Generalization of derivatives to higher dimensions:
* limit of difference quotient: partial derivatives,
* linearization: total derivative.
## Partial derivatives
*Definition*: let $D \subseteq \mathbb{R}^n$ ($n=2$ for simplicity) and let $f: D \to \mathbb{R}$ and $\mathbf{a} \in D$, if the limit exists the partial derivates of $f$ are
*Theorem*: suppose that two mixed $n$th order partial derivatives of a function $f$ involve the same differentations but in different orders. If those partials are continuous at a point $\mathbf{a}$ and if $f$ and all partials of $f$ of order less than $n$ are continuous in a neighbourhood of $\mathbf{a}$, then the two mixed partials are equal at the point $\mathbf{a}$. We have for $n=2$
*Definition*: let $D \subseteq \mathbb{R}^n$ ($n=2$ for simplicity) and let $f: D \to \mathbb{R}$, determining an affine linear approximation of $f$ around $\mathbf{a} \in D$
let $\mathbf{r}(t) = \big(x(t),\; y(t) \big)^T$ be a parameterization of the level curve of $f$ such that $\mathbf{r}(0) = \mathbf{a}$. Then for all $t$ near $0$, $f(\mathbf{r}(t)) = f(\mathbf{a})$. Differentiating this equation with respect to $t$ using the chain rule, we obtain
*Definition*: let $D \subseteq \mathbb{R}^n$ and let $f: D \to \mathbb{R}$ with $\mathbf{v} \in D$ and $\|\mathbf{v}\| = 1$ a unit vector. The directional derivative is then the change of $f$ near a point $\mathbf{a} \in D$ in the direction of $\mathbf{v}$
*Definition*: let $D \subseteq \mathbb{R}^n$ and let $\mathbf{f}: D \to \mathbb{R}^m$, with $f_i: D \to \mathbb{R}$, with $i = 1, \dotsc, m$ being the components of $\mathbf{f}$.
* $\mathbf{f}$ is continuous at $\mathbf{a} \in D$ $\iff$ all $f_i$ continuous at $\mathbf{a}$,
* $\mathbf{f}$ is partially/totally differentiable at $\mathbf{a}$ $\iff$ all $f_i$ are partially/totally differentiable at $\mathbf{a}$.
The linearization of every component $f_i$ we have
with $D\mathbf{f}(\mathbf{a})$ the Jacobian of $\mathbf{f}$.
*Definition*: the Jacobian is given by $\big[D\mathbf{f}(\mathbf{a}) \big]_{i,\;j} = \partial_j f_i(\mathbf{a}).$
### Chain rule
Let $D \subseteq \mathbb{R}^n$ and let $E \subseteq \mathbb{R}^m$ be sets and let $\mathbf{f}: D \to \mathbb{R}^m$ and let $\mathbf{g}: E \to \mathbb{R}^k$ with $\mathbf{f}$ differentiable at $\mathbf{x}$ and $\mathbf{g}$ differentiable at $\mathbf{f}(\mathbf{x})$. Then $D\mathbf{f}(\mathbf{x}) \in \mathbb{R}^{m \times n}$ and $D\mathbf{g}\big(\mathbf{f}(\mathbf{x})\big) \in \mathbb{R}^{k \times m}$.
Then if we differentiate $\mathbf{g} \circ \mathbf{f}$ we obtain