Derivative with respect to Space

• In the fall of 1972 President Nixon announced that the rate of increase of inflation was decreasing. This was the first time a sitting president used the third derivative to advance his case for re-election. (Hugo Rossi)
• Calculus required continuity, and continuity was supposed to require the infinitely little; but nobody could discover what the infinitely little might be. (Bertrand Russell)
• Who has not been amazed to learn that the function $u(t) = e^t$, like a phoenix rising from its own ashes, is its own derivative? (Francois le Lionnais)
• I recoil with dismay and horror at this lamentable plague of functions which do not have derivatives.(Charles Hermite)
• Senate Panel Approves Tougher Rules on Derivatives.

Derivative with respect to x

We now extend to a function $u(x)$ depending on position $x=(x_1,x_2,x_3)$ instead of time $t$.

Definition: A function $u:\Omega\rightarrow R$ where $\Omega\subset R^3$, said to be differentiable in $\Omega$ with derivative or gradient

• $\nabla u(x) \equiv (\frac{\partial u}{\partial x_1}(x),\frac{\partial u}{\partial x_2}(x),\frac{\partial u}{\partial x_3}(x))$,

if for some positive constant $C_u$ and all $x\in\Omega$,

• $\vert u(x+dx)-u(x) -\nabla u(x)\cdot dx \vert\le C_u\vert dx\vert^2\quad\mbox{for } \vert dx\vert\quad\mbox{small}$.

In particular, choosing $dx=(dx_1,0,0)$, we have

• $\vert u(x_1+dx_1,x_2,x_3)-u(x_1,x_2,x_3) -\frac{\partial u}{\partial x_1}(x)dx_1 \vert\le C_u\vert dx_1\vert^2\quad\mbox{for } \vert dx_1\quad\mbox{small}$.

which means that $\frac{\partial u}{\partial x_1}(x)$ is the derivative of $f(x)$ with respect to $x_1$, with $x_2$ and $x_3$ kept constant, referred to as the partial derivative with respect to $x_1$.

The definition directly generalizes to real-valued function $u(x)$ of d-vector variable $x=(x_1,x_2,....,x_d)$, where the variable components can have have some other meaning than position. In the case $d=1$, that is with latex $u(x)$ a function of one variable $x\in R$,we often use $u^\prime (x)$ to denote the derivative, thus with the defining relation

• $\vert u(x+dx)-u(x) -u^\prime (x)dx \vert\le C_u\vert dx\vert^2\quad\mbox{for } \vert dx\vert\quad\mbox{small}$.

Seeking the derivative as the slope of the tangent for a function f(x) of one real variable x.

Vector-valued function of vector variable

The definition of derivative directly generalize to an $m$-vector-valued function $u(x)=(u_1(x),....,u_m(x))$ of an $n$-vector variable $x=(x_1,x_2,....,x_n)$:

Definition: A function $u:R^n\rightarrow R^m$ is differentiable with derivative $u^\prime (x)$ if for som positive constant $C_u$

• $\vert u(x+dx)-u(x) - u^\prime (x) dx \vert\le C_u\vert dx\vert^2\quad\mbox{for } x,\, x+ dx\in R^n$.

Here the derivative $u^\prime (x)$ is an $m\times n$ matrix.

We shall use this derivative below when solving an equation $u(x)=0$ where $u:R^n\rightarrow R^n$ is a differentiable function with non-singular derivative $u^\prime (x)$ using Newton’s method.