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.