# Math781, 2018 fall

Week-1 possible quiz problems:

- With usual notations what kind of objects are ? (number? function? vector? vector field? etc)
- Define “derivation at the point ”. The set of those forms a …………………… It is denoted …………………… Define “derivation of the algebra ”. The set of those forms a …………………… It is denoted ……………………
- Problem 2.4 from the Tu-book.

Week-2 possible quiz problems:

- Define , describe a basis in both. Define the tensor product on , the wedge product on . Give an 8-term expression for the wedge product of .
- Define differential form. Define , prove the expansion of in terms of .
- Tu: problem 3.8 (page 33) [“-covector” = “linear -form”].

Week-3 possible quiz problems:

- List what operations we have on differentiable forms, and how they interact (e.g. vs addition of forms, of a -product, of a of a form, etc).
- Adding all compatible charts to an atlas results a maximal atlas. For this claim we needed a lemma. Phrase and prove that lemma.
- Describe a complex 1-manifold structure on .

Week-5 possible quiz problems:

- Define . Explain why . Prove the transition matrix formula between the bases and .
- Recall the relation between tangent vectors and curves. Prove that for smooth, if then .
- Prove that the derivative of Lie-group-multiplication is Lie-algebra-addition. What is the derivative of the inverse map in a Lie group?

Week-6 possible quiz problems:

- Define regular value. State the regular level set theorem. Define transversality. State the transversality theorem.
- State and prove the statement about the tangent space at of .
- State and prove the statement about the tangent space at of .

Week-8 possible quiz problems:

- page 162: 14.10
- page 162: 14.12 (the formula was given in class, you need to carry out the calculation to prove it )
- page 162: 14.11 (using 14.12)

October 30 possible quiz problems:

- Prove that .
- page 162: 14.12 (the formula was given in class, you need to carry out the calculation to prove it)
- State the defining axioms of an exterior differentiation on a manifold.

November 6 possible quiz problems:

- page 219: 19.8(a).
- Prove Cartan’s formula.
- page 234: 20.7

November 13 possible quiz problems:

- State and prove Key Calculation 1.
- State and prove Key Calculation 2.
- State and prove Key Calculation 3.

November 20 possible quiz problems:

- Prove Stokes theorem for .
- Calculate the integral .