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 for \vert dx_1\vert 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 generalises 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 u(x) a function of one variable x\in R, we sometimes use u^\prime (x) to denote the derivative (following Newton), thus with the defining relation

  • \vert u(x+dx)-u(x) -u^\prime (x)dx \vert\le C_u\vert dx\vert^2 for  \vert dx\vert 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) (or Du(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) = Du(x) is an m\times n matrix.

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

Read More

Charles Babbage’s Analytical Engine 1871

Next: Rules of Differentiation   Previous: Derivate with respect to Time.

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