Lecture 1

Structure of course
Multivariable functions (recall)
Multiple integrals
Differential equations
Line integrals

Multivariable functions.

Level curves

Multivariable functions.
Partial derivatives and partial differentials

Partial derivatives
Notations of partial derivatives
(in relation to х)
Analogically
partial derivatives
(in relation to y)
Notations of partial derivatives (in relation to y)

Examples

Total increment

Definition. The linear part of formulas (4)
and (5) is called total differential and denotes as
or
(1.6)
(1.6*)
Each of addends in formula 6 is called partial differential. So, the total differential is the sum of partial differentials. And independent increments are equal to its differentials (as for function od one variable).