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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 dz r The divergence is the flux per unit volume. If F = F r e r + F e + F z k , then the flux out of this cube is, approximately (using the linear approximation), Flux [( r + dr ) F r ( r + dr,,z )- rF r ( r,,z )] d dz + [ F ( r, + d,z )- F ( r,,z )] dr dz + [ F z ( r,,z + dz )- F z ( r,,z )] dr rd ( rF r ) r dr d dz + F dr d dz + F z z rdr d dz. Thus, the flux per unit volume is div F = 1 r r ( rF r ) + 1 r F + 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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