# Pen & Paper Homework I

Due Friday, November 08.

This assignment covers topics from exterior calculus and surface curvatures. For exercises 2- 5, a careful read of the extra course notes, especially section 2.0.3 The Exterior Algebra, will be necessary. Feel free to ask any questions by writing a blog post!

1. (4 pts. ) Show that the mean curvature is indeed the mean curvature over all directions in the tangent space
$H = \frac{1}{2\pi}\int_0^{2\pi}\kappa^N(X^\theta)d\theta$
where $$\theta \in [0,2\pi]$$ parameterizes the unit circle of directions $$X$$ . Hint: use principal curvature coordinates.

2. (4 pts. ) Course Notes Ex. 2.2 : Let $$V=\mathbb R^4$$ and define the $$2$$-form $$\alpha = u_{12} e_1^\flat\wedge e_2^\flat + u_{24} e_2^\flat\wedge e_4^\flat + u_{34} e_3^\flat \wedge e_4^\flat$$ and the $$1$$-form $$\beta = w_2 e_2^\flat + w_3 e_3^\flat$$. Compute $$\alpha \wedge \beta$$ and $$\alpha \wedge \alpha$$ .

3. (4 pts. ) Course Notes Ex. 2.3 : Prove that $$e_I^\flat(e_{\hat{I}})=\delta_{I\hat{I}}$$, i.e., takes on the value $$1$$ when $$I=\hat{I}$$ and $$0$$ otherwise .

4. (4 pts. ) Course Notes Ex. 2.4 : Let $$\alpha$$, $$\beta$$, and $$\gamma$$ be $$k$$-, $$l$$-, and $$r$$-forms respectively. Show that the wedge product is associative, $$(\alpha \wedge \beta)\wedge \gamma = \alpha \wedge(\beta\wedge\gamma)$$, distributive over addition (for $$l=r$$), $$\alpha \wedge(\beta + \gamma) = \alpha\wedge \beta + \alpha \wedge \gamma$$, and anti-commutative, $$\alpha \wedge \beta = (-1)^{kl}\beta \wedge \alpha$$ .

5. (4 pts. ) Course Notes Ex. 2.6 : Instead of stating the properties that define the exterior derivative we could also give a working definition. Let $$\omega = \sum_I w_I d x^I$$ and define $$d$$ by
$d\omega = \sum_I d(w_I) \wedge dx^I$
and $$d$$ applied on a 0-form $$f$$ is defined by
$df = \sum_{i=1}^n \frac{\partial f}{\partial x^i}dx^i.$
Now one needs to show that this amounts to the usual differential for functions, is a linear operator, satisfies $$d^2 = 0$$, and the product rule. Show that this is true. To show $$d^2=0$$ first show that this is true for functions using the fact that for multiple partial derivatives their order does not matter.
.

Total: 20 pts.