The Jacobian Determinant: How exactly does it work?
<p>Named after German mathematician Carl Gustav Jacob Jacobi, the Jacobian determinant (or just the Jacobian) is required when evaluating multiple integrals after undergoing a change of variables.</p>
<p>To understand the Jacobian, it may help to consider the analogous case in single variable calculus where a change of variable is used. Consider we have the integral of some function <em>f</em>(<em>x</em>), and we wish to undergo the change of variables of <em>x = g</em>(<em>u</em>). Now, using the fact that <em>x = g</em>(<em>u</em>), we can get the following equation:</p>
<p>Which if rearranged will give <em>dx = g’</em>(<em>u</em>)<em>du</em>. Substituting this into the original integral, we will get the following:</p>
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