202104181519 Homework 1 (Q4)

i. Find the infinitesimal small vector $\mathbf{dr}$ in the cylindrical coordinate induced by an infinitesimal small changes of $\mathrm{d}\rho$ , $\mathrm{d}\theta$ , and $\mathrm{d}z$ in terms of $\rho$ , $\theta$ , $z$ , $\mathrm{d}\rho$ , $\mathrm{d}\theta$ , $\mathrm{d}z$ and the corresponding unit vector.

ii. $f(u_1, u_2, u_3)$ is defined in $\mathbf{r}=(u_1, u_2, u_3)$ coordinate. Its gradient is defined

$\displaystyle{\lim_{\Delta l_i\to 0}\sum_{i=1}^{3}\frac{\Delta f_i}{\Delta l_i}\hat{\mathbf{u}}_l}$

where $\Delta l_i$ and $\Delta f_i$ are respectively the changes in length and functional value induced purely by the infinitesimal change in $u_i$ . $\hat{\mathbf{u}}_l$ is the unit vector of $\mathbf{u}_i$ . Thus find the gradient of $f$ in cylindrical coordinate.

Solution.

(The solution below is based on the manuscript of 2015-2016 PHYS2155 Methods of Physics II Homework Solutions.)

i.

$\mathbf{dr}=\mathrm{d}\rho\,\hat{\boldsymbol{\rho}}+\rho\,\mathrm{d}\theta\,\hat{\boldsymbol{\theta}}+\mathrm{d}z\,\hat{\mathbf{z}}$

Compare to the figure below.

ii.

$\begin{aligned} \nabla f & = \lim_{\Delta l_i\to 0}\sum_{i=1}^{3}\frac{\Delta f_i}{\Delta l_i}\hat{\mathbf{u}_i} \\ & = \lim_{\Delta\rho\to 0} \frac{\Delta f_\rho}{\Delta \rho}\,\hat{\boldsymbol{\rho}} + \lim_{\Delta\theta\to 0} \frac{\Delta f_\theta}{\rho\Delta\theta}\,\hat{\boldsymbol{\theta}} + \lim_{\Delta z\to 0}\frac{\Delta f_z}{\Delta z}\,\hat{\mathbf{z}} \\ & = \frac{\partial f}{\partial \rho}\,\hat{\boldsymbol{\rho}} + \frac{1}{\rho}\frac{\partial f}{\partial \theta}\,\hat{\boldsymbol{\theta}} + \frac{\partial f}{\partial z}\,\hat{\mathbf{z}} \end{aligned}$

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