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Differential forms: exam 1 discussion

1. Simplify: $$(a)\, 2dxdydz+3dydxdz-dydxdz,$$ $$(b)\, (x¹dx²+x²dx²)(x²dx¹dx²+x²dx¹dx³+x¹dx²dx³).$$ (a) Compute the exterior derivative of the following discrete differential 1-form in R²:

...	⋅	0	⋅	0	⋅	0	⋅	...
...	-1		0		1		2	...
...	⋅	-1	⋅	0	⋅	1	⋅	...
...	-1		0		1		2	...
...	⋅	0	⋅	0	⋅	0	⋅	...

(b) Is this form exact?

2. Prove that $dx_{i₁}...dx_{i_{k}}=0$ if and only if $i_{s}=i_{r}$ for some $s≠r$.

3.(a) Define discrete differential forms for the triangular grid on the plane. (b) Define the exterior derivative of 0- and 1-forms and prove that $dd=0$.

4. (a) Describe the vector spaces of discrete differential forms for the complex below:

(b) Evaluate the cohomology of the complex (just the answer).

5. (a) Given a smooth function $F:R→R$, what happens to the forms $1,dx$ under $F$? Hint: you can think of F as a change of variables. (b) Given a smooth function $F:R²→R²$, what happens to the forms $1,dx,dy,dxdy$ under $F$? (c) Given a function $F:R₁→R₂$, what kind of function does $F$ generate between $Ω^{k}(R₁)$ and $Ω^{k}(R₂)$?

Discrete forms:

Basis of the $0$-form: $\varphi_1, \ldots, \varphi_6$ defined byt $\varphi_1(v_j) = \delta_{ij}$

Basis of the $1$-form: $\psi_1, \ldots, \psi_5$ defined by $\psi_i(\varphi_j) = \delta_{ij}$.

$C^2 = 0$

Co-homology: $H^0 = {\bf R}^2$, $H^1 = {\bf R}$, $H^2 = 0$, etc.

#6: $F \colon {\bf R} \rightarrow {\bf R}$ smooth. ($1 \rightarrow ?$ and $dx \rightarrow ?$)

$y = F(x)$ $\leftarrow F(dx)?$