Differential of a Map I have the following map that embeds the Torus $T^2$ into $\mathbb{R}^3$:
$$f(\theta, \phi)=(cos\theta(R+rcos(\phi)),sin\theta(R+rcos(\phi)), rsin\phi)$$
noting that $0<r<R$.
I want to compute the differential of $f$, $f_*$, that maps $T_P(T^2)$ to $T_{f(p)}(\mathbb{R}^3)$.
This topic is extremely confusing to me.  I am not sure how to really approach the problem at all.  I believe that if $v\in T_p(T^2)$, then I choose a smooth curve $g:\mathbb{R}\to T^2$ s.t. $g(0)=p$ and $g'(0)=v$, then $df(p)v=\frac{d}{dt}f(g(t))$ at $t=0$.
I don't really know what to do with all this.  I don't know where to go.  If someone has a good example or a good source to look at that would help explain this problem, or if someone could help me with this problem that would be greatly appreciated. Thank you in advance.

The differential of the map is given by the Jacobian. Basically what you want to do is take all of the partial derivatives of the coordinate functions and assemble them into a matrix. As you said this matrix should be a transformation from $T_p(T^2) \to T_{f(p)}(\mathbb{R}^3)$ so we want a $3 \times 2$ matrix. The matrix will look like: $  \begin{bmatrix}
\frac{\partial\theta}{\partial x} & \frac{\partial\phi}{\partial x} \\
\frac{\partial\theta}{\partial y} & \frac{\partial\phi}{\partial y}\\
\frac{\partial\theta}{\partial x} & \frac{\partial\phi}{\partial x} \end{bmatrix} $. For example $\frac{\partial\theta}{\partial x} = -sinθ(R+rcos(ϕ))  $ If you'd like to know the differential at a particular point on the torus, just plug in the $(\theta, \phi)$ coordinates.
