Vector Analysis by Louis Brand

Author:Louis Brand
Language: eng
Format: epub
Publisher: Dover Publications
Published: 2006-05-26T04:00:00+00:00

n being an external normal with respect to the region V. The uniform approach of r2f to zero means that

a suitably large value. If the radius r of ∑ is now chosen > R, we have

and hence the surface integral over ∑ approaches zero as r → ∞. At the same time V → V∞, and the theorem is proved.

Example 1. When f = r, div r = 3; then from (1)

where p is the perpendicular from O on the tangent plane to S at the end point of r. Thus the volume within S is expressed as a surface integral over S.

For a cone of base B, altitude h, take O at the vertex. Then p = 0 over the lateral surface, p = h over the base, and .

For a sphere of radius a, take O at the center. Then p = a and .

Example 2. Solid Angle. The rays from a point O through the points of a closed curve generate a cone; and the surface of a unit sphere about O intercepted by this cone is called the solid angle Ω 1 of the cone.

Let us apply the divergence theorem to the vector

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