Editing Jacobian matrix and determinant (section)

=== Example 3: spherical-Cartesian transformation ===

The transformation from [[spherical coordinate system|spherical coordinates]] {{math|(''ρ'', ''φ'', ''θ'')}}<ref>Joel Hass, Christopher Heil, and Maurice Weir. ''Thomas' Calculus Early Transcendentals, 14e''. Pearson, 2018, p. 959.</ref> to [[Cartesian coordinate system|Cartesian coordinates]] (''x'', ''y'', ''z''), is given by the function {{math|'''F''': '''R'''<sup>+</sup> × [0, ''π'') × [0, 2''π'') → '''R'''<sup>3</sup>}} with components

<math display="block">\begin{align}
x &= \rho \sin \varphi \cos \theta ; \\
y &= \rho \sin \varphi \sin \theta ; \\
z &= \rho \cos \varphi .
\end{align}</math>

The Jacobian matrix for this coordinate change is

<math display="block">\mathbf J_{\mathbf F}(\rho, \varphi, \theta)
= \begin{bmatrix}
  \dfrac{\partial x}{\partial \rho} & \dfrac{\partial x}{\partial \varphi} & \dfrac{\partial x}{\partial \theta} \\[1em]
  \dfrac{\partial y}{\partial \rho} & \dfrac{\partial y}{\partial \varphi} & \dfrac{\partial y}{\partial \theta} \\[1em]
  \dfrac{\partial z}{\partial \rho} & \dfrac{\partial z}{\partial \varphi} & \dfrac{\partial z}{\partial \theta}
  \end{bmatrix}
= \begin{bmatrix}
      \sin \varphi \cos \theta & \rho \cos \varphi \cos \theta & -\rho \sin \varphi \sin \theta \\
      \sin \varphi \sin \theta & \rho \cos \varphi \sin \theta & \rho \sin \varphi \cos \theta \\
      \cos \varphi & - \rho \sin \varphi & 0
  \end{bmatrix}.</math>

The [[determinant]] is {{math|''ρ''<sup>2</sup> sin ''φ''}}. Since {{math|''dV'' {{=}} ''dx'' ''dy'' ''dz''}} is the volume for a rectangular differential volume element (because the volume of a rectangular prism is the product of its sides), we can interpret {{math|''dV'' {{=}} ''ρ''<sup>2</sup> sin ''φ'' ''dρ'' ''dφ'' ''dθ''}} as the volume of the spherical [[differential volume element]]. Unlike rectangular differential volume element's volume, this differential volume element's volume is not a constant, and varies with coordinates ({{math|''ρ''}} and {{math|''φ''}}). It can be used to transform integrals between the two coordinate systems:

<math display="block">\iiint_{\mathbf F(U)} f(x, y, z) \,dx \,dy \,dz = \iiint_U f(\rho \sin \varphi \cos \theta, \rho \sin \varphi\sin \theta, \rho \cos \varphi) \, \rho^2 \sin \varphi \, d\rho \, d\varphi \, d\theta .</math>