Differentiation Rules for Scalar Fields

Theorem: Chain Rule for Scalar Fields

Let $f:D\subseteq {\mathbb{R}}^n\to \mathbb{R}$ be a real scalar field and let $\mathbf{r}:[a;b]\to D$ be a curve parameterisation.

If $\mathbf{r}$ is differentiable and $f$ is partially differentiable, then the derivative of the composition $f\circ \mathbf{r}$ is the dot product of $f$‘s gradient and $\mathbf{r}$‘s derivative.

$\frac{\mathrm{d}}{\mathrm{d}t}f(\mathbf{r}(t))=\nabla f(\mathbf{r}(t))\cdot {\mathbf{r}}^{\prime }(t)$

NOTE

The composition $f\circ \mathbf{r}$ is a real function.

PROOF

TODO

Theorem: Product Rule

Let $f,g:\mathcal{D}\subseteq {\mathbb{R}}^n\to \mathbb{R}$ be real scalar fields.

If $f$ and $g$ are differentiable at $\mathbf{a}\in \mathcal{D}$, then the product $fg:\mathcal{D}\subseteq {\mathbb{R}}^n\to \mathbb{R}$ is also differentiable at $\mathbf{a}\in \mathcal{D}$ with

$$\mathrm{d}(fg{)}_a=g(a)\mathrm{d}{f}_a+f(a)\mathrm{d}{g}_a$$

IMPORTANT

You should remember that $\mathrm{d}(fg{)}_a$ is a function and so are $\mathrm{d}{f}_a$ and $\mathrm{d}{g}_a$

PROOF

TODO

Theorem: Quotient Rule

Let $f,g:\mathcal{D}\subseteq {\mathbb{R}}^n\to \mathbb{R}$ be real scalar fields.

If $f$ and $g$ are differentiable at $\mathbf{a}\in \mathcal{D}$ and $g(a)\ne 0$, then the quotient $f/g:\mathcal{D}\subseteq {\mathbb{R}}^n\to \mathbb{R}$ is also differentiable at $\mathbf{a}\in \mathcal{D}$ with

$$\mathrm{d}(f/g{)}_a=\frac{g(a)\mathrm{d}{f}_a-f(a)\mathrm{d}{g}_a}{g(a{)}^2}$$

IMPORTANT

You should remember that $\mathrm{d}(f/g{)}_a$ is a function and so are $\mathrm{d}{f}_a$ and $\mathrm{d}{g}_a$

PROOF

TODO