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Drdd depending on coordinate system dvol a ha dxa

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Unformatted text preview: sin θ ) ∂ er · ∂ r er + 1 eθ ∂r eθ + eφ · cos θeφ r s1 θ � ∂� � � � in r� cos θ r sin θ 2 �� � · eφ = (er · � ∂ � ∂ r )eφ er · ∂ r er � ∂� = �v �·� = ∂ vr ∂r ∂ � � ∂� ∂ θ )eθ + (eφ · ∂ φ )eφ 1 �∂� r + eφ ( ∂ φ · eφ ) = + (eθ · � + eθ · � + 2 vr + 1 ( ∂vθθ + r r∂ cos θ sin θ vθ ) + 1 r sin θ · 0 ∂ vφ ∂φ ∂ product rule w/ ∂ r product rule w/ ∂∂θ product rule w/ ∂∂φ Σa (ha dxa )ea � ea · eb = δab �� ⎧ ⎪dxdydz ⎪ ⎪ ⎪ ⎪ ⎪2 ⎨r sin θdrdθdφ (depending on coordinate system) d(vol) = Πa (ha dxa ) = ⎪.... ⎪ ⎪ ⎪ ⎪ ⎪ ⎩.... d� = x Jacobian... okay...?? ⎧ ⎪da�e = h dx · h · dx ⎪ || �a b b i i ⎪ ⎨ da = r2 dΩer � ⎪r ⎪ ⎪ ⎩dΩ = sin θdθdφ d(vol) = dxdydz = dvdudw ||J || eb × ei �� where ||J || is the Jacobian. Spherical C...
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This note was uploaded on 01/23/2014 for the course PHYS 8.022 taught by Professor Shaw during the Fall '06 term at MIT.

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