University Mathematics Handbook (2015)

VII. Differential Calculus of Multivariable Functions

Chapter 5. Differentiability. Taylor's Formula

5.1  Differentiability of Functions of Two-Variable

a.  Function is differentiable at point , if its general addition can be represented the following way:

when .

b.  If function is differentiable at point , then it is continuous at that point.

The inverse proposition is incorrect.

c.  If function is defined on the neighborhood of , and has partial derivatives , continuous in that neighborhood, then it is differentiable at .

5.2  Differentiability of Function

a.  Let be an addition to , , and let

be an addition to function at .

If addition can be represented in the following way:

when , , then function is said to be differentiable on .

b.  If function is continuous on , and has partial derivatives continuous in the neighborhood of , then it is differentiable at .

5.3  Differential

a.  The linear part of in relation to is the differential of and is denoted as :

when .

b.  The differential of two-variable function is

c.  Second-order differential:

d.  Differential Operator , follows the rule: .

The power of operator is a binomial, such as:

-th differential is

5.4  Taylor's Formula

If function belongs to class in the neighborhood of point , then there exists point in the interval between and , such that holds

when .

Explicit formula for a 2-variable function: