PX153 - I7 - other coordinate systems

spherical coordinates

coordinate transformations:

\begin{aligned} x & = r \sin θ \cos ϕ \\ y & = r \sin θ \sin ϕ \\ z & = r \cos θ \end{aligned}

what is $d V$ ?
$d V = r^{2} \sin θ d θ d ϕ d r$

I = ∭_{V} f (\vec{r}) d V = ∭ f (r, θ, ϕ) r^{2} \sin θ d θ d ϕ d r

eg: what is the volume of the sphere with radius, $a$ ? $I = \int_{0}^{a} \int_{0}^{2 π} \int_{0}^{π} [f (\vec{r}) = 1] r^{2} \sin θ d θ d ϕ d r = \frac{4 π}{3} a^{3}$
aside : surface area of a sphere
- if the radius is increased from $a$ to $a + δ a$ , the volume of the sphere increases by the volume of the spherical shell with thickness $δ a$ , and area of the surface of a sphere: $Δ V = S . A . \times Δ a$ :
$S . A . = \frac{Δ V}{Δ a} = \frac{d}{d a} (\frac{4 π}{3} a^{3}) = 4 π a^{2}$

cylindrical coordinates

coordinate transformations:

\begin{aligned} x & = r \cos θ \\ y & = r \sin θ \\ z & = z \end{aligned}

what is $d V$ ?
$d V = r d r d θ d z$
eg: what is the volume of a cylinder with height, $L$ , and radius, $a$ ?

I = \int_{0}^{L} \int_{0}^{a} \int_{0}^{2 π} r d r d θ d z = π a^{2} L

2D polar coordinates

coordinate transformations:

\begin{aligned} x & = r \cos θ \\ y & = r \sin θ \end{aligned}

eg: integrate the gaussian function over all space.
- gaussian function:

f (x) = e^{- x^{2}}

I = \int_{- \infty}^{+ \infty} e^{- x^{2}} d x

- trick:

\begin{aligned} I^{2} & = \int_{- \infty}^{+ \infty} e^{- x^{2}} d x \int_{- \infty}^{+ \infty} e^{- y^{2}} d y \\ = \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} e^{- y^{2}} d y d x \end{aligned}

- switching to polar coordinates:

\begin{aligned} I^{2} & = \int_{0}^{\infty} \int_{0}^{2 π} r e^{- r^{2}} d θ d r \\ = 2 π \int_{0}^{\infty} r e^{- r^{2}} d r \\ = 2 π {[- \frac{1}{2} e^{- r^{2}}]}_{0}^{\infty} \\ = π \\ ∴ I & = \sqrt{π} \end{aligned}