I'm taking a Diferential Manifolds course and I don't understand how to compute $DF_a$ in order to apply the following theorem:
Let $F:U \rightarrow \mathbb{R}^m$ be a $C^\infty$ function on an open set $U \subseteq \mathbb{R}^{n+m}$ and let $c \in R^m$.
Assume that for each $a \in F^{-1}(c)$, the derivative $DF_a:\mathbb{R}^{n+m} \rightarrow \mathbb{R}^m$ is surjective.
Then, $F^{-1}(c)$ has the structure of an $n$-dimensional manifold which is Hausdorff and has a countable basis of open sets.
For example, if we take $F:\mathbb{R}^{n+1} \rightarrow \mathbb{R}$ defined by $F(x)=\sum_{k=1}^{n+1}x_{k}^{2}$, why is $DF_a(h)=2\sum_{k=1}^{n+1}a_kh_k$?
And more, if we take the function defined by $F(A)=AA^{T}$, why is $DF_A(H)=HA^T+AH^T$?
Thanks!
If $F$ is $C^1$ then $DF$ is, in standard coordinates, given by the matrix of the partial derivatives of $F$. With this information it is easy to calculate the first example.
Sometimes it is easier to use the definiton of differentiabilty. A linear map is (in the finite dimensional case) always differentiable and coincides with it's derivative. Using this and the fact that matrix multiplication and transposition $(A, B) \rightarrow AB^T$ is a composition of linear maps, it is easy to see that the second result is true.
In short, you need to know the definition of differentiability and the rules for differentiating (chain rule, product rule, relation between partial and total differentiability,... -- the things which are taught in calculus) and, of course, should practise a lot.