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Unformatted text preview: Change of variables in the integral; Jacobian • Element of area in Cartesian system, dA = dxdy • We can see in polar coordinates, with x = r cos θ , y = r sin θ , r 2 = x 2 + y 2 , and tan θ = y / x , that dA = rdrd θ • In three dimensions, we have a volume dV = dxdydz in a Carestian system • In a cylindrical system, we get dV = rdrd θ dz • In a spherical system, we get dV = r 2 drd φ d (cos θ ) • We can find with simple geometry, but how can we make it systematic? • We can define the Jacobian to make this more straightforward and automatic Patrick K. Schelling Introduction to Theoretical Methods The Jacobian • In a Cartesian system we find a volume element simply from dV = dxdydz • Now assume x → x ( u , v , w ), y → y ( u , v , w ), and z → z ( u , v , w ) • We have in the Cartesian system d ~ r = ˆ idx + ˆ jdy + ˆ kdz • We can then find the total differentials dx , dy , and dz from dx = ∂ x ∂ u du + ∂ x ∂ v dv + ∂ x ∂ w dw dy = ∂ y ∂ u du + ∂ y ∂ v dv + ∂ y ∂ w dw dz = ∂ z ∂ u du + ∂ z ∂ v dv + ∂ z ∂ w dw Patrick K. Schelling Introduction to Theoretical Methods Jacobian continued • We can define ~ A to be along a direction such that dv = dw = 0, then in the Cartesian system ~ A = ˆ i ∂ x ∂ u + ˆ j ∂ y ∂ u + ˆ k ∂ z ∂ u du • Likewise ~ B will be along a direction with du = dw = 0, then in the Cartesian system we see, ~ B = ˆ i ∂ x ∂ v + ˆ j ∂ y ∂ v + ˆ k ∂ z ∂ v dv • Finally ~ C will be along a direction where du = dv = 0, then in the Cartesian system we see, ~ C = ˆ i ∂ x ∂ w + ˆ j ∂ y ∂ w + ˆ k ∂ z ∂ w dw Patrick K. Schelling Introduction to Theoretical Methods Jacobian continued • The volume element made by these vectors is dV = ~ A · ( ~ B × ~ C ), which is simply the determinant ∂ x ∂ u ∂ y ∂ u ∂ z ∂ u ∂ x ∂ v ∂ y ∂ v ∂ z ∂ v ∂ x ∂ w ∂ y ∂ w ∂ z ∂ w dudvdw = Jdudvdw • Here the determinant is the Jacobian J • We have to be careful! The J found above might be negative, so in general we take  J  • Notice also that we can interchange rows and columns (i.e. take the transpose) and the determinant is unchanged, so J = ∂ ( x , y , z ) ∂ ( u , v , w ) = ∂ x ∂ u ∂ x ∂ v ∂ x ∂ w ∂ y ∂ u ∂ y ∂ v ∂ y ∂ w ∂ z ∂ u ∂ z ∂ v ∂ z ∂ w Patrick K. Schelling Introduction to Theoretical Methods Example: Volume element in cylindrical coordinates • We know that dV = dxdydz in Cartesian coordinates, and also dV = rdrd cos θ dz in cylindrical coordinates, but let’s prove it!...
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 Spring '03
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 Vector Calculus, Polar coordinate system, Patrick K. Schelling

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