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.

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