Feasibility of Introducing Exterior Algebra in Honors Calc 3 Class? 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very negative and I'm leaning towards writing my own lecture notes so that I won't be bound by what I see as a rather outdated approach to multivariable calculus.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. 
 A: 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.
