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# Differential forms: review questions

### From Mathematics Is A Science

- Define smooth k-forms in ${\bf R}^{3}$ in terms of $dx,dy$ etc. Give examples. Define the operations and their properties.
- Define multiplication of forms. Give examples. State and prove the property of multiplication of forms ${\phi \psi =(-1)^{?}\psi \phi .}$
- Define the exterior derivative of smooth $k$-forms. Compute the general form of the exterior derivative of a 2-form in ${\bf R}^{3}.$
- State and prove the Leibniz rule for exterior derivative.
- State and prove the main properties of the space of smooth differential forms.
- Prove $dd=0$ for smooth forms.
- Define De Rahm cohomology. Give examples.
- Define closed and exact forms. Give the main properties of the spaces related to them.
- If $d\varphi =d\psi ,$ prove that $\phi -\psi $ is closed.
- Define simple connectedness. Give examples. State the theorem about the closed and exact forms on a simply connected region in the plane.
- Define discrete forms in the plane. Give $dx,dy,dxdy.$
- State the Fundamental Theorem of Calculus for discrete forms.
- Define multiplication of discrete forms and state its main properties.
- Define exterior derivative of discrete forms in the plane.
- Prove $dd=0$ for discrete forms in the plane.
- State and prove the Product Rule (Leibniz) for discrete forms of degree 1 in the plane.
- Define cubical cohomology. Give examples. State main theorems.
- Define closed and exact discrete forms in the plane. Give the main properties of the spaces related to them.
- State the fundamental correspondence for ${\bf R}^{3}.$
- Define the Hodge * operator for smooth forms. Give examples.
- Define the Hodge * operator for discrete forms. Give examples.
- Compute $d(Fdx+Gdy+Hdx)$ for discrete forms.
- Define a smooth manifold. Give examples and non-examples.
- Define an atlas of a smooth manifold.
- Define the tangent bundle of a smooth manifold. Give examples.
- State the Implicit Function Theorem for $f:{\bf R}^{2}\rightarrow {\bf R}.$
- Prove that every smooth form is $R^{3}$ can be represented as $Fdx+Gdy+Hdx.$
- Define the integral of smooth 0-form.
- Define the integral of a smooth 1-form. State main properties.
- Define oriented 0- and 1-manifolds.
- How do you construct a cochain from a smooth form?
- Prove that $\int_{-C}\omega =-\int_{C}\omega $ for 1-forms.
- Define smooth forms on smooth manifolds.
- What is the standard basis of $\Omega ^{2}(M)$ where $M$ is a 3-manifold.
- State the Stokes theorem. Give applications.
- Draw the Mobius band in the 3D grid.