University Mathematics Handbook (2015)

VII. Differential Calculus of Multivariable Functions

Chapter 4. Partial Derivatives

4.1  Definition of Partial Derivatives

Let function be defined on domain and point on .

Selecting variables , we construct a function of one variable, .

Adding to a at , we get addition :

If the limit exists, it is called the partial derivative of function with respect to on . It is denoted as either, , , , or .

Similarly, we defined partial derivatives with respect to

4.2  Geometric Description of Partial Derivatives

Let be a surface defined by the function , and let point be on surface when .

Fixing , we get a one-variable function defining the line of intersection between plane and surface . Therefore, defines the slope of tangent line at point to curve and (Figure 1).

Figure 1

That is to say, describes the rate of change of function towards the -axis. Similarly, describes the rate of change of function towards the -axis, and equals which is the slope of tangent line (Figure 2).

Figure 2

The equation of tangent line to curve at is

The equation of tangent line to curve at point is

Through these tangent lines, passes a plane expressed by the equation

This is a tangent plane to surface at point .

Vector is the normal to the surface at (Figure 3).

Figure 3

4.3  Higher-Order Partial Derivatives

a.  Partial derivative by of function , is also a function of variables. If it is partially differentiable on , the result is second-order partial derivatives, denoted by:

, , or ,

If , derivative is called a second-order mixed partial derivative.

b.  If function is defined on domain and has partial derivatives continuous in the neighborhood of , then, at that point, the mixed partial derivatives are equal: .

c.  Function is said to be of to class on domain if it is continuous on . Function belongs to class if it is continuous and has continuous partial derivatives up to and including -th order, on .

d.  If function is defined on -dimensional domain and belongs to class , then the value of -th order mixed derivatives is independent of the order of derivation.