Questions tagged [vector-calculus]

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

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Why Massively Multivariate open online calculus class (M2O2C2) on Coursera was discontinued?

I know that MOOCs were generally unsuccessful. However, I felt that M2O2C2 in Coursera was a great (at least my favorite) course and it's a pity it was removed. Does anyone have any info - will it ...
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6 votes
1 answer
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Multivariable Calculus Project Ideas

Next semester, I am going to teach a small section of advanced high school students a class of Multivariable Calculus (it's about 3-4 students that have completed AP Calculus BC). Multivariable ...
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  • 1,992
7 votes
1 answer
122 views

Resources for Teaching Parameterization of Curves/Surfaces

In classes like Calc 3 or Computer Graphics, I want my students to be comfortable describing common curves and surfaces parametrically (such as lines, triangles, circles, or surfaces of revolution). ...
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6 votes
1 answer
196 views

More advanced (free) alternatives to Geogebra and Math3D?

I teach vector calculus. I love both Math3D and Geogebra. But I have reached a limit in terms of what these programs can do. Some examples of features that I wish Math3D had: Draw vector fields with ...
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7 votes
3 answers
1k views

Is there a study that compares 8-week vs 16-week math classes?

I see a push toward having undergraduate curriculums built around 8-week classes. This is mostly in the online education in the USA. Recently I have seen a number of these in sophomore or junior-level ...
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11 votes
2 answers
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Why are so many online sources "wrong" about directional derivatives?

I noticed many seemingly reputable online sources have "incorrect" description of directional derivatives for real-valued functions in several variables. Here, by "incorrect" I ...
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19 votes
11 answers
5k views

Why do we still teach the determinant formula for cross product? And is it as bad as I think it is?

The cross product is an important vector operation in in any serious multivariable calculus course. In most textbooks that I'm aware of, right after the definition, we always introduce the ...
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10 votes
6 answers
2k views

What is the right notation to use in multivariable chain rules?

The following "chain rule" is in my multivariable calculus course: If $f$ depends on $x$ and $y$, but $x$ and $y$ depend on $t$, then $\frac{df}{dt} = \frac{\partial f}{\partial x} \frac{d ...
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5 votes
4 answers
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Examples of real-life vector fields for vector calculus

My two main ones are Electrostatic force field $\mathbf{E}\left(\mathbf{r}\right)=\frac{Q}{4\pi\epsilon_0 \left|\left|\mathbf{r}\right|\right|^3}\mathbf{r}$ and Gravitational force field, $\mathbf{F}\...
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4 votes
1 answer
183 views

Low-tech ways of visualizing multivariable and vector calculus

One way, which is the most obvious, is do sketches of 3d shapes that tend to be the ones that we can all draw (like rectangle, cone, cylinder, sphere, etc.) Another way is by analogy so even if we can'...
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5 votes
2 answers
288 views

What is a good way to teach Taylor expansion of multi-variable calculus?

I found teaching Taylor expansion for multivariable functions rather challenging. It is a bit complicated to prove and to to compute. So what happened to me last year was that my students simply ...
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15 votes
1 answer
425 views

Analogies for grad, div, curl, and Laplacian?

I want to try making some calculation-less questions about vector calculus identities that are solely based upon picture diagrams of vector fields, or fields that could be sketched out by hand. The ...
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2 votes
1 answer
146 views

Advanced textbook for vector fields [closed]

I am currently reading Spivak Calculus on Manifolds and Munkres Analysis on Manifolds. I am looking for a more advanced text, especially on vector fields as they relate to the great conserved fields ...
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3 votes
1 answer
135 views

Refreshing math knowledge

How do I refresh advanced math I learned at a graduate level? I once was able to do the full solution of a particle in a parabolic well and other advanced math, however 20 years later I'm struggling ...
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  • 141
10 votes
4 answers
420 views

A fun, one-day topic for a vector analysis course

I am currently teaching a course in "vector analysis", following Colley's book. So far we have reviewed multivariable calculus (a prereq for the course), and discussed: the derivative in general; ...
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  • 201
5 votes
2 answers
223 views

Undergraduate Vector Calculus Notation Mess

Question 1: What are your arguments in favor of the big array of different notations used in the context of undergraduate vector calculus: line integrals, surface integrals (of scalars and fields), ...
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8 votes
4 answers
532 views

Easy examples of correspondence between global and local, as preparation for Gauss's theorem and Stokes's theorem

I'm teaching freshman electricity and magnetism this semester, and as usual in this type of course, I will need to teach my students a lot of vector calculus before they see it in a math course. The ...
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5 votes
2 answers
211 views

In a typical 3rd-semester multivariate calculus course in the US, what kind of area integrals do students actually learn to do?

I teach mostly physics and a little math at a California community college. I've never taught the multivariate calculus course, but I have taught the electricity and magnetism course for which the ...
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1 vote
4 answers
251 views

Analogy for nested loops/integrals

In teaching students how to do iterated integrals, I would like to find some analogy using a finite task nested inside another finite task. It would be especially nice if it satisfied the following ...
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7 votes
2 answers
276 views

Vector calculus texts that are free-as-in-speech?

I'm looking around for a text that covers vector calculus and multivariable calculus, and that is also "free as in speech," not just "free as in beer." In other words, I'm looking for texts that are ...
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6 votes
0 answers
84 views

Long-form, multi-step, skills-integrating applied mathematics problems in calculus I, II, III

When recently teaching Calculus II to college students, I instructed my students to read and be ready to work through the first 8 or so questions of James Walsh's climate modeling differential ...
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3 votes
0 answers
430 views

A proof based Multivariable Calculus and Linear Algebra

May I know how can I teach a proof-based Multivariable Calculus and linear algebra as a single course? While there are quite a few known books in the field such as: 1) Vector Calculus, Linear Algebra ...
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2 votes
1 answer
468 views

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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6 votes
3 answers
382 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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10 votes
1 answer
222 views

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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6 votes
4 answers
603 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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  • 433
7 votes
3 answers
221 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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  • 1,881
21 votes
3 answers
573 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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7 votes
2 answers
3k 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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5 votes
4 answers
496 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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17 votes
5 answers
403 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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7 votes
3 answers
558 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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0 votes
1 answer
81 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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1 vote
1 answer
168 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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5 votes
3 answers
152 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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17 votes
3 answers
870 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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  • 421
18 votes
5 answers
3k 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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  • 281
18 votes
3 answers
1k 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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17 votes
1 answer
557 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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  • 1,988
6 votes
0 answers
202 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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  • 1,016
11 votes
2 answers
2k 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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  • 2,879
10 votes
2 answers
583 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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7 votes
4 answers
446 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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  • 2,879
8 votes
1 answer
234 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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  • 2,879
12 votes
3 answers
432 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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13 votes
3 answers
11k 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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14 votes
3 answers
4k views

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 ...
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