17 votes

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

What's wrong with this?: $$ \frac{df}{dt} = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt} + \frac{\partial f}{\partial t} \frac{dt}{dt}$$ with $\frac{dt}{dt}...
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  • 4,656
17 votes

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

My answer is probably not very useful when teaching in high school. I'll just mention here a few reasons why this definition is in fact a good one, and why it's a good idea to teach this formula at a ...
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15 votes

What is a good physical example of Stokes' Theorem?

I don't know if this is what you are looking for, but for Green's theorem, when you discuss how you can change the integral $\int 1 dA$ into the line integral $\int x dy - y dx$, you can discuss ...
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15 votes

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

Below are instructions for drawing various quadric surfaces. They're not sketches but drawings designed to look (sort of) pretty and be easy to draw. You can see all of them in one YouTube playlist if ...
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14 votes

Good examples of Lagrange multiplier problems

A place that Lagrange Multipliers comes up is in the proof of the real spectral theorem. Namely, let $A$ be a symmetric matrix. Define $f: \mathbb{R}^n \to \mathbb{R}$ by $f(v) = v^\top A v$. If ...
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14 votes

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

I think you're correct, strictly speaking, but overreacting. I just checked two calculus books, Rogawski and Strang. Rogawski defines the cross product as the "symbolic" determinant (his ...
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  • 1,234
13 votes
Accepted

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

I might seem picky, but I would first refrain from saying that $\nabla f$ is a vector. It is a vector field. This might be considered a common abuse of vocabulary, but using it amounts to assuming ...
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13 votes
Accepted

Polymorphic functions in vector calculus

By coincidence, I'm teaching multi-variable Calculus for the first time this semester, and have given some thought to how to handle this precise issue. This seems to me to be closely related to an ...
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  • 16.1k
13 votes

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

Sure, give up on the mixed-object determinant formula. Instead use tensor arithmetic. Let $\epsilon_{ijk}$ be the completely antisymmetric symbol. Also, while we're at it, let's give up on ...
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12 votes

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

This is somewhat of a hypothesis rather than a definitive answer, but one reason why vector calculus may no longer be the first proof based course at many colleges is because vector calculus involves ...
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11 votes
Accepted

What is a good physical example of Stokes' Theorem?

I always think of $\int_D \nabla \times v = \int_{\partial D} v$ in terms of water flow. You have a bunch of water flowing around: It's velocity at a given position $(x,y)$ is given by the vector ...
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11 votes
Accepted

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

Do you ever introduce the Jacobian matrix, or the derivative of a function at a point as a linear map? This clarifies everything. If not, and you must restrict yourself to "traditional ...
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10 votes

Multivariable limits

A problem I used to assign on take-home tests back when I was teaching gifted high school students (late 1990s) might be of interest. Show that $f(x,y) =y^{x}$ has the same limit along any 1st ...
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10 votes

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

I guess you are aware of it, but since you don't say so explicitly: the chain rule on the top of your post does not apply to your example, since that $f$ is not a function of $x,y$ but of $x,y,t$. ...
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8 votes
Accepted

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

(This is not bounty-worthy; just consider these illustrated comments.) Smith & Minton's textbook Calculus emphasizes drawing curves in each coordinate plane, and a few cross-sections (echoing ...
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8 votes

Hands-on demonstration ideas for multivariate calculus

Here's a crafty but perhaps crazy way to convey some ideas to a class. Have all the students gather on the football field (or another field) in a grid on a mildly windy day. Each student carries a ...
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8 votes

Polymorphic functions in vector calculus

I don't have much to add to mweiss' nice answer in terms of teaching suggestions. But I would like to add my own point of view on what the abuse of notation $\mathbf{r}=\mathbf{r}(s)=\mathbf{r}(t)$ ...
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8 votes

Examples of real-life vector fields for vector calculus

Air speed/direction on a weather map) is a very intuitive one. There's also other fluid velocity (and flux) vector fields in various chemE, mechE, and nukeE applications. I personally think the air ...
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  • 284
8 votes

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

In all scenarios, unless the school judges the success of the course by some external metric it is very difficult to objectively compare different methods of instruction. The plain fact is that ...
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7 votes

Surfaces and volumes for vector calculus

I think the situation is clearer in $\mathbb{R}^2$ than in $\mathbb{R}^3$. Draw an open curve and ask students how much area it enclosed. Presumably they will recognize that the question has no real ...
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  • 16.1k
7 votes

What is a good physical example of Stokes' Theorem?

The divergence theorem or Gauß’s theorem can be nicely linked to reality: In terms of hydrodynamic flows you could start with the following statement: The total signed water flow through the ...
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  • 2,366
7 votes
Accepted

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

Search arxiv.org for "unit vectors i, j, k" and you will find examples of research papers where the notation is used. Many are from physics communities investigating phenomena in three-dimensional ...
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  • 7,575
7 votes

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

I'm not sure this a research setting, but the cross product relations $i \times j = k$, $j \times k = i$, etc are nice for illustrating the connection between quaternions (where $ij=k, jk=i$, etc) and ...
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  • 179
7 votes

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

One idea that comes to mind is the fact that a connected graph has a Eulerian circuit if and only if every vertex has even degree. Another idea that comes to mind is just the Fundamental Theorem of ...
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  • 7,575
7 votes

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

If we study the tangent line to $y=f(x)$ at $(a,f(a))$ as given by: $$ y = L^a_f(x) = f(a)+f'(a)(x-a) $$ then you can verify that a little input interval $(a-h,a+h)$ is transferred to the output ...
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7 votes

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

I had a lecturer who thought the same as you, so suggested we just learnt the definition of the cross product (which he derived along the same lines as James S Cook's answer). Nobody liked his ...
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  • 79
7 votes

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

this makes no sense But it does. A determinant exists for any $n\times n$ matrix for which the Leibniz formula$$\det A:=\sum_{\sigma\in S_n}\epsilon_\sigma\prod_{i=1}^nA_{i\sigma_i}$$is fully ...
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  • 499

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