91Ó°ÊÓ

The Jacobian Determinant is a special function that helps us understand how areas or volumes transform under a change of variables. It's particularly useful in coordinate transformations of integrals.
When we switch from one set of variables to another, the Jacobian determinant tells us how an infinitesimal area changes. In simple terms, it shows the scale factor by which the area transforms.
To calculate the Jacobian determinant, we need partial derivatives of the new variables with respect to the old ones. For example, in our transformation equations, we have:

• Let’s consider the transformations: \( x = \frac{u+v}{2} \) and \( y = \frac{v-u}{2} \).
• The Jacobian matrix \( J \) for this transformation looks like this: \[ J = \begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{vmatrix} = \begin{vmatrix} \frac{1}{2} & \frac{1}{2} \ -\frac{1}{2} & \frac{1}{2} \end{vmatrix} \]
• When you find the determinant of this matrix, you calculate: \[ \frac{1}{2}*\frac{1}{2} - (-\frac{1}{2}*\frac{1}{2}) = \frac{1}{4} + \frac{1}{4} = \frac{1}{2} \]
• This tells us that the area in the new coordinate system is scaled by a factor of \frac{1}{2}.

Coordinate transformation is changing from one coordinate system to another. This technique simplifies complex integrals by making them easier to evaluate.
For instance, in our problem, we switched from \( (x, y) \) coordinates to \( (u, v) \) coordinates with the following transformations:

• \( u = x - y \)
• \( v = x + y \)

We then rewrite the limits of the integral according to these new variables. The original region \( 0 \le y \le 1 \) and \( 0 \le x \le 1 - y \) changes to:

• Bottom-left corner: \( (u,v) = (0,0) \)
• Top-right corner: \( (u,v) = (1,1) \)

With the transformed coordinates and limits, the integral is now easier to evaluate. The transformed integral becomes: \[ \int_{0}^{1} d v \int_{0}^{v} e^{\frac{u}{v}} \left( \frac{1}{2} \right) d u \]

Double integrals extend the concept of an integral to functions of two variables. They are used to compute volumes and areas.

• When you set up a double integral, you integrate over a two-dimensional region. This involves an inner and outer integral.
• In our example, the original double integral was, \[ \int_{0}^{1} d y \int_{0}^{1-y} e^{(x-y) /(x+y)} d x \].
• After changing variables, this becomes two nested integrals: one inside the other. The limits are much simpler after transformation.

By using new variables and simplifying limits, the double integral becomes: \[ \int_{0}^{1} d v \int_{0}^{v} e^{\frac{u}{v}} \left( \frac{1}{2} \right) d u \].
We can then first solve the inner integral:\[ \int_{0}^{v} e^{\frac{u}{v}} d u = v \left[ e^{\frac{u}{v}} \right]_{0}^{v} = v(e^{1} - 1) \].
And finally solve the outer integral:\[ \int_{0}^{1} v(e - 1) \left( \frac{1}{2} \right) d v = \frac{e-1}{2} \int_{0}^{1} v d v = \frac{e-1}{2} \left[\frac{v^2}{2} \right]_{0}^{1} = \frac{e-1}{4} \].
This approach shows how coordinate transformations and Jacobian determinants simplify double integrals.