 ... ... @@ -133,13 +133,40 @@ understood: the area swept by an angle difference $d\varphi$ We find: $$\int_0^{2\pi} d\varphi \int_0^r_0 r dr = \\ 2\pi \int_0^r_0 r dr = 2 \pi \frac{1}{2} r_0^2 = \pi r_0^2, \int_0^{2\pi} d\varphi \int_0^r_0 r dr =\\ 2$$ 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 ... ...