Examples8.4

Examples8.4 - 1 Example 1 Find a formula for the divergence...

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Unformatted text preview: 1 Example 1. Find a formula for the divergence of a vector field F in cylindrical coordinates. Solution. We proceed along the same lines as the discussion in the text at the end of § 8.4. Consider the situation of the Figure. x y z dr rd θ dz θ r The divergence is the flux per unit volume. If F = F r e r + F θ e θ + F z k , then the flux out of this cube is, approximately (using the linear approximation), Flux ≈ [( r + dr ) F r ( r + dr,θ,z )- rF r ( r,θ,z )] dθ dz + [ F θ ( r,θ + dθ,z )- F θ ( r,θ,z )] dr dz + [ F z ( r,θ,z + dz )- F z ( r,θ,z )] dr · rdθ ≈ ∂ ( rF r ) ∂r dr dθ dz + ∂F θ ∂θ dr dθ dz + ∂F z ∂z rdr dθ dz. Thus, the flux per unit volume is div F = 1 r ∂ ∂r ( rF r ) + 1 r ∂F ∂θ + ∂F z ∂z . ♦ Example 2. (a) Use Gauss’ theorem to show that ZZ S 1 ( ∇ × F ) · n dS = ZZ S 2 ( ∇ × F ) · n dS, where S 1 and S 2 are two surfaces having a common boundary....
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Examples8.4 - 1 Example 1 Find a formula for the divergence...

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