Commit 74c42e0c authored by Michael Wimmer's avatar Michael Wimmer

add missing homework

parent be7850ec
Pipeline #20839 passed with stages
in 51 seconds
......@@ -134,7 +134,7 @@ understood: the area swept by an angle difference $d\varphi$
$$
\int_0^{2\pi} d\varphi \int_0^{r_0} r dr =\\
2\pi \int_0^r_0 r dr = 2 \pi \frac{1}{2} r_0^2 = \pi r_0^2,
2\pi \int_0^{r_0} r dr = 2 \pi \frac{1}{2} r_0^2 = \pi r_0^2,
$$
which is indeed the area of a circle with radius 0.
......@@ -163,7 +163,7 @@ involving the chain rule for a function of several variables.
as well as $g_i(x_1, x_2, \ldots, x_n)$ for $i=1,2,\ldots, n$. Then
$$\frac{\partial}{\partial x_i} = \sum_{j=1}^n
\frac{partial f}{\partial y_j} \frac{\partial g_j}{\partial x_i}$$
\frac{\partial f}{\partial y_j} \frac{\partial g_j}{\partial x_i}$$
We start by replacing the function $f(x, y)$ by a function in polar coordinates
$f(r, \varphi)$, and ask what is $\frac{\partial}{\partial x} f(r, \varphi)$. When
......@@ -185,8 +185,8 @@ are two different functions from the fact that they use different coordinates.
With this information, we can now apply the chain rule:
$$ \frac{partial}{\partial x} f(r, \varphi) =
\frac{\partial f}{\partial r} \frac{partial r(x, y)}{\partial x} +
$$ \frac{\partial}{\partial x} f(r, \varphi) =
\frac{\partial f}{\partial r} \frac{\partial r(x, y)}{\partial x} +
\frac{\partial f}{\partial \varphi} \frac{\partial \varphi(x,y)}{\partial x}
$$
......@@ -344,7 +344,17 @@ Problems
3. Find the spherical coordinates of the points
$${\bf r} = (3/2, \sqrt{3}/2, 1).$$
2. [:smirk:] From the transformation from polar to Cartesian
2. [:grinning:] *Geometry and different coordinate systems}*
What geometric objects do the following boundary conditions describe?
1. $r<1$ in cylindrical coordinates,
2. $\varphi=0$ in cylindrical coordinates,
3. $r=1$ in spherical coordinates,
4. $\theta = \pi/4$ in spherical coordinates,
5. $r=1$ and $\theta=\pi/4$ in spherical coordinates.
3. [:smirk:] From the transformation from polar to Cartesian
coordinates, show that
$$\frac{\partial}{\partial x} = \cos\varphi \frac{\partial}{\partial r} - \frac{\sin\varphi}{r} \frac{\partial}{\partial \varphi}$$
and
......@@ -363,3 +373,20 @@ Problems
\frac{\partial}{\partial \vartheta}\left( \sin\vartheta \frac{\partial\psi(r,\vartheta, \varphi)}{\partial \vartheta}\right).$$
This is however even more tedious (you do not have to show this).
5. [:grinning:] *Integration and coordinates I*
Compute the area of the spherical cap defined by $r=r_0$ and $\theta <\theta_0$.
6. [:smirk:] *Integration and coordinates II*
In 2D we can define a shape by specifying a function $r(\varphi)$:
![image](figures/shape_polar.svg)
(Of course, here we need to have $r(0) = r(2\pi)$.)
Show that the area of this shape is given by
$$
\int_0^{2\pi} \frac{1}{2}\left[r(\varphi)\right]^2 d\varphi
$$
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