Ordinary calculus provides tools for understanding rates of change (via derivatives), total quantities (via integration), and the total change (via the fundamental theorem of calculus). Exterior calculus generalizes these ideas to \(n\)-dimensional quantities that arise throughout geometry and physics. Our first lecture on exterior calculus studies the *exterior derivative*, which describes the rate of change of a differential form, and (together with the Hodge star) generalizes the gradient, divergence, and curl operators from standard vector calculus.

## 3 thoughts on “Slides—Exterior Calculus I: Differentiation”

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Possible typos:

Slide 14: In coordinates, $ d\phi(X) := \cdots $ should be $ d\phi := \cdots $? We are just defining the form and not evaluating it on the vector field.

Slide 31 and 32: it seems div X is missing a # on the outside.

How does Hessian come into the picture of exterior derivative?

So far I have a vague sense that the information contained in a Hessian matrix of a scalar function is split into

the curl and divergence of gradient. The laplacian described in the slides seems to be the diagonal of the Hessian, whereas the curl just plays with the off-diagonal entries.

But the picture is still quite muddy. I searched on wikipedia but the descriptions are not beginner friendly enough to parse.

Differential forms are antisymmetric tensors whereas the Hessian is symmetric. So you can’t express the Hessian of a function via the exterior derivative and Hodge star, which take differential forms to differential forms. You can however express it as the covariant derivative of the exterior derivative, i.e., for a scalar function u the Hessian is

\(\text{Hess}(u) = \nabla du,\)

where \(\nabla\) denotes the covariant derivative. To define covariant differentiation we will have to discuss concepts like parallel transport, which will come later in the course. One thing to say here is that, like the gradient (and unlike the exterior derivative) the covariant derivative depends on the choice of inner product. Which provides another way of understanding why the Hessian cannot be expressed purely in terms of differential forms (which are completely topological).