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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
(1) |
has the form of:
(2) |
since for the point (1)
(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,
(4) |
the first approximation is the tangent plane:
(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:
(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
(9) |
The following example will be used to illustrate the tangent plane.
(10) |
The section of this function with a constant z plane is
(11) |
which is an ellipse for zc<3, a point for zc=3 and non existing for zc>3. Therefore
(12) |
The sections with constant x or y values are parabolas:
(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:
(14) |
For the maximum (12), the tangent plane becomes
(15) |
which corresponds to a stationary point (9).
This example (9) and some display of the tangent planes are shown in Fig. Elliptic paraboloid.