Lecture ¹13

# Lecture ¹13. The integrals over the surface

Exercises for Lecture #13

The purpose of work:

The acquaintance with the integrals over the surface and its properties.

Theoretical information and examples

Content

2.10 Surface Integrals

2.10.1 Surface Integral of the first type

2.10.2 Surface Integral of the second type

2.10.3 Stokes' formula

2.10 Surface Integrals

2.10.1 Surface Integral of the first type. Let f(x, y, z) be a continuous function and z =j(x, y) a smooth surface S.

The surface integral of the first type is the limit of the integral sum

,

where DSi is the area of the i-th element of the surface S, the point (xi, yi, zi) belongs to this element and the maximum diameter of the subdivision elements tends to zero.

The value of this integral does not depend on the choice of the side of the surface S over which the integration is performed.

If a projection s of the surface S onto the xy-plane is single-valued, i.e., if every straight line parallel to the z-axis intersects the surfaces at only one point, then the appropriate surface integral of the first type may be calculated from the formula

Example 1. Compute the surface integral

where S is the surface of the cube 0 £ x £ 1, 0 £ y £ l, 0 £ z £ 1.

Let us compute the sum of the surface integrals over the upper edge of the cube (z = l) and over its lower edge (z = 0):

The required surface integral is obviously three times larger and equal to

2.10.2 Surface Integral of the second type. If P = P(x, y, z), Q =Q(x, y, z), R = R(x, y, z) are continuous functions and S+ is one side of the smooth surface S characterized by the direction of the normal n {cos a, cos b, cos g}, then the corresponding surface integral of the second type is:

When we go to the other side S- of the surface, this integral reverses its sign.

If the surface S is represented implicitly as F(x, y, z) = 0, then the direction cosines of the normal to this surface are given by

where

and the choice of sign in front of the the root should be brought into agreement with the side of the surface S.

2.10.3 Stokes' formula. If the functions P=P(x,y,z), Q=Q(x,y,z), R=R(x,y,z) are continuously differentiable and C is a closed contour bounding a two-sided surface S, we have Stokes' formula

where cos a, cos b, cos g are the direction cosines of the normal to the surface S and the direction of the normal is defined so that on the side of the normal the contour S is traced counter-clockwise (in a right hand coordinate system).

There are the other record of Stokes' formula: where S is an oriented smooth surface that is bounded by a simple, closed, smooth boundary curve C with positive orientation.