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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Summer 2020 • Question 8

Question

Math 6510 • Differentiable Manifolds Summer 2020 • Question 8

Let \(V\) be a continuous vector field on the unit ball \(B^n\subset \mathbb {R}^n\) which is nowhere zero. Prove that there are points \(x,y\in S^{n-1}=\partial B^n\) and positive numbers \(a,b>0\) such that \(V(x)=ax\) and \(V(y)=-by\) (so the vector field points outward at \(x\) and inward at \(y\)).

Raw LaTeX

Let \(V\) be a continuous vector field on the unit ball \(B^n\subset\R^n\)
which is nowhere zero. Prove that there are points
\(x,y\in S^{n-1}=\partial B^n\) and positive numbers \(a,b>0\) such that
\(V(x)=ax\) and \(V(y)=-by\) (so the vector field points outward at \(x\) and inward
at \(y\)).

Topic Tags

Left-Invariant Vector Fields Vector Fields and Flows