]> First Approximation

# First Approximation

### of a function with many variables.

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:

where the index 0, denotes at point x0.

The first approximation of a function of n variables

around the point

 ${\left({x}_{1},....,{x}_{n}\right)}_{0}=\left({x}_{1,0},....,{x}_{n,0}\right)$ (1)

has the form of:

 ${y}_{1}={y}_{0}+\sum _{k=1}^{n}{\left(\frac{\partial y}{\partial {x}_{k}}\right)}_{0}\left({x}_{k}-{x}_{k,0}\right)$ (2)

since for the point (1)

 $\begin{array}{l}{y}_{1}={y}_{0}\text{ }\text{and}\\ \frac{\partial {y}_{1}}{\partial {x}_{k}}={\left(\frac{\partial y}{\partial {x}_{k}}\right)}_{0}\text{ }k=1,\text{\hspace{0.17em}}...,\text{\hspace{0.17em}}n\end{array}\right\}$ (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,

 $z=z\left(x,y\right)$ (4)

the first approximation is the tangent plane:

 ${z}_{1}={z}_{0}+{\left(\frac{\partial z}{\partial x}\right)}_{0}\left(x-{x}_{0}\right)+{\left(\frac{\partial z}{\partial y}\right)}_{0}\left(y-{y}_{0}\right)$ (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:

 $Ax+By+Cz=D$ (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

 $z={z}_{0}\text{ }\text{where}\text{\hspace{0.28em}}{z}_{0}\text{\hspace{0.28em}}\text{is}\text{\hspace{0.28em}}z\text{\hspace{0.28em}}\text{at that point}$ (9)

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

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

The section of this function with a constant z plane is

 ${\left(\frac{x}{4}\right)}^{2}+{\left(\frac{y}{3}\right)}^{2}=1-\frac{{z}_{c}}{3}\text{ }\text{with}\text{ }{z}_{c}=\text{constant}$ (11)

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

 $\begin{array}{l}x=y=0\\ z=3\end{array}\right\}\text{ }\text{is}\text{\hspace{0.17em}}\text{a}\text{\hspace{0.17em}}\text{point}\text{\hspace{0.17em}}\text{of}\text{\hspace{0.17em}}\text{maximum}$ (12)

The sections with constant x or y values are parabolas:

 $\begin{array}{l}z=-\frac{{y}^{2}}{3}+3\left[1-{\left(\frac{{x}_{c}}{4}\right)}^{2}\right]\text{ }{x}_{c}=\text{constant}\\ z=-\frac{3}{16}{x}^{2}+\left[3-\frac{{y}_{c}^{2}}{3}\right]\text{ }\text{\hspace{0.28em}}{y}_{c}=\text{constant}\end{array}\right\}$ (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:

 $z={z}_{0}-\frac{3{x}_{0}}{8}\left(x-{x}_{0}\right)-\frac{2{y}_{0}}{3}\left(y-{y}_{0}\right)$ (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.