1.) Read first page of
Calculating area/volume
and take a
look at the second page while listening to the videos below. Note the time of each video is given in parenthesis. For example, the video in item (2) is 7 minutes 30 seconds.
|*| is used for both determinant and absolute value.
Parallelograms are used to approximate $\Delta A$ = area of "boxes").
At 5:13, he writes the transpose of the Jacobian matrix. But taking the transpose does not change the determinant, so if you just need the determinant, it doesn't matter if your rows correspond to functions $x(u, v)$ and $y(u, v)$ or if your columns correspond to these functions.
3.) View some of the videos listed below to see some examples. Note in these examples u and v are chosen in order to simplify the region over which one is integrating. In some cases this also simplifies the function, but often the main goal in multivariable calculus is to do a change of variable to simplify the region over which one is integrating.
In the ellipse examples, one is asked to integrate over an elliptical region:
$\frac{(x - c)^2}{a^2} +\frac{(y - d)^2}{b^2} \leq 1$
One can change this into integrating over a circular region: $u^2 + v^2 \leq 1$
by letting $u = \frac{(x - c)}{a}$ and $v = \frac{(y - d)}{b}$
Note to calculate the Jacobian, $\frac{\partial(x, y)}{\partial(u, v)}$, we solve for $x$ and $y$ in terms of $u$ and $v$.
Option 1:
View Jacobians II: Two examples, L. Sadun (7:21). He does a nice job explaining how to choose $u$ and $v$ when one is asked to integrate over a region which is a parallelogram.
Option 2:
If you prefer the examples at a slower pace where algebra steps are not skipped, you can instead listen to the following two videos after listening to the first couple of minutes of Jacobians II (and all of Jacobians 1):
Example 1: Ellipse, Mathispower4u (9:38) and
Example 2, Mathispower4u (7:59)
Note the region in example 2 is a parallelogram with sides $<0, 4>$ and $<2, 2>$ (you can find these vectors by looking at the graph and finding where $x = 0$ and $y = x + 4$ intersect and where $x = 2$ and $y = x$ intersect)
Thus,
$< x, y> = u<2, 2> + v<0, 4>$. Note that if $ 0 \leq u, v \leq 1$, then $< x, y>$ must lie within the parallelogram with sides $<2, 2>$ and $<0, 4>$.
Hence $x = 2u$ and $y = 2u + 4v$.
Note to calculate the Jacobian, $\frac{\partial(x, y)}{\partial(u, v)}$, we need $x$ and $y$ in terms of $u$ and $v$. In the parallelgram examples, it was easy to get $x$ and $y$ in terms of $u$ and $v$. But in many cases, we first determine $u$ and $v$ in terms of $x$ and $y$, and then solve for $x$ and $y$ in terms of $u$ and $v$ such as in the ellipse examples.
4.) Reread page 2 of
Calculating area/volume
to see how the 2-dimensional case easily extends to the 3-dimensional case.