View source for Jacobian

==Multiple integration==
It can be proved <ref>T. M. Apostol, ''Mathematical Analysis'', Addison-Wesley, 2nd ed. (1974), sec. 15.10</ref> that
:<math>
\int_{\mathbf{f}(\mathbf{t})} \phi(\mathbf{x})\; \mathrm{d}\mathbf{x} 
=\int_T \phi\big(\mathbf{f}(\mathbf{t})\big)\; \mathbf{J}_\mathbf{f}(\mathbf{t})\;\mathrm{d}\mathbf{t}.
</math>
As an example we consider the spherical polar coordinates mentioned above. Here '''x''' = '''f'''('''t''') &equiv; '''f'''(''r'', &theta;, &phi;) covers all of <math>\mathbb{R}^3</math>, while ''T'' is the region {''r'' > 0, 0 < &theta;<&pi;, 0 <&phi; <2&pi;}. Hence the theorem states that
:<math>
\iiint\limits_{\mathbb{R}^3} \phi(\mathbf{x})\; \mathrm{d}\mathbf{x} =
\int\limits_{0}^\infty \int\limits_0^\pi \int\limits_0^{2\pi} \phi\big(\mathbf{x}(r,\theta,\phi)\big)\; r^2\sin\theta \;
\mathrm{d}r \mathrm{d}\theta \mathrm{d}\phi .
</math>