Definition: (Jacobian) If  is the transformation from the  -plane to the  -plane defined by the equations  and  , then the Jacobian of is denoted by  or by  and is defined by
 ,
i.e. the determinant of the matrix of the partial derivatives (also known as the Jacobian matrix). Naturally, this can be generalized to more variables.
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Theorem: If the transformation , maps the region  in the -plane into the region  in the -plane, and if the Jacobian is nonzero and does not change sign on , then (with appropriate restrictions on the transformation and the regions) it follows that
 .
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Problem: Evaluate  , where is the region in the first quadrant enclosed by the trapezoid with vertices  .
Solution: The bounding lines can be written as  ,  ,  , and  . Now consider the transformation  and  . In the -plane, the bounding lines of the new region can now be written as  ,  ,  , and  .
We can write  and  as functions of  and  : simply  and  . So the Jacobian  .
Then our original integral becomes  . And this is equivalent to
![\displaystyle \frac{1}{2} \int_S \int e^{v/u} dA = \frac{1}{2} \int_1^4 \int_{-u}^u e^{v/u} dv du = \frac{1}{2} \int_1^4 \big[ u e^{v/u} \big]_{v=-u}^u du = \frac{1}{2} \int_1^4 u\left(e-\frac{1}{e}\right) du = \frac{15}{4}\left(e-\frac{1}{e}\right)](http://www.wangsblog.com/jeffrey//pictures/e5ffc613c9ce441f0774195e29fdc1e9.gif "\displaystyle \frac{1}{2} \int_S \int e^{v/u} dA = \frac{1}{2} \int_1^4 \int_{-u}^u e^{v/u} dv du = \frac{1}{2} \int_1^4 \big[ u e^{v/u} \big]_{v=-u}^u du = \frac{1}{2} \int_1^4 u\left(e-\frac{1}{e}\right) du = \frac{15}{4}\left(e-\frac{1}{e}\right)") .
QED.
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Comment: Note that the above theorem is probably very important in multivariable calculus, as it is the equivalent to -substitution in one variable, which we all know is the ultimate integration technique. It functions in the same way, giving you a lot more flexibility on the function you are integrating and the region you are integrating on.
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Practice Problem: Evaluate  , where is the rectangular region enclosed by the lines  ,  ,  ,  .
|
and 
, show that there is a continuous function 
that satisfies 
.
. This immediately gives
.
is a solution to this, as well as any constant multiple of it. Now try 
and 
, resulting in
.
. That’s pretty convenient.
.
does not always have a solution (take 
and 
for example). But that’s ok and I’ll worry about it some other time. In any case we have found a function for whenever the equation 
where 
.
.
,
.
.
, find an approximate local minimum.
to immediately give the solution. We seek an algorithm that will lead us to a local minimum (hopefully a global minimum as well).
. Consider the following process (for 
), known as gradient descent:
.
, where 
is a constant that can be determined by a 
oriented counterclockwise. If 
and 
are continuous and have continuous first partial derivatives on some open set containing 
.
, where 
and 
.
and 
. From these, we have 
and 
. Then we obtain
.
, so it remains a matter of calculation to get the answer. First, we evaluate the inner integral to get![\displaystyle \int_0^1 \int_{x^2}^{\sqrt{x}} (y^2-x^2) dy dx = \int_0^1 \left[\frac{y^3}{3}-x^2y\right]_{x^2}^{\sqrt{x}} dx = \int_0^1 \left(\frac{x^{3/2}}{3}-x^{5/2}-\frac{x^6}{3}+x^4\right)dx](http://www.wangsblog.com/jeffrey//pictures/c3332ec8db9094ba053721e2ebdcadb5.gif "\displaystyle \int_0^1 \int_{x^2}^{\sqrt{x}} (y^2-x^2) dy dx = \int_0^1 \left[\frac{y^3}{3}-x^2y\right]_{x^2}^{\sqrt{x}} dx = \int_0^1 \left(\frac{x^{3/2}}{3}-x^{5/2}-\frac{x^6}{3}+x^4\right)dx")
.![\displaystyle \int_0^1 \left(\frac{x^{3/2}}{3}-x^{5/2}-\frac{x^6}{3}+x^4\right)dx = \left[\frac{2x^{5/2}}{15}-\frac{2x^{7/2}}{7}-\frac{x^7}{21}+\frac{x^5}{5}\right]_0^1 = \frac{2}{15}-\frac{2}{7}-\frac{1}{21}+\frac{1}{5} = 0](http://www.wangsblog.com/jeffrey//pictures/d47fb50d217ebd922b1645f5847a9066.gif "\displaystyle \int_0^1 \left(\frac{x^{3/2}}{3}-x^{5/2}-\frac{x^6}{3}+x^4\right)dx = \left[\frac{2x^{5/2}}{15}-\frac{2x^{7/2}}{7}-\frac{x^7}{21}+\frac{x^5}{5}\right]_0^1 = \frac{2}{15}-\frac{2}{7}-\frac{1}{21}+\frac{1}{5} = 0")
.
and 
.
whose boundary is a piecewise smooth simple closed curve 
of 
and 
.