Partial Derivatives - Differential Calculus of Multivariable Functions - University Mathematics Handbook

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.