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# Functions of Several Variables

Generally, the graph of y = f ( x) gives a curve in the xy -plane.

Functions of Several Variables

Recall that given a function f of x ; we sketch the graph of y = f ( x) to understand it better. Generally, the graph of y = f ( x) gives a curve in the xy -plane. Also, the derivative f ŌĆ▓(a) of f at x = a represents the slope of the tangent at x = a , to the graph of f . In the introduction we have seen the need for considering functions of more than one variable. Here we shall develop some concepts to understand functions of more than one variable. First we shall consider functions of two variables. Let F ( x, y) be a function of x and y . To obtain graph F , we graph z = F ( x , y) in the xyz -space. Also, we shall develop the concepts of continuity, partial derivatives of a function of two variables.

Let us look at an example, g ( x, y ) = 30 ŌłÆ x2 ŌłÆ y2 , for x , y Ōłł ŌäØ. Given a point ( x , y) Ōłł ŌäØ2, then z = 30 ŌłÆ x2 ŌłÆ y2 gives the z coordinate of the point on the graph. Thus the point (x , y, 30 ŌłÆ x2 ŌłÆ y2 ) lies 30 ŌłÆ x2 ŌłÆ y2 high just above the point ( x , y) in xy -plane. For instance, for (2, 3) Ōłł ŌäØ2 , the point (2, 3, 30 ŌłÆ 22 ŌłÆ 32 ) = (2, 3,17) lies on the graph of g . If we fix the value y = 3 , then g ( x, 3) = ŌłÆx2 + 21 which is a function that depends only on x variable; so its graph must be a curve. Similarly, if we fix value x = 2 , then we have g (2, y) = 26 ŌłÆ y2 which is a function that depends only on y . In each case the graph, as the resulting function being quadratic, will be a parabola. The surface we obtain from z = g( x , y) is called paraboloid.

Note that g ( x, 3) = 21ŌłÆ x2 represents a parabola; which is obtained by intersecting the surface of z = 30 ŌłÆ x2 ŌłÆ y2 with the plane y = 3 [see Fig. 8.5). Similarly g (2, y) = 26 ŌłÆ y2 represents a parabola; which is obtained by intersecting the surface of z = 30 ŌłÆ x2 ŌłÆ y2 with the plane x = 2 [see Fig. 8.6). Following graphs describes the above discussion.

z = 30 - x2 - y2 , z = 30 - x2 - y2 In the same way, given a function F of a two variables say x , y , we can visualize it in the three space by considering the equation z = F ( x , y) . Generally, this will represent a surface in ŌäØ3.

Tags : Differentials and Partial Derivatives | Mathematics , 12th Maths : UNIT 8 : Differentials and Partial Derivatives
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12th Maths : UNIT 8 : Differentials and Partial Derivatives : Functions of Several Variables | Differentials and Partial Derivatives | Mathematics