivariable calculus that discusses applications of higher-dimensional integrals that feel contemporary ...
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0answers
148 views

How is cooperative learning being used in vector calculus, and what are the origins of this work?

I'm doing some research about cooperative learning in vector calculus. It seems like what cooperative learning in calculus is referred to varies over time. In 1987, there was an MAA book, Calculus ...
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2answers
1k views

Advanced Calculus vs. Analysis for a first proof-based course

Question: Why was advanced calculus removed as the first proof-based course in favor of real analysis in most curriculums? I regularly see in advanced calculus books either that: its purpose is, ...
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2answers
394 views

Open Source Math Software in Multivariate Calculus

I am teaching calculus III in the upcoming semester. The course is fairly standard, just a brief run-down: Test 1: covers vectors and coordinate systems as well as the calculus of space curves ...
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4answers
165 views

Multivariable limits

Multivariable limits are harder than their one-variable counterparts, and textbooks examples usually focus on limits that don't exist when approaching from different straight lines. This gives the ...
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1answer
175 views

Surfaces and volumes for vector calculus

We'll reach vector calculus very soon and the following problem presents itself: how can I help students distinguish curves, surfaces and volumes as separated entities? I've seen they hold the ...
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3answers
321 views

Hands-on demonstration ideas for multivariate calculus

In teaching Calculus III geometry plays a very important role. It is crucial that students get a good sense of how to visualize curves, surfaces, coordinate axis, frames to curves, vector fields and ...
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1answer
7k views

Applications of Vector Calculus to Economics/Finance

I will be teaching a course focusing on multivariable integration soon, for the millionth time. The most difficult topic in such a course is certainly Vector Calculus, by which I mean line and surface ...
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3answers
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What is a good physical example of Stokes' Theorem?

I find it useful to give physical examples of theorems, especially in vector calculus - for example $\nabla f$ being the direction of maximum ascent on a surface $f$. What is a good example for ...