Lecture №12

# Lecture №12. Green's Theorem. Applications of line integrals.

Exercises for Lecture #11

The purpose of work:

The acquaintance with the Green's formula for a plane and applications of line integrals.

Theoretical information and examples

Content

2.9.4 Green's formula for a plane

2.9.5 Applications of line integrals

2.9.4 Green's formula for a plane. If C is the boundary of a region S and the functions P(x, y) and Q(x, y) are continuous together with their first order partial derivatives in the closed region S + C, then Green's formula applies:

where the circulation about the contour C is chosen so that the region S remains on the left hand side.

2.9.5 Applications of line integrals.

l) An area bounded by the closed contour C is

(the direction of circulation of the contour is chosen counter-clockwise).

The following formula for the area is more convenient for applications:

2) The work of a force with the components X=X(x,y,z), Y=Y(x,y,z), Z=Z(x,y,z) (or, accordingly, the work of a force field) along a path C is expressed by the integral

If the force has a potential, i.e., if there exists a function U =U(x, y, z) (a potential function or a force function) such that

then the work, irrespectively of the shape of the path C, is equal to

where (x1, y1, z1) is the initial and (x2, y2, z2) is the final point of the path.