The Jacobian of a $f':\Bbb{R}^n\to\cal{L}(\Bbb{R}^n;\Bbb{R}^n)$ I was given a function $f:\Bbb{R}^n\to\Bbb{R}^n$ that is $\cal{C}^{\infty}$ and was asked to calculate the Jacobian matrix of its derivative $f':\Bbb{R}^n\to\cal{L}(\Bbb{R}^n,\Bbb{R}^n)$. I know from a previous question I made the nature of the derivative as a multilinear function, and that $f''(x)$ cam be presented as a 3-dimensional array
I have calculated all of it's partial second derivatives but now I don't know how to determine the Jacobian matrix itself. For $Jf$ I know its entries are just the partial derivatives. But how to use the second partial derivatives to determine $Jf'$ I'm not quite seeing. Can someone help?
 A: Let $x, h\in R^n$. Then $Jf(x)h:R^n\times R^n\rightarrow R^n$.
Then we can take the differential of this $d(Jf(x)h)=d(Jf(x))h+Jf(x)dh$
You don't need the part $Jf(x)dh$, it isn't used in the Taylor theorem, for example. So just set $dh=0$. What you need is this part $d(Jf(x))h$. Denote the matrix $G(x)=Jf(x)$.
As the differential of a matrix equals to the matrix of the differentials, we have $dG(x)=(\nabla G_{i,j}(x)\cdot dx)_{i,j}$, where $\nabla G_{i,j}(x)$ means the gradient of the partial derivative $(f_i)'_{x_j}(x)$, $\cdot$ denotes the dot product.
Denote $l=dx$. Your second derivative is the bilinear map $f''(x)(l, h)=(\nabla G_{i,j}(x)\cdot l)_{i,j}h$.
For $n=2$ this is
$$
f''(x)(l, h)=
\begin{bmatrix}
    \nabla (f_1)'_{x_1}(x)\cdot l & \nabla (f_1)'_{x_2}(x)\cdot l \\
    \nabla (f_2)'_{x_1}(x)\cdot l & \nabla (f_2)'_{x_2}(x)\cdot l
\end{bmatrix}
\begin{bmatrix}
    h_1 \\
    h_2
\end{bmatrix}
$$
You can rearrange it to get vector$\times$tensor$\times$vector multiplication.
For the Taylor theorem you have
$$f(x+h)=f(x)+Jf(x)h+\frac{1}{2}f''(h,h)+o(||h||^2)$$
