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MATH1011 - Lecture 16
Skeleton Slides
\left| \det\left( \frac{\partial \mathbf{g}(u,v)}{\partial(u,v)} \right) \right| \, du\, dv,$$
where the substitution is:
$$\begin{aligned} (x, y) = \mathbf{g}(u,v) = (\phi(u,v), \psi(u,v)), \\ \\ dx\,dy = \left| \det\left( \frac{\partial \mathbf{g}(u,v)}{\partial(u,v)} \right) \right| \, du\,dv, \\ \\ S = \mathbf{g}^{-1}(R) \end{aligned}$$
### Change of Coordinates in Triple Integrals
$$\iiint\limits_R f(x, y, z) \, dx\, dy\, dz = \iiint\limits_S f(\mathbf{g}(u,v, w))
\left| \det\left( \frac{\partial \mathbf{g}(u,v, w)}{\partial(u,v,w)} \right) \right| \, du\, dv\, dw,$$
where the substitution is:
$$\begin{aligned} (x, y, z) = \mathbf{g}(u,v,w) = (\phi(u,v,w), \psi(u,v, w)), \\ \\ dx\,dy\, dz = \left| \det\left( \frac{\partial \mathbf{g}(u,v,w)}{\partial(u,v,w)} \right) \right| \, du\,dv\, dw, \\ \\ S = \mathbf{g}^{-1}(R) \end{aligned}$$
### Polar Coordinates for Double Integrals
- Each point $(x, y)$ in the plane (apart from $(0, 0)$) can be described uniquely by a pair of polar coordinates $r, θ$ where: $r = |(x, y)|$, $θ$ is the angle between $(x, y)$ and $x ≥ 0$, we choose $0 ≤ θ < 2π$
- $x = r\cos\theta$
- $y = r\sin\theta$
$$\iint\limits_R f(x, y) \, dx\, dy = \iint\limits_S f(r\cos\theta,r\sin\theta)\textcolor{#00ddff}{r}\, dr\, d\theta$$
- **Tip:** Integrate over the angle $\theta$ first!
### Cylindrical Coordinates
- Adding on to polar coordinates, we can describe any point $(x, y, z)$ in space by a triple of *cylindrical coordinates* $r, θ, ξ$
- $ξ$ just adds a vertical component onto our polar coordinate
### Spherical Coordinates
- Each point (x, y, z) in space (except for the $z$-axis) can be described uniquely by a triple of *spherical coordinates* $ρ, θ, \phi$
- $x = ρ \cos θ \sin \phi$
- $y = ρ \sin θ \sin \phi$
- $z = ρ \cos \phi$
- Example diagram:
![[SphericalCoordinates.png|450]]
<font color="#ffff00">The Jacobian</font> will be equal to $ρ^2 \sin \phi$
> If the region $R$ of integration has a spherical symmetry around $(0, 0, 0)$ (for instance sphere, wedge of sphere) using spherical coordinates will often simplify the computations a lot
- Examples of using these coordinate changes can be found in the [[MATH1011-Wk9 (Annotated).pdf#page=55|slides]]
> Remember for a sphere: $0 \leq \phi \leq \pi$, not $2\pi$!
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> [[MATH1011-Wk9 (Annotated).pdf|Annotated Slides]]