$f:S^3 \rightarrow \mathbb{R}$ is defined as $f(x,y,z,w)=x+zw$, where $S^3= \{(x,y,z,w) | x^2 +y^2 +z^2 +w^2 =1\}$
I tried using a stereographic chart but that got ugly. The function is so simple I was hoping there would be a better way to compute it. Thanks.
You mean the map $f_* : TS^3 \to T\mathbb{R}$? $f$ extends to a function $\mathbb{R}^4 \to \mathbb{R}$, so just find the differential of that.
In more detail, let $g$ be said extension, and let $i$ be the inclusion $S^3 \to \mathbb{R}^4$. Then,
$$ g_* = f_* \circ i_* $$
and $i_*$ is very simple.
You mean the exterior derivative $\mathrm{d}f$? $f$ extends to a scalar field on $\mathbb{R}^4$, so find the exterior derivative of that.
In more detail, let $g$ be said extension, and let $i$ be the inclusion $S^3 \to \mathbb{R}^4$. Then,
$$ i^*\mathrm{d}g = \mathrm{d}f $$
and $i^*$ is very simple.