46
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 ...
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20
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 ...
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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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 ...
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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 ...
- 8,873
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 ...
- 23k
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
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 ...
- 17k
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 ...
- 10.4k
12
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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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 ...
- 23k
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 $x,y,t$. ...
- 1,630
10
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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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 ...
- 28.7k
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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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 ...
- 10.4k
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 ...
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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 $\mathbf{r}=\mathbf{r}(s)=\mathbf{r}(t)$ ...
- 1,630
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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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 ...
- 8,550
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 ...
- 10.4k
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" ...
- 28.7k
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 ...
- 521
6
votes
Good examples of Lagrange multiplier problems
A question with two constraints might make the method seem preferable to finding a parameterization (which I assume is the "easier" technique you refer too in the OP).
For example, maximizing $f(x,y,...
- 23k
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 ...
- 8,550
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 ...
- 20.2k
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....
- 5,532
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 ...
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