Page 427 - Advanced engineering mathematics

12.8 The Divergence Theorem of Gauss 407

2
Here we have introduced the symbol ∇ , defined by
2
2
2
∂ u ∂ u ∂ u
2
∇ u = + + .
∂x 2 ∂y 2 ∂z 2
2
∇ is called the Laplacian,or Laplacian operator, and is read “del squared.”

Now the heat equation is
∂u 2
μρ =∇K ·∇u + K∇ u.
∂t
If K is constant then its gradient vector is zero and this equation simplifies to
∂u K
2
= ∇ u.
∂t μρ
In the case of one space dimension, u = u(x,t) and we often write this equation as
2
∂u ∂ u
= k
∂t ∂x 2
in which k = K/μρ.

The steady-state heat equation occurs when ∂u/∂t = 0. In this case we get ∇ u =0, which
2
is called Laplace’s equation.

SECTION 12.8 PROBLEMS

In each of Problems 1 through 8, evaluate either F · ndσ planes x = 0, y = 0, z = 0 and the planes x = 4,

y = 2, z = 3.
or div(F)dV , whichever is easier.
2
2
2
2
2
M 7. F = x i + y j + z k, is the cone z = x + y for
2
2
x + y ≤ 2, together with the top cap consisting of
√
1. F = xi + yj − zk, is the sphere of radius 4 about the points (x, y, 2) with x + y ≤ 2.
2
2
(1,1,1).
z
2
8. F = x i − e j + zk, is the surface bounding the
2. F=4xi−6yj+ zk, is the surface of the solid cylin- cylinder x + y ≤ 4,0 ≤ z ≤ 2, including the top and
2
2
2
2
der x + y ≤ 4,0 ≤ z ≤ 2, including the end caps of bottom caps of the cylinder.
the cylinder.
9. Let be a smooth closed surface and F a vector
3. F = 2yzi − 4xzj + xyk, is the sphere of radius 5 field that is continuous with continuous first and sec-
about (−1,3,1) ond partial derivatives throughout and the region it

3
3
3
4. F = x i + y j + z k, is the sphere of radius 1 about bounds. Evaluate (∇× F) · ndσ.
the origin. 10. Let be a piecewise smooth closed surface bounding
2 2 2 a region M. Show that
5. F=4xi− zj+ xk, is the hemisphere x + y + z =
1, z ≥ 0, including the base consisting of points 1
2
2
(x, y,0) with x + y ≤ 1. volume of M = R · ndσ
3
6. F = (x − y)i + (y − 4xz)j + xzk, is the surface
of the rectangular box bounded by the coordinate where R = xi + yj + zk.
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October 14, 2010 14:53 THM/NEIL Page-407 27410_12_ch12_p367-424