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Questions tagged [vector-calculus]

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

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3 answers
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how to parametrize these bands related with spinors? [closed]

In a recently viewed educational video focused on the concept of spinors (available at this link:https://youtu.be/b7OIbMCIfs4?si=5ZZLxdGotxAj6YwP ), an intriguing visual representation caught my ...
Humberto José Bortolossi's user avatar
7 votes
5 answers
273 views

Why do some (pre-) calculus text allow $r<0$ in polar coordinates?

Form this question, I was surprised to learn that it is common for calculus textbooks in the US to allow $r<0$ when discussing polar coordinates. This answer by Dan Fox summarizes some mathematical ...
Kostya_I's user avatar
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8 votes
4 answers
3k views

Any examples of calculus sequence that deemphasizes calculation tricks?

I'm considering creating a series of classes that explore deeper ideas in calculus without overemphasizing the various computational tricks used in integration and differentiation. My vision is ...
Bilbo's user avatar
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1 vote
1 answer
117 views

Best Free Direction Field Plotter?

Can you recommend one for a first or second year calculus course? Ideally the website that can plot direction fields: is free is 100% WYSIWYG (does not require any coding or markup or anything of the ...
BravoMath's user avatar
  • 564
5 votes
8 answers
9k views

What do you do in order to drag out lectures?

I posted earlier about how I was surprised that a typical Calculus 1 course that meets 3-4 hours each week for 15 weeks only barely manages to reach the fundamental theorem by the end of the course. ...
mrwillparker's user avatar
11 votes
1 answer
1k views

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 ...
Emil N's user avatar
  • 113
6 votes
1 answer
689 views

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 ...
ruferd's user avatar
  • 2,099
7 votes
1 answer
135 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). ...
TomKern's user avatar
  • 4,435
6 votes
1 answer
370 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 ...
1Teaches2Learn's user avatar
9 votes
3 answers
2k 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 ...
Maesumi's user avatar
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12 votes
2 answers
681 views

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 ...
user avatar
25 votes
11 answers
6k 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 ...
user avatar
11 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 ...
Chris Cunningham's user avatar
6 votes
5 answers
5k views

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}\...
user avatar
4 votes
1 answer
206 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'...
user avatar
5 votes
2 answers
798 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 ...
user avatar
15 votes
1 answer
500 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 ...
user avatar
2 votes
1 answer
166 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 ...
AlfredG's user avatar
  • 21
3 votes
1 answer
163 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 ...
ppaulojr's user avatar
  • 141
10 votes
4 answers
474 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; ...
academic's user avatar
  • 201
5 votes
2 answers
313 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), ...
Behnam Esmayli's user avatar
8 votes
4 answers
545 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 ...
user avatar
5 votes
2 answers
309 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 ...
user avatar
1 vote
4 answers
316 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 ...
user avatar
7 votes
2 answers
302 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 ...
user avatar
6 votes
0 answers
89 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 ...
Sinister Cutlass's user avatar
3 votes
0 answers
716 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 ...
Sophia's user avatar
  • 31
2 votes
1 answer
529 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 ...
MathNovice's user avatar
6 votes
3 answers
437 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 ...
Mike Pierce's user avatar
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10 votes
1 answer
265 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 ...
Mike Pierce's user avatar
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6 votes
4 answers
764 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 ...
shuhalo's user avatar
  • 443
8 votes
3 answers
242 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 ...
Nate Bade's user avatar
  • 1,941
22 votes
3 answers
704 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 $\...
Henry Towsner's user avatar
7 votes
2 answers
5k 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 ...
mweiss's user avatar
  • 17.4k
5 votes
4 answers
620 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 ...
danilocn94's user avatar
17 votes
5 answers
453 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 $\...
Chris Cunningham's user avatar
7 votes
3 answers
602 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 (...
danilocn94's user avatar
0 votes
1 answer
89 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 ...
Wilfred V's user avatar
1 vote
1 answer
176 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 ...
Allyn Shell's user avatar
5 votes
3 answers
166 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 ...
Dirk's user avatar
  • 2,991
18 votes
3 answers
1k 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 ...
rdt2's user avatar
  • 431
19 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 ...
Seth's user avatar
  • 291
19 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 ...
Brian Rushton's user avatar
18 votes
1 answer
669 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 ...
KCd's user avatar
  • 3,536
6 votes
0 answers
208 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 ...
James S.'s user avatar
  • 1,023
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, ...
Mark Fantini's user avatar
  • 3,040
10 votes
2 answers
653 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 ...
James S. Cook's user avatar
11 votes
4 answers
691 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 ...
Mark Fantini's user avatar
  • 3,040
8 votes
1 answer
251 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 ...
Mark Fantini's user avatar
  • 3,040
12 votes
3 answers
514 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 ...
James S. Cook's user avatar