[DEFINITIONS] Derivatives and level-set geometry

Function of several variables

A function of several variables is a map

f:D\to\mathbb{R},\qquad D\subseteq\mathbb{R}^n, \qquad \mathbf{x}=(x_1,\ldots,x_n)\mapsto f(\mathbf{x}),

where $n\ge2$ and $D$ is the domain of allowed inputs.

When $n=2$ , the graph of $f$ is the surface $z=f(x,y)$ in $\mathbb{R}^3$ . A function of three variables has a graph in $\mathbb{R}^4$ , which is harder to visualise, but the coordinate and vector formulas remain the same.

Level set

For $F:D\to\mathbb{R}$ , with $D\subseteq\mathbb{R}^n$ and $c\in\mathbb{R}$ , the level set of $F$ at level $c$ is

\{\mathbf{x}\in D:F(\mathbf{x})=c\}.

For example, the graph $z=x^2-y^2$ is the level set $x^2-y^2-z=0$ and the unit sphere is the level set $x^2+y^2+z^2=1$ . The function defining the level set need not be the same function whose graph we are studying.

[1]:Constrained extrema
[2]:Gradient normal to a level set

Partial derivative

For $f:D\to\mathbb{R}$ with $D\subseteq\mathbb{R}^n$ , at an interior point $\mathbf{x}\in D$ , the partial derivative with respect to $x_i$ is the one-variable derivative obtained by varying $x_i$ and holding every other coordinate fixed:

\frac{\partial f}{\partial x_i}(\mathbf{x}) = \lim_{h\to0} \frac{ f(x_1,\ldots,x_i+h,\ldots,x_n)-f(\mathbf{x}) }{h},

when the limit exists.

The ordinary sum, product, quotient and chain rules still apply. Treat every variable other than the one being differentiated as a constant. For two variables we write $f_x=\partial f/\partial x$ and $f_y=\partial f/\partial y$ .

Gradient

For $g:D\to\mathbb{R}$ with $D\subseteq\mathbb{R}^n$ , at points where all first partial derivatives exist, the gradient is the vector-valued function

\nabla g(\mathbf{x}) = \begin{pmatrix} \dfrac{\partial g}{\partial x_1}(\mathbf{x})\\ \vdots\\ \dfrac{\partial g}{\partial x_n}(\mathbf{x}) \end{pmatrix}.

When $g$ is differentiable, the gradient has a second interpretation. For a unit direction $\mathbf u$ , the rate of change along the line $\mathbf x+t\mathbf u$ is

D_{\mathbf u}g(\mathbf x) =\left.\frac{d}{dt}g(\mathbf x+t\mathbf u)\right|_{t=0} =\nabla g(\mathbf x)\cdot\mathbf u.

The Cauchy–Schwarz inequality gives $|D_{\mathbf u}g|\le\|\nabla g\|$ . When the gradient is non-zero, the maximum directional derivative is attained by taking $\mathbf u$ in the gradient direction (and the most negative rate in the opposite direction). Thus $\nabla g$ points in the direction of steepest increase and its length is that maximum directional rate.

The linear map that gives first-order change is the total derivative (or derivative map). At a differentiable point it is represented by the gradient:

Dg(\mathbf{x})[\mathbf{h}] =\nabla g(\mathbf{x})\cdot\mathbf{h}, \qquad D_{\mathbf{u}}g(\mathbf{x})=Dg(\mathbf{x})[\mathbf{u}].

Differentiability means this one linear map approximates change in every direction. The coordinate partial derivatives alone do not establish that, so they do not guarantee the directional-derivative formula above.

Definitions from Derivatives and level-set geometry

Function of several variables

Level set

Partial derivative

Gradient

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