![]() The names used for the functions in the limits of integration pertain to just the cell in which a particular iteration is displayed. Table 7.6.1 In spherical coordinates, the six iterations of a triple integralĪs in Table 7.3.1, lower-case letters are used for lower limits of integration upper-case for upper limits. ∫ &rho = c &rho = C ∫ &phi = &phi &rho &phi = &Phi &rho ∫ &theta = &theta &rho, &phi &theta = &Theta &rho, &phi f &rho, &phi, &theta &rho 2 sin &phi d &theta d &phi d &rho ∫ &phi = b &phi = B ∫ &rho = &rho &phi &rho = &Rho &phi ∫ &theta = &theta &rho, &phi &theta = &Theta &rho, &phi f r, &phi, &theta &rho 2 sin &phi d &theta d &rho d &phi ∫ &rho = c &rho = C ∫ &theta = &theta &rho &theta = &Theta &rho ∫ &phi = &phi &rho, &theta &phi = &Phi &rho, &theta f &rho, &phi, &theta &rho 2 sin &phi d &phi d &theta d &rho ∫ &theta = a &theta = A ∫ &rho = &rho &theta &rho = &Rho &theta ∫ &phi = &phi &rho, &theta &phi = &Phi &rho, &theta f &rho, &phi, &theta &rho 2 sin &phi d &phi d &rho d &theta ∫ &phi = b &phi = B ∫ &theta = &theta &phi &theta = &Theta &phi ∫ &rho = &rho &phi, &theta &rho = &Rho &phi, &theta f &rho, &phi, &theta &rho 2 sin &phi d &rho d &theta d &phi ∫ &theta = a &theta = A ∫ &phi = &phi &theta &phi = &Phi &theta ∫ &rho = &rho &phi, &theta &rho = &Rho &phi, &theta f &rho, &phi, &theta &rho 2 sin &phi d &rho d &phi d &theta ![]() Table 7.6.1, analogous to Table 7.3.1 for Cartesian coordinates, lists the six possible iterations for a triple integral in spherical coordinates. (See Example 7.6.9 for the explicit calculation of this Jacobian.) The relevant Jacobian is &rho 2 sin &phi. X = &rho cos &theta sin &phi, y = &rho sin &theta sin &phi, z = &rho cos &phi In particular, for spherical coordinates defined by the equations Is the Jacobian of the transformation from R ′ to R. ∂ x, y, z ∂ a, b, c = &verbar x a x b x c y a y b y c z a z b z c &verbar Where R is a region in Cartesian xyz -space, R ′ is its image under the invertible mapping defined by the equations ∫ ∫ ∫ R f x, y, z dv = ∫ ∫ ∫ R ′ f x a, b, c, y a, b, c, z a, b, c ∂ x, y, z ∂ a, b, c dv ′ Extending this discussion to the triple integral leads to the "formula" Section 5.6 details changing coordinates in a double integral. Section 7.6: Integration in Spherical Coordinates
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