Explicit pushforwards and pullbacks

 Preliminary exam prep

Here we consider a map $f:M\to N$ between manifolds of dimension $m$ and $n$, respectively, and the maps that it induces. Let $p\in M$ with $x_1,\dots,x_m$ a local chart for $U\owns p$ and $y_1,\dots,y_n$ a local chart for $V\owns f(p)$. Induced from $f$ are the differential (or pushforward) $df$ and the pullback $df^*$, which are duals of each other:
\[
\begin{array}{r c l}
df_p\ :\ T_p M & \to & T_{f(p)}N \\[10pt]
df\ :\ TM & \to & TN \\
\alpha & \mapsto & (\beta\mapsto \alpha(\beta\circ f)) \\[10pt]\\\\
\end{array}
\hspace{1cm}
\begin{array}{r c l}
df^*_p\ :\ T_{f(p)}^* N & \to & T_p^*M\\[10pt]
df^*\ :\ T^*N & \to & T^*M \\
\omega & \mapsto & \omega\circ f\\[10pt]
\bigwedge ^k T^*N & \to & \bigwedge^k T^*M\\
\omega\ dy_1\wedge\cdots \wedge dy_k & \mapsto & (\omega \circ f)\ d(y_1\circ f)\wedge \cdots \wedge d(y_k\circ f)
\end{array}
\]
These maps may be described by the diagram below.

Example: For example, consider the map $f:\R^3\to \R^3$ given by $f(x,y,z) = (x-y,3z^2,xz+yz)$, with the image having coordinates $(u,v,w)$. With elements
\[
2x\frac\dy{\dy x} - 5z\frac\dy{\dy y}\in TM,
\hspace{2cm}
2uv+\sqrt w-5\in C^\infty(N),
\hspace{2cm}
\cos(uv)\in T^*N,
\]
we have
\begin{align*}
df_p\left(2x\frac\dy{\dy x} - 5z\frac\dy{\dy y}\right)(2uv+\sqrt w-5) & = \left(2x\frac\dy{\dy x} - 5z\frac\dy{\dy y}\right)\left(6(x-y)z^2+\sqrt{xz+yz}-5\right)(p),\\
df_p^*\left( \cos(uv)\right) & = \cos((x-y)3z^2), \\
\left(\textstyle\bigwedge^2 df_p^*\right)(\cos(uv)du\wedge dw) & = \cos((x-y)3z^2)d(3z^2)\wedge d(xz+yz) \\
& = \cos((x-y)3z^2)\left(-6z^2\ dx\wedge dz -6z^2\ dy \wedge dz\right).
\end{align*}