lec5-1 - Change of variables in the integral; Jacobian...

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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 , y = r sin , r 2 = x 2 + y 2 , and tan = y / x , that dA = rdrd In three dimensions, we have a volume dV = dxdydz in a Carestian system In a cylindrical system, we get dV = rdrd dz In a spherical system, we get dV = r 2 drd d (cos ) 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 = idx + jdy + 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 = i x u + j y u + k z u du Likewise ~ B will be along a direction with du = dw = 0, then in the Cartesian system we see, ~ B = i x v + j y v + k z v dv Finally ~ C will be along a direction where du = dv = 0, then in the Cartesian system we see, ~ C = i x w + j y w + 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 dz in cylindrical coordinates, but lets prove it!...
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lec5-1 - Change of variables in the integral; Jacobian...

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