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Just to provide some background, I have been given the chance to teach a section of the honors Calc 3 class at my school next semester (well technically, I'd be overseen by a professor but he told me I could teach it how I wanted as long as I didn't compromise the integrity of the class) and if I do it, I really want the students to understand the material and enjoy it. My experience with Stewart was very negative and I'm leaning towards writing my own lecture notes so that I won't be bound by what I see as a rather outdated approach to multivariable calculus. One concept that I think causes a ton of confusion is that of the "cross product" and "normality", and I think, with sufficiently motivated students, this can be avoided by introducing the wedge product early on and building from there. Just as a disclaimer, this topic is very far away from where I am currently working so consider this a rough sketch.

Just as an example of why I think introducing this concept is helpful, one can write a plane as a vector equation given by (normal form) $\mathbf{n} \cdot (\mathbf{x}-\mathbf{x_{0}})=0$ but this is really just a multivector equation masquerading as a vector equation since the cross product in $\mathbb{R}^{3}$ is really a bivector (where $\mathbf{u},\mathbf{v} \in \mathbb{R}^{3}$) $\mathbf{n}=\mathbf{u} \times \mathbf{v} \cong \mathbf{u} \wedge \mathbf{v}$. Then by the correspondance of the "scalar triple product" and trivectors $\mathbf{n} \cdot (\mathbf{x}-\mathbf{x_{0}}) \cong \mathbf{u} \wedge \mathbf{v} \wedge (\mathbf{x}-\mathbf{x}_{0})=0$ since (pseudo-)vectors and bivectors correspond via Hodge duality and $\Lambda^{3}(\mathbb{R}^{3})$ is the highest grade of the exterior algebra for $\mathbb{R}^{3}$, this equation gives us a plane in $\mathbb{R}^{3}$.

Geometrically, defining a plane as $\mathbf{u} \wedge \mathbf{v} \wedge (\mathbf{x}-\mathbf{x}_{0})=0$ is much more intuitive to me since the essence of a plane can be characterized by the fact that it has no volume. I remember being very frustrated in my introductory classes (especially Calc 3) where we were given all of these geometric ideas and then had completely unintuitive formulas like $4x-4y+z-7=0$ shoved down our throats, whereas a formula like $(4x-4y+z-7) \mathbf{e}_{1} \wedge \mathbf{e}_{2} \wedge \mathbf{e}_{3}=0$ communicates the geometric intuition behind this formula.

One advantage to this approach is that it easily generalizes to $\mathbb{R}^{n}$ and it avoids unnecessary dependence on a particular choice of coordinates. The disadvantage is that for those students who do not go on to pursue mathematics further will likely only encounter the cross product in physics/engineering classes. On the other hand, they might gain insight into the pitfalls of this concept, especially since it seems that very few students understand why we differentiate between left and right handed systems.

Any thoughts/experience with introducing these concepts early on? They don't seem to be so advanced that a good student would be completely lost but I could also understand that the average student might not pick up on the motivation behind it.

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  • $\begingroup$ If you do adopt this approach, I would be interested to read your thoughts on how you think it went afterwards. $\endgroup$ – ziggurism Feb 12 '17 at 19:15
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    $\begingroup$ I strongly disagree with your claim that $4x - 4y + z - 7 = 0$ is somehow "totally unintuitive" compared to $(4x-4y+z-7) \mathbf{e}_{1} \wedge \mathbf{e}_{2} \wedge \mathbf{e}_{3}=0$. In particular, the "dot product" presentation of a plane via its normal vector is pretty neat. $\endgroup$ – Omnomnomnom Feb 12 '17 at 19:15
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    $\begingroup$ @Omnomnomnom What I was trying to communicate is that a formula like $4x-4y+z-7=0$ is unintuitive in the sense that the expression simply doesn't communicate the essence of a plane. On the other hand when I see something like $e_{1} \wedge e_{2} \wedge e_{3} =0$, that immediately registers in my mind as being representative of a plane. That being said, it is largely a matter of taste but the bigger issue to me is that if a student is going to spend a semester learning something, it should generalize easily rather than being dependent on the specific properties of $\mathbb{R}^{3}$. $\endgroup$ – Wavelet Feb 12 '17 at 19:35
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    $\begingroup$ @Wavelet but from the perspective of many students, there isn't anything other than $\Bbb R^3$. I think in order to have an appreciation for why generalization is a good thing to do, a student needs to know how things are supposed to work in $\Bbb R^3$. Otherwise, you're asking them to bear with you through a lot of formalisms that, while intuitive to us, are not intuitive coming out of high school. $\endgroup$ – Omnomnomnom Feb 12 '17 at 19:39
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I'd say that it's not a good idea to introduce exterior algebras as a primary way of thinking about things. First of all, anyone in a calc-3 class will likely do (or has already done) some kind of physics/engineering, so they already understand the "unnatural" machinery of a dot-product. Second, it takes a lot more effort to learn all the formalisms of a wedge product and exterior derivatives than it does to add the cross-product to your toolbox.

Just because different ideas turn out to be examples of the same thing, doesn't mean that we should start out teaching them as such. It's very easy, having come to the point where you're familiar with the ideas behind exterior algebras and $\Bbb R^n$ generalizations, to say that life would have been so much easier if your teacher had just told you all the things you know now. However, the straight line from single variate calculus to exterior algebras is long and unintuitive. If you just jump right into the formalisms necessary for $\Bbb R^n$ without first spending time focusing on $\Bbb R^2$ and $\Bbb R^3$, then you don't give students the chance to leverage their existing intuitions.

I like how Colley handles this in her "Vector Calculus": at the end of the text, after all the $\Bbb R^2,\Bbb R^3$ Stokes' theorems, she has a chapter on vector analysis in higher dimensions in which she introduces exterior algebras and recasts the Stokes' and divergence theorems as instances of the generalized Stokes' theorem. When I had a class on this text, the instructor spent the last week and change covering this last chapter, but didn't test the class on any of it.

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  • $\begingroup$ These are all good points, my current inclination is to introduce these concepts gently and let the students begin to digest them but not stress them out by testing them on things I wouldn't have been expected to know back then. I do think that would be a barrier to learning and probably disastrous. $\endgroup$ – Wavelet Feb 12 '17 at 19:39
  • $\begingroup$ @Wavelet that sounds reasonable. I would suspect, however, that many students are going to ignore what you say about the $\wedge$ properties and jump right to "okay, okay, but what's the formula? How do I find the answer?" Note also that, as a teacher, one has a tendency to underestimate the time it takes to absorb things. $\endgroup$ – Omnomnomnom Feb 12 '17 at 19:44
  • $\begingroup$ @Wavelet and I'm sure you'll have a student ask "but why would I throw around all these $\wedge$s and stuff if I can just use this neat cross-product formula that I learned for physics?", and they will not be swayed. After all, $\Bbb R^n$ is only really a thing for $n \leq 3$, right? $\endgroup$ – Omnomnomnom Feb 12 '17 at 19:47

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