Geometry Second Semester 2010/11 Homework 2

1. Sketch the following vector fields X = (·, X(·)) on R^{2} where X is defined as follows.

(a) X(p) = (0, 1)

(b) X(p) = −p

(c) X(x_{1}, x_{2}) = (x_{2}, −x_{1})

(d) X(x_{1}, x_{2}) = (x_{2}, x_{1})

(e) X(x_{1}, x_{2}) = (−2x_{2}, \frac{1}{2}x_{1})

2. Find and sketch the gradient of f: R^{2} → R for each of the following examples.

(a) f(x_{1}, x_{2}) = x_{1} + x_{2}

(b) f(x_{1}, x_{2}) = x_{1}^{2} + x_{2}^{2}

(c) f(x_{1}, x_{2}) = x_{1} − x_{2}^{2}

(d) f(x_{1}, x_{2}) = (x_{1}^{2} − x_{2}^{2})/4

3. The divergence of a smooth vector field X on U ⊂ R^{n+1}, where

X(p) = (p, X_{1}(p), X_{2}(p), …, X_{n+1}(p)) for p ∈ U

is a function div(X): U → R defined by div(X) = ∑_{i=1}^{n+1} ∂_{i}X_{i}. Find the divergence of each of the vector fields in Exercises 1 and 2.

4. Find the integral curve through (1, 1) of each of the vector fields in Exercise 1.

5. Find the integral curve through the general point (a, b) of each of the vector fields in Exercise 1.

6. A smooth vector field X on an open set U ∈ R^{n+1} is said to be complete if for each p ∈ U the maximal integral curve of X through p has domain equal to R. Determine which of the following vector fields are complete.

(a) X(x_{1}, x_{2}) = (x_{1}, x_{2}, 1, 0), U = R^{2}

(b) X(x_{1}, x_{2}) = (x_{1}, x_{2}, 1, 0), U = R^{2} \ {(0, 0)}

(c) X(x_{1}, x_{2}) = (x_{1}, x_{2}, −x_{2}, x_{1}), U = R^{2} \ {(0, 0)}

(d) X(x_{1}, x_{2}) = (x_{1}, x_{2}, 1 + x_{1}^{2}, 0), U = R^{2}

7. Recall that a subset U of R^{N} is said to be open if for each q ∈ U there exists an ε > 0 such that B_{ε}(q) ⊂ U. Here B_{ε}(q) := {p ∈ R^{n} | ∥p − q∥ < ε} is a ball of radius ε with centre q. Prove that any union of open sets is itself an open set. That is, let Γ be any given index set and suppose {S_{γ}}_{γ∈Γ} is a collection of open sets. Show that

⋃_{γ∈Γ} S_{γ}

is an open set.

8. Suppose that U is an open set in R^{n+1} and let p ∈ U. Given a smooth vector field X on U, let α: I → U be the maximal integral curve of X through p. Show that if β: Ĩ → U is any integral curve of X with β(t_{0}) = p for some t_{0} ∈ Ĩ, then β(t) = α(t − t_{0}) for all t ∈ Ĩ.

9. Let U be an open set in R^{n+1}, X be a smooth vector field on U and α: I → U the maximal integral curve of X through a point. Suppose that α(0) = α(t_{0}) for some t_{0} ∈ I \ {0}. Show that I = R and α(t + t_{0}) = α(t) for all t ∈ R.

10. Consider the vector field X(x_{1}, x_{2}) = (x_{1}, x_{2}, 1, 0) on R^{2}. For t ∈ R and p ∈ R^{2}, let φ_{t}(p) = α_{p}(t), where α_{p} is the maximal integral curve of X through p.

(a) Show that, for each t, φ_{t} is a one-to-one transformation from R^{2} onto itself. (That is, show that φ_{t}: R^{2} → R^{2} is injective and surjective.) Geometrically, what does this transformation do?

(b) Show that φ_{0} is the identity, φ_{t_{1}+t_{2}} = φ_{t_{1}} ∘ φ_{t_{2}} for all t_{1}, t_{2} ∈ R and φ_{−t} = φ_{t}^{−1} for all t ∈ R. (This shows that t ↦ φ_{t} is a homeomorphism from the additive group of real numbers into the group of one-to-one transformations of the plane.)

11. Repeat Exercise 10 for the following vector fields.

(a) X(x_{1}, x_{2}) = (x_{1}, x_{2}, −x_{2}, x_{1})

(b) X(x_{1}, x_{2}) = (x_{1}, x_{2}, x_{1}, x_{2})

(c) X(x_{1}, x_{2}) = (x_{1}, x_{2}, x_{2}, x_{1})

12. Let X be any smooth vector field on an open set U ⊂ R^{n+1}. Let φ_{t}(p) = α_{p}(t), where α_{p} is the maximal integral curve of X through p. Use the uniqueness of integral curves to show that φ_{t_{1}+t_{2}} = φ_{t_{1}} ∘ φ_{t_{2}} and φ_{−t} = φ_{t}^{−1} for all t, t_{1}, t_{2} for which all the terms are defined. (φ_{t} is called the local 1-parameter group associated to X.)