Chapter 15.7.pdf - MA211 Week 13 LECTURES The Divergence...

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1 MA211 Week 13: LECTURES The Divergence Theorem
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3 The Divergence Theorem Let G be a solid whose surface σ is oriented outward. If F ( x, y, z ) = f ( x, y, z ) i + g ( x, y, z ) j + h ( x, y, z ) k where f, g, and h have continuous first partial derivatives on some open set containing G, and if n is the outward unit normal on σ , then Also called the Gauss’s Theorem . In this section we are going to relate surface integrals to triple integrals. We will do this with the Divergence Theorem.
Use the Divergence Theorem to find the outward flux of the vector field F ( x, y, z ) = z across the sphere x 2 + y 2 + z 2 = a Example 1 k 2 .
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5 Use the Divergence Theorem to find the outward flux of the vector field F ( x, y, z ) = 3 x i + 2 y j + z 2 k across the unit cube shown below.

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