]> First Approximation

First Approximation

of a function with many variables.

by Samuel Dagan (Copyright © 2007)

As known from functions with one variable, the tangent line at a point x₀ is used, as a first approximation (y₁) of the function y = y(x) , around this point:

$y_{1} = y_{0} + {(\frac{d y}{d x})}_{0} (x - x_{0})$

where the index 0, denotes at point x₀.

The first approximation of a function of n variables

$y = y (x_{1}, ...., x_{n})$

around the point

${(x_{1}, ...., x_{n})}_{0} = (x_{1, 0}, ...., x_{n, 0})$	(1)

has the form of:

$y_{1} = y_{0} + \sum_{k = 1}^{n} {(\frac{\partial y}{\partial x_{k}})}_{0} (x_{k} - x_{k, 0})$	(2)

since for the point (1)

$\begin{array}{l} y_{1} = y_{0} and \\ \frac{\partial y_{1}}{\partial x_{k}} = {(\frac{\partial y}{\partial x_{k}})}_{0} k = 1, ..., n \end{array}}$	(3)

As seen from (2), the first approximation itself is in general a function of n variables. Care should be taken that the function and the derivatives are continuous at the point (1).

In the case of a function of two variables,

$z = z (x, y)$	(4)

the first approximation is the tangent plane:

$z_{1} = z_{0} + {(\frac{\partial z}{\partial x})}_{0} (x - x_{0}) + {(\frac{\partial z}{\partial y})}_{0} (y - y_{0})$	(5)

It is called "tangent", since its derivatives are equal to these of the function at the point (x, y, z)₀, but is it a plane?

A plane in a three dimensional space is defined as any linear combination of the Cartesian coordinates:

$A x + B y + C z = D$	(6)

where at least one of the constant factors of the coordinates (A,B,C) does not vanish. Indeed any intersection of (6) with the planes of constant x, y or z, yields a straight line, as can be easily seen. From the definition (6) it follows that (5) is a plane.

A point of a function of many variables with vanishing and continuous derivatives is called a stationary point. As in the case of a function of a single variable, a stationary point indicates that for an infinitesimal deviation from this point, the function remains constant. In the case of two variables (5), the tangent plane at a stationary point is

$z = z_{0} where z_{0} is z at that point$	(9)

The following example will be used to illustrate the tangent plane.

$z = 3 [1 - {(\frac{x}{4})}^{2} - {(\frac{y}{3})}^{2}]$	(10)

The section of this function with a constant z plane is

${(\frac{x}{4})}^{2} + {(\frac{y}{3})}^{2} = 1 - \frac{z_{c}}{3} with z_{c} = constant$	(11)

which is an ellipse for z_c<3, a point for z_c=3 and non existing for z_c>3. Therefore

$\begin{array}{l} x = y = 0 \\ z = 3 \end{array}} is a point of maximum$	(12)

The sections with constant x or y values are parabolas:

$\begin{array}{l} z = - \frac{y^{2}}{3} + 3 [1 - {(\frac{x_{c}}{4})}^{2}] x_{c} = constant \\ z = - \frac{3}{16} x^{2} + [3 - \frac{y_{c}^{2}}{3}] y_{c} = constant \end{array}}$	(13)

The shape is similar to a paraboloid of revolution, except that instead of circles there are ellipses, and therefore it is called an elliptic paraboloid.

The tangent plane at point (x,y,z)₀, according to (5) is:

$z = z_{0} - \frac{3 x_{0}}{8} (x - x_{0}) - \frac{2 y_{0}}{3} (y - y_{0})$	(14)

For the maximum (12), the tangent plane becomes

$z = 3$	(15)

which corresponds to a stationary point (9).

This example (9) and some display of the tangent planes are shown in Fig. Elliptic paraboloid.