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lecture7 notes - 8.022 Lecture Notes Class 7 So exactly...

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8.022 Lecture Notes Class 7 - 09/18/2006 So exactly what is curvilinear? And what’s this orthonormal stuff? (=1?) Gradient in Spherical Let f ( x ) = f ( r, θ, φ ) df = f dx · = f ( e r dr + e θ rdθ + e φ r sin φdφ ) · = dr + + dr So , 1 1 f = e r + e θ + e φ dr r dθ r sin θ Divergence Let v = v a e a where v a and A are scalars. ( B + A ( ( AB ) = A ) · · B ) v ) = ( ) e a + v a ( e a ) · ( v a · · e r = ( ) e r + ( 1 ) e r + ( e φ 1 ) e r · e r · ∂r e θ · r ∂θ r sin φ ∂φ e r = 0 e r = e θ e r = sin θe φ ∂θ 1 1 · e r = 0 + e θ · r · e θ + e φ · r sin θ · sin θe φ 1 1 2 = + = r r r = ( ) e θ + ( 1 ) e θ + ( 1 ) · e θ e r · r e θ · e φ · e θ r sin θ = e r + 1 e θ e θ + cos θe φ 1 e r · r e φ · r sin θ cos θ = r sin θ
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2 e φ = ( ) e φ + ( ) e θ + ( ) e φ · e r · ∂r e θ · ∂θ e φ · ∂φ = e r + 1 + e φ ( e φ ) = 0 e r · e θ · r · v = ∂v r + 2 1 ( θ + cos θ v θ ) + 1 φ · r v r + r sin θ r sin θ · product rule w/ product rule w/ product rule w/ dx = d ( vol ) = Π a ( h a dx a ) =
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lecture7 notes - 8.022 Lecture Notes Class 7 So exactly...

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