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 $\alpha =u_{12}{e}_1^{\mathrm{♭}}\wedge e_2^{\mathrm{♭}}+u_{24}{e}_2^{\mathrm{♭}}\wedge e_4^{\mathrm{♭}}+u_{34}{e}_3^{\mathrm{♭}}\wedge e_4^{\mathrm{♭}}$ and the $1$-form $\beta =w_2{e}_2^{\mathrm{♭}}+w_3{e}_3^{\mathrm{♭}}$. Compute $\alpha \wedge \beta$ and $\alpha \wedge \alpha$ .

3. (4 pts. ) Course Notes Ex. 2.3 : Prove that $e_I^{\mathrm{♭}}(e_{\hat{I}})={\delta }_{I\hat{I}}$, 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
$$d\omega =\sum _{I}d(w_I)\wedge d{x}^I$$
and $d$ applied on a 0-form $f$ is defined by
$$df=\sum _{i=1}^n\frac{\partial f}{\partial x^i}d{x}^i.$$
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.
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Total: 20 pts.