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Mathematic Methods HW Solutions 51

Mathematic Methods HW Solutions 51 - 9.3 See 8.2 9.5 ∇ U...

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Unformatted text preview: Chapter 10 51 9.3 See 8.2 9.5 ∇ U = e r ∂U ∂r + e θ parenleftbigg 1 r ∂U ∂θ parenrightbigg + e φ parenleftbigg 1 r sin θ ∂U ∂φ parenrightbigg ∇ · V = 1 r 2 ∂ ∂r ( r 2 V r ) + 1 r sin θ ∂ ∂θ (sin θ V θ ) + 1 r sin θ ∂V φ ∂φ ∇ 2 U = 1 r 2 ∂ ∂r parenleftbigg r 2 ∂U ∂r parenrightbigg + 1 r 2 sin θ ∂ ∂θ parenleftbigg sin θ ∂U ∂θ parenrightbigg + 1 r 2 sin 2 θ ∂ 2 U ∂φ 2 ∇ × V = 1 r sin θ bracketleftbigg ∂ ∂θ (sin θ V φ )- ∂V θ ∂φ bracketrightbigg e r + 1 r sin θ bracketleftbigg ∂V r ∂φ- sin θ ∂ ∂r ( rV φ ) bracketrightbigg e θ + 1 r bracketleftbigg ∂ ∂r ( rV θ )- ∂V r ∂θ bracketrightbigg e φ 9.6 See 8.11 9.7 See 8.12 9.8 See 8.13 9.9 See 8.14 9.10 Let h = ( u 2 + v 2 ) 1 / 2 represent the u and v scale factors. ∇ U = h- 1 parenleftbigg e u ∂U ∂u + e v ∂U ∂v parenrightbigg + k ∂U ∂z ∇ · V = h- 2 bracketleftbigg ∂ ∂u ( hV u ) + ∂ ∂v ( hV v ) bracketrightbigg +...
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