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47 votes

What do you do in order to drag out lectures?

What you are describing is so far outside of my, and I suspect most educators, experience that it appears to be literally incredible. The very strongest Universities in the country, with some of the ...
Steven Gubkin's user avatar
21 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 ...
Michał Miśkiewicz's user avatar
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}...
Adam's user avatar
  • 5,733
16 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 ...
Thierry's user avatar
  • 1,527
16 votes

Any examples of calculus sequence that deemphasizes calculation tricks?

As you undertake the journey, you should read about the Calculus Reform movement of the 1990s. A unifying idea of the efforts was to increase conceptual understanding by using symbolic computational ...
user52817's user avatar
  • 11k
15 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 ...
mweiss's user avatar
  • 17.4k
15 votes
Accepted

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

I was the primary author of the M2O2C2 content. So glad that you enjoyed the course! My friend Jim Fowler wrote the backend code (called Ximera). All of the files for M2O2C2 were in an earlier, not ...
Steven Gubkin's user avatar
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 ...
James S. Cook's user avatar
12 votes
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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 ...
Benoît Kloeckner's user avatar
12 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 ...
Steven Gubkin's user avatar
11 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 $x,y,t$. ...
Michael Bächtold's user avatar
11 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 ...
jezza's user avatar
  • 219
10 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 ...
James S. Cook's user avatar
10 votes
Accepted

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

There is still a simple geometric meaning for polar coordinates when we allow for negative values of $r$. (It's how I was taught polar coordinates.) Given an ordered pair $(r, \theta)$, Start at the ...
Justin Hancock's user avatar
9 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 ...
guest's user avatar
  • 304
8 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 ...
J.G.'s user avatar
  • 531
8 votes

What do you do in order to drag out lectures?

What do I do? First, explain the product rule. Let's say 3 minutes on that. Then write a random product of 2 functions and differentiate it. 2 more minutes. Then write a product of 2 functions and ask ...
fedja's user avatar
  • 3,909
8 votes

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

It would be nice if we distinguished between "polar coordinates" and "polar parameterizations". Let $\Omega \subset \mathbb{R}^2$ be an open set which does not contain any loop ...
Steven Gubkin's user avatar
7 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 $\mathbf{r}=\mathbf{r}(s)=\mathbf{r}(t)$ ...
Michael Bächtold's user avatar
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 ...
Jonathan's user avatar
  • 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 ...
user52817's user avatar
  • 11k
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 ...
James S. Cook's user avatar
7 votes

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

I agree with the OP: the determinant is only a mnemonic device, but (a) students may not know how to evaluate a determinant, so it's useless as a mnemonic, and (b) it is not a "legal" ...
Joseph O'Rourke's user avatar
6 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 ...
user52817's user avatar
  • 11k
6 votes
Accepted

Justifying the multi-variable chain rule to students

You can tell them the values of some of the partial derivatives without giving them the entire function. This prevents the students from creating the composite function, since they don't even know one ...
Chris Cunningham's user avatar
6 votes

Applications of Vector Calculus to Economics/Finance

As far as I know, vector calculus is applied by financial analysts in exotic derivatives pricing. The Black-Scholes Model is actually a special form of Schrödinger equation. Thus, if you want to ...
user10611's user avatar
6 votes
Accepted

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

For 2nd year engineering students, you should not be wasting time trying to prove the expansion, except perhaps giving a heuristic argument for the quadratic approximation in the case of two variables....
Dave L Renfro's user avatar
6 votes
Accepted

Low-tech ways of visualizing multivariable and vector calculus

I don't know if this is what you're looking for exactly, but I've run forms of this activity a few times when introducing partial/directional derivatives. Supply the class with materials: square grids ...
Nick C's user avatar
  • 9,639

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