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Differential forms: exams

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Tests for Differential forms: course.

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:

Describe the vector spaces of discrete differential forms for the complex below.png

(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 $Ω^{k}(R₂)$?

Differential forms: exam 1 discussion


1. Prove that the following is or is not a (smooth) 2-manifold: (a) point, (b) circle, (c) torus, (d) $R³$.

2. Evaluate $∫_{C}dx+dy$, where C is the upper half of the unit circle oriented counter-clockwise.

3. Suppose $d:Ω^{k}(R²)→Ω^{k+1}(R²)$ is the exterior derivative. Suppose $D:Ω^{k}(R²)→Ω^{k+1}(R²)$ is an "alternative" exterior derivative; it satisfies the following: (a) the linearity, (b) the Product Rule, and (c) $DD=0$. Assume that $D=d$ on $Ω⁰(R²)$ and prove that $D=d$ on $Ω¹(R²)$.

4. Any differential equation of the form: $$\frac{dy}{dx}=\frac{f(x)}{g(y}$$ can be solved by "separation of variables": $$g(y)dy=f(x)dx$$ followed by integration. Explain the relation between these two equations. Hint: this is a one-dimensional situation.

5. In Calculus 1, definite integral is used to produce anti-derivatives. In what sense does integration of 1-forms produce anti-derivatives?