Homework 2 Problem 1

Problem

Consider $M = \mathbb{T}^2$ with local coordinates $(\theta,\phi)$ and $N = \mathbb{S}^3$ with local coordinates $(a,b,c)$.

Let $f : M \rightarrow N$ be given in local coordinates by

(1)
\begin{align} a = cos(2\theta + \phi), b = sin(\theta + 3\phi), c = 4-5\phi. \end{align}

If the point $q \in M$ has coordinates $(\theta,\phi) = (\pi,0)$ and $v = 6 \frac{\partial}{\partial \theta} + 7 \frac{\partial}{\partial \phi} \in T_q M$, compute $(T_q f)v$.

Solution

This problem is asking, "if you've got a vector on the torus, what does the corresponding vector look like on the sphere?" where the function $f$ is the thing that tells you how stuff on the torus corresponds with stuff on the sphere. Since we basically just need to know how coordinates change (the tangent map), the procedure is:

  1. compute the Jacobian of the transformation $f$
  2. evaluate the Jacobian at the point $q$ (since our vector $v$ is rooted at $q$)
  3. transform $v$ by the Jacobian at $q$

So let's do it:

(2)
\begin{align} J_f = \left[ \begin{array}{cc} \frac{\partial a}{\partial \theta} & \frac{\partial a}{\partial \phi} \\ \frac{\partial b}{\partial \theta} & \frac{\partial b}{\partial \phi} \\ \frac{\partial c}{\partial \theta} & \frac{\partial c}{\partial \phi} \end{array} \right] = \left[ \begin{array}{cc} -2 \sin(2\theta + \phi) & -\sin(2\theta + \phi) \\ \cos(\theta + 3\phi) & 3 \cos(\theta+3\phi) \\ 0 & -5 \end{array} \right] \end{align}
(3)
\begin{align} J_f \Big |_q = \left[ \begin{array}{cc} 0 & 0 \\ -1 & -3 \\ 0 & -5 \end{array} \right] \end{align}
(4)
\begin{align} \left( J_f \Big |_q \right) v = \left[ \begin{array}{cc} 0 & 0 \\ -1 & -3 \\ 0 & -5 \end{array} \right] \left[ \begin{array}{c} 6 \\ 7 \end{array} \right] = \left[ \begin{array}{c} 0 \\ -27 \\ -35 \end{array} \right] = -27 \frac{\partial}{\partial b} - 35 \frac{\partial}{\partial c}. \end{align}

Scott agrees!

Discussion

The thing I like about LaTeX is that even if it's wrong, it still looks official…

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