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Linear Approximation of Functions of Several Variables

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z overview
Suppose that the function $ f$ has continuous partial derivatives. Then in linear (i.e. in first) approximation $ o\left( \left\vert \Delta x \right\vert \right)$

$\displaystyle \Delta f(x) = f(x+\Delta x) - f(x) \approx f^\prime(x)\Delta x\, . $

To stress the limit process $ \vert\Delta x\vert\to0$ one often writes

$\displaystyle df = \frac{\partial f}{\partial x_1} dx_1 + \cdots + \frac{\partial f}{\partial x_n} dx_n\, $

with differentials $ df$ and $ dx_i$.

The linear function

$\displaystyle l(x) = f(x_0) + f^\prime(x) (x - x_0) $

is called the linear approximation of $ f$ at $ x_0 .$

If $ f = f(x,y,z)$ is a scalar function of three variables $ x,y$ and $ z$, then the linear approximation at $ (x_0, y_0, z_0)$ is given by

$\displaystyle l(x,y,z) = f(x_0,y_0,z_0) + f_x(x_0,y_0,z_0)(x - x_0) + f_y(x_0,y_0,z_0)(y - y_0) + f_z(x_0,y_0,z_0)(z - y_0) .$

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  automatically generated 8/ 4/2008