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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 ^ ) .

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