Geometry Sample Questions

Annual Exam 2009

This is a collection of questions taken from previous 224 exams, exercise sheets, as well as some questions from previous 211 and 221 exams. I’ve taken some past schols questions, but also ignored some, as the end of year geometry exam tends to be slightly easier than the schol exam. However, the end of year is often similar to the schol paper from that year (and often some questions are nearly identical), so it is worth looking at.

I’ve used Chris Blair’s list of geometry questions as a guide, and some questions were taken directly from there. So thanks Chris.

Questions

1. Let M be a vector space with basis u1, ··· , un. Define the dual space M ∗ and prove that the coordinate functions u1, ··· , un form a basis.

2. Let (.|.) be a symmetric, non-degenerate scalar product on M with ˜ components gij with respect to the basis ui. Let f 7→ f be the operation of raising the index. Findu ˜i. Let a scalar product (.|.) be defined on M ∗ by: (f|h) = (f˜|h˜). Show that the components of (.|.) on M ∗ with i ij respect to u are the matrix g inverse to gij.

3. Let M be a vector space with basis u1, ··· , un. Show that ui ⊗ uj are a basis for M ⊗ M.

4. Show that there is a linear isomorphism x 7→ x˜ from M to M ∗ such that

1 hx,˜ fi = hf, xi

5. Define the contraction of the tensor

T : M × M ∗ × M → K

with respect to the first and second indices, and show that your defini- tion is coordinate independent (that is, independent of choice of basis).

6. (i) Define tensor, tensor product and tensor contraction. (ii) Show how the components of a tensor change under change of basis. i (iii) If T is a tensor with components aj, write down the components of T ⊗ T ⊗ T . Show that contracting three times will give (traceT )3, (traceT )(traceT 2) or traceT 3.

7. Let S be a tensor of degree s and T a tensor of degree t. Then

(i) A[(AS) ⊗ T ] = A[S ⊗ T ] = A[S ⊗ (AT )] (ii) A(S ⊗ T ) = (−1)stA(T ⊗ S)

8. Let T : M → M be a linear operator in a finite dimensional vector space M. Define the tensor product and wedge product of tensors over M. Define the push-forward and pull-back operators on the tensors and show that they preserve the tensor and wedge product.

9. Let ui be a basis for a finite dimensional vector space M. Prove that

ui ∧ uj ∧ uk

with i < j < k, is a basis for the space M(3).

10. Show that the vectors x1, ··· , xr are linearly independent if and only if x1 ∧ · · · ∧ xr is non-zero.

11. Show that the linearly independent vectors x1, ··· , xr generate the same subspace as the vectors y1, ··· , yr if and only if x1 ∧ · · · xr is a scalar multiple of y1 ∧ · · · ∧ yr.

2 12. Show that the matrix equation Ax = b is equivalent to x1c1 +··· xncn = b where ci are the columns of A. Use the wedge product to derive Cramer’s rule.

13. Let M be a finite dimensional real vector space with non-degenerate symmetric scalar product. Prove that the volume form

vol = u1 ∧ · · · ∧ un

is independent of choice of standard basis u1, ··· , un. 14. Define the Hodge star operator and show that

1 r r+1 n ∗u ∧ · · · ∧ u = sr+1 ··· snu ∧ · · · ∧ u

where si = (ui|ui). 15. State and prove the implicit function theorem.

16. Let f be the function defined on the space of non-singular n × n real matrices given by

f(A) = A−1

Prove that f is differentiable and find its derivative.

17. Let f be a C2 function of two independent variables. Prove that

∂2f ∂2f = ∂x∂y ∂y∂x

∂f ∂f 18. Let f be a real valued function on an open set in 2. Let and R ∂x ∂y exist and be continuous. Prove that f is differentiable.

19. Show that the function

x2 − y2 f(x, y) = 2xy x2 + y2

3 if (x, y) 6= (0, 0) and f(0, 0) = 0 is not C2 on any open set containing (0, 0).

20. Define the following:

• n-dimensional coordinate system • C∞ compatible coordinates • C∞ manifold • velocity vector at t • tangent space at a point • tangent vector field • tangent and cotangent bundles • differential of a function ∂f i • partial derivative ∂yi to a coordinate system y • push-forward of tangent vectors • pull-backs of differential r-forms

21. Prove that if yi are coordinates on a manifold X, and ω be a differential r-form. Define the differential dω with respect to the coordinates yi. Prove the Leibniz property and the property that ddω = 0. Prove that dω is independent of choice of basis.

22. Prove the chain rule for maps on mainfolds.

23. Prove that the differential commutes with the pull-back.

24. State and prove the Poincare lemma.

25. State and prove the theorem of Stokes.

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