Past exam of the options II course of the University of Oxford / 2011 / F / F2 / a

Writing $f=(f_1,\dots ,f_m)$, the $j$th partial derivative at $x$ is the vector limitwhen it exists. The total derivative is a linear map $Df(x):{\mathbb{R}}^n\to {\mathbb{R}}^m$ satisfyingThe Jacobian matrix has entries $J_f(x{)}_{ij}={\partial }_j{f}_i(x)$. If the total derivative exists, its matrix in the standard bases is precisely $J_f(x)$, because substitution of $h=t{e}_j$ identifies every column. Existence of the partial derivatives alone does not in general imply existence of the total derivative.

For a scalar-valued differentiable function, its gradient is the column vectorIt represents the total derivative by the Euclidean inner product:Thus $J_f(x)=\nabla f(x{)}^T$ is a row matrix; the gradient and the total derivative represent the same first-order information but have different types.

The chain rule states that $f\circ g$ is differentiable at $x$ andIn terms of Jacobian matrices,The factors have dimensions $k\times m$ and $m\times n$, respectively, giving the required $k\times n$ Jacobian matrix.