[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 $y$.

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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 \[ \left\lvert\pd{(X,Y,Z)}{(x,y,z)}\right\rvert = \text{det}A. \]

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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 $3/2 \pi a^2$.

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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.

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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?

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