]> 6.5 The Tangent Plane and the Gradient Vector

6.5 The Tangent Plane and the Gradient Vector

We define differentiability in two dimensions as follows. A function f of two variables is differentiable at argument ( x 0 , y 0 ) if the surface it defines in ( x , y , f ) space looks like a plane for arguments near ( x 0 , y 0 ) .

(Given any positive numerical criterion, there is a circle around ( x 0 , y 0 ) within which its graph differs from the plane by less than that criterion.)

Recall that a plane in variables f , x and y is defined by a linear equation that can be put in the form

f ( x , y ) = a ( x x 0 ) + b ( y y 0 ) + f ( x 0 , y 0 )      (A)

The plane that f resembles here is called the tangent plane to f at ( x 0 , y 0 ) and the function it represents is called the linear approximation to f defined at ( x 0 , y 0 ) .

The quantities a and b are called the partial derivatives of f with respect to x and with respect to y , and written as follows

a = f x ; b = f y

Here a is the directional derivative of f in the direction of the x axis, i ^ , while b is the directional derivative of f in the direction of the y axis, j ^ .

The linear approximation to f ( x , y ) at arguments x 0 and y 0 which describes the tangent plane to f at ( x 0 , y 0 ) therefore takes the form

f L ( x , y ) = f x | ( x 0 , y 0 ) ( x x 0 ) + f y | ( x 0 , y 0 ) ( y y 0 ) + f ( x 0 , y 0 ) = f ( x 0 , y 0 ) + ( grad f ) · ( r r 0 )

where the vector grad f , called the gradient vector to f at ( x 0 , y 0 ) , is the vector whose components are the partial derivatives of f in the x and y directions at the point ( x 0 , y 0 )

grad f | ( x 0 , y 0 ) = ( f x | ( x 0 , y 0 ) , f y | ( x 0 , y 0 ) )

We generally do not write out the cumbersome subscripts that indicate the point ( x 0 , y 0 ) at which the gradient and linear approximation are defined, because they are so cumbersome, and simply write

grad f = ( f x , f y ) = f x i ^ + f y j ^

Notice that we can write grad f as ( x i ^ + y j ^ ) f or f with the symbol , called "del" representing the combination: ( x i ^ + y j ^ ) .