Commit ea6e722e authored by Michael Wimmer's avatar Michael Wimmer

add moe text

parent dc650409
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......@@ -133,13 +133,40 @@ understood: the area swept by an angle difference $d\varphi$
We find:
\int_0^{2\pi} d\varphi \int_0^r_0 r dr = \\
2\pi \int_0^r_0 r dr = 2 \pi \frac{1}{2} r_0^2 = \pi r_0^2,
\int_0^{2\pi} d\varphi \int_0^r_0 r dr =\\
which is indeed the area of a circle with radius 0.
## Cylindrical coordinates
## Converting derivatives between coordinate systems
Often, in physics important equations involve derivatives given in terms
of Cartesian coordinates. One prominent example are equations of the form
$$\left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2)\right)
f(x, y) = \ldots.$$
The derivative operator $\left(\frac{\partial^2}{\partial x^2} +
\frac{\partial^2}{\partial y^2)\right)$ is so common it has its own name:
the Laplacian (here for two-dimensional space).
Such an equation is universal, but for particular situations it might be
advantageous to use a different coordinate system, such as polar coordinates
for a system with rotational symmetry. The question then is: How does the
corresponding equation look like in a different coordinate system?
There are different ways to find the answer. Here, we will focus on
directly deriving the transformed equation through an explicit calculation
involving the chain rule for a function of several variables.
!!! info Chain rule for a multi-variable function
Let $f$ be a function of $n$ variables: $f(y_1, y_2, \ldots, y_n)$,
as well as $g_i(x_1, x_2, \ldots, x_n)$ for $i=1,2,\ldots, n$. Then
$$\frac{\partial}{\partial{x_i} = \sum_{j=1}^n
\fac{partial f}{\partial y_j} \frac{\partial g_j}{\partial x_i}$$
# Cylindrical coordinates
Three dimensional systems may have axial symmetry. An example is an
electrically charged wire of which we would like to calculate the
......@@ -172,7 +199,7 @@ $$d s^2 = d l^2 + d z^2 = d r^2 + r^2 d \varphi^2 + d z^2.$$
The volume element is consequently given as
$$dV = r dr d\varphi dz.$$
## Spherical coordinates
# Spherical coordinates
For problems with spherical symmetry, we use *spherical coordinates*.
These work as follows. For a point $\bf r$ in 3D space, we can specify
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