[Multivariable Calculus] The Jacobian

1. Jacobian computation

Recall the definition of parabolic planar co-ordinates $x = \frac{1}{2}(u^2 - v^2)$ and $y = uv$. Find $|\pd{(u,v)}{(x,y)}|$ in terms of $x$ and $y$.


2. Linear transformations

If the co-ordinates $(X,Y,Z)$ are related to $(x,y,z)$ by the matrix transformation $(X,Y,Z)^T = A(x,y,z)^T$ then show that \[ \left\lvert\pd{(X,Y,Z)}{(x,y,z)}\right\rvert = \text{det}A. \]


3. Area of Cardioid

Let $r$ and $\theta$ denote polar co-ordinates and $a > 0$ is constant. The curve with polar equation \[ r = a(1 + \cos\theta), \quad 0 \leq \theta \leq 2\pi \]

is called a cardioid. Sketch the curve and show that the area bounded by it equals $3/2 \pi a^2$.


4. Computing an area

First, let $a > 0$ and sketch the curve \[ x^{2/3} + y^{2/3} = a^{2/3} \]

in the first quadrant. Next, by using the transformation $x = u\cos^3 v$ and $v = u\sin^3 v$, determine the equation of the curve in terms of the new coordinates, calculate the Jacobian, and then fine the area underneath the curve in the first quadrant.


5. Elliptical coordinates

Evaluate the integral \[ \iint_D \left[ 3 - \frac{1}{2}\left( \frac{x^2}{a^2} + \frac{y^2}{b^2}\right)\right] \, \de{x} \, \de{y}, \]

where $D$ is the region $x^2/a^2 + y^2/b^2 \leq 4$. What does this integral represent in terms of a volume beneath a surface?