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Divergence Theorem and Variations

# Divergence Theorem and Variations - Divergence Theorem and...

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Divergence Theorem and Variations Theorem. If V is a solid with surface ∂V ii F · ˆ n dS = iii V ∇ · F dV f ˆ n dS = V f dV ˆ n × F dS = V ∇ × F dV where ˆ n is the outward unit normal of . Memory Aid. All three formulae can be combined into ˆ n ˜ F dS = V ∇ ∗ ˜ F dV where can be either · , × or nothing. When = · or = × , then ˜ F = F . When is nothing, ˜ F = f . Proof: The Frst formula is the divergence theorem and was proven in class. To prove the second formula, assuming the Frst, apply the Frst with F = f a , where a is any constant vector. f a · ˆ n dS = V ∇ · ( f a ) dV = V b ( f ) · a + f ∇ · a B dV = V ( f ) · a dV To get the second line, we used vector identity # 8. To get the third line, we just used that a is a constant, so that it is annihilated by all derivatives. Since a is a constant, we can factor it out of both integrals, so a · f ˆ n dS = a · V f dV = a · ±ii f ˆ n dS V f dV ² = 0 March 3, 2013 Divergence Theorem and Variations 1

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