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Differential forms: review questions

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