Linear approximation

Set

context	$X,Y$ … Banach spaces with topology
context	$\mathcal O$ … open in $X$
context	$x\in\mathcal O$
context	$f:\mathcal O\to Y$
definiendum	$J_x^f$
postulate	$J_x^f$ … bounded linear operator from $X$ to $Y$
postulate	$\mathrm{lim}_{h\to 0}\ \Vert f(x+h)-f(x)-J_x^f(h)\Vert / \Vert h\Vert\ =\ 0$

Discussion

The approximation of the value of $f$ at a point $x$ is $f(x+d)\sim f(x)+J_x^f(d)$ .

The operators $J_x^f$ are actually both functional in $x$ and in $f$ and this is how we can define general differentiation operators, see Fréchet derivative.

Examples

For $X=\mathbb R^n,Y=\mathbb R^m$ and the Euclidean metric, we can take the Open subsets of ℝⁿ $\ \mathfrak J_o$ , and then if

$f({\bf x})= f^i(x^1,\dots,x^n)\cdot {\bf e}_i$

we find, for all values ${\bf x}$ where the limit in the definition is indeed zero, the operator $J_{\bf x}^f:R^n,Y\to R^m$ is given by the so called

Jacobi matrix

$(J_{\bf x}^f)_m^k=\frac{\partial f^k(x^1,\dots,x^n)}{\partial x^m}$

where $\frac{\partial g(x)}{\partial x}$ is given by Real function derivative.

For a map $f:\mathbb R^2\to\mathbb R^2$ , if

$f(x,y)=\langle u(x,y),v(x,y)\rangle$

the derivative at $x$ in direction $d=\langle d^1,d^2\rangle$ evaluates to

$J_{\bf x}^f(d)=\left\langle\left\langle\frac{\partial u}{\partial x},\frac{\partial u}{\partial y}\right\rangle,\left\langle\frac{\partial v}{\partial x},\frac{\partial v}{\partial y}\right\rangle\right\rangle\cdot\langle d^1,d^2\rangle=\left\langle\frac{\partial u}{\partial x}\cdot d^1+\frac{\partial u}{\partial y}\cdot d^2,\frac{\partial v}{\partial x}\cdot d^1+\frac{\partial v}{\partial y}\cdot d^2\right\rangle$ .

For a function $f:\mathbb R\to\mathbb R$ we have

$J_x^f(d) = f'(x)\cdot d$

It's in a sense remarkable that the linear approximations of real functions at a point $x$ is just a scalar multiplication by another real function $f'$ . As seen above, this isn't generally true in higher dimensions, where one needs matrix multiplication.

The field $\mathbb C$ is $\mathbb R^2$ as a vector space. So to compute the linear approximation of a function $u(x,y)+i v(x,y)$ , where $x,y$ and the functions $u,v$ are real, we must only identify it with the vector field $\langle u(x,y),v(x,y)\rangle$ .

Much more on this in the article holomorphic function.

Reference

Wikipedia: Derivative of a function

Parents

Context

Continuous function, Banach space, ε-δ function limit

Element of

Bounded linear operator

Open subsets of ℝⁿ