---
# System prepended metadata

title: HW6
tags: [Mechanics, ' ss', ' ncu']

---

---
tags: Mechanics, ncu, ss
author: N0-Ball
title: HW6
GA: UA-208228992-1
---

# HW6

[ToC]

## Preknowledge

**For Cylindrical Coordinate System**
$$
\begin{aligned}
\nabla = \hat r \frac{\partial}{\partial r} + \hat \theta \frac{1}{r}\frac{\partial}{\partial \theta} + \hat z \frac{\partial}{\partial z} 
\end{aligned}
$$
$$
\begin{aligned}
\frac{d \hat r}{d \theta} &= -sin(\theta) \hat x + cos(\theta) \hat y &= \hat \theta \\[1em]
\frac{d \hat \theta}{d \theta} &= -cos(\theta) \hat x + -sin(\theta) \hat y &= - \hat r
\end{aligned}
$$

**For Spherical Coordinate System**

$$
\nabla = \hat r \frac{\partial }{\partial r} + \hat \theta \frac{1}{r} \frac{\partial }{\partial \theta} + \hat \phi \frac{1}{r \sin \theta } \frac{\partial }{\partial \phi}
$$

$$
\begin{aligned}
\frac{\partial \hat r}{\partial \theta} &= -sin\theta\ \hat z + cos\theta\ (cos\phi\ \hat x + sin\phi\ \hat y) &= \hat \theta \\[1em]
\frac{\partial \hat r}{\partial \phi} &= sin\theta\ (-sin\phi \hat x + cos\phi\ \hat y) &= sin\theta\ \hat \phi \\[1em]
\frac{\partial \hat \theta}{\partial \theta} &= -cos\theta\ \hat z + -sin\theta\ (cos\phi\ \hat x + sin\phi\ \hat y) &= - \hat r\\[1em]
\frac{\partial \hat \theta}{\partial \phi} &= cos\theta\ (-sin\phi\ \hat x + cos\phi\ \hat y) &= cos\theta\ \hat \phi \\[1em]
\frac{d \hat \phi}{d \phi} &= -cos\phi\ \hat x - sin\phi\ \hat y \\[1em]
&= -sin\theta\ [cos\theta\ \hat z + sin\theta\ (cos\phi\ \hat x + sin\phi\ \hat y)] - cos\theta\ [-sin\theta\ \hat z + cos\theta\ (cos\phi\ \hat x + sin\phi\ \hat y)] &= -sin\theta\  \hat r - cos\theta\ \hat \theta
\end{aligned}
$$

## 7.

:::info
**Proof that**
$$
\nabla \cdot A(r) = \frac{\partial A_r}{\partial r} + \frac{A_r}{r} + \frac{1}{r}\frac{\partial A_\theta}{\partial \theta} + \frac{\partial A_z}{\partial z}
$$
:::

$$
\nabla \cdot A(r) = (\hat r \frac{\partial}{\partial r}  + \hat \theta \frac{1}{r}\frac{\partial}{\partial \theta} + \hat z \frac{\partial}{\partial z}) \cdot (A_r \hat r + A_\theta \hat \theta + A_z \hat z)
$$

**For $\hat r$**
$$
\begin{split}
&\hat r \cdot (\frac{\partial A_r}{\partial r} \hat r + A_r \frac{\partial \hat r}{\partial r} + \frac{\partial A_\theta}{\partial r} \hat \theta + A_\theta \frac{\partial \hat \theta}{\partial r} + \frac{\partial A_z}{\partial r} \hat z + A_z \frac{\partial \hat z}{\partial r}) \\[1em]
&= | \hat r |^2 \frac{\partial A_r}{\partial r} + \hat r \cdot \hat 0 A_r + \hat r \cdot \hat \theta \frac{\partial A_\theta}{\partial r} + \hat r \cdot \hat 0 A_\theta + \hat r \cdot \hat z \frac{\partial A_z}{\partial r} + \hat r \cdot \hat 0 A_z \\[1em]
&= 1 \frac{\partial A_r}{\partial r} + (0)A_r + (0) \frac{\partial A_\theta}{\partial r} + (0) A_\theta + (0) \frac{\partial A_z}{\partial r} + (0) A_z \\[1em]
&= \frac{\partial A_r}{\partial r}
\end{split}
$$

**Similary For $\hat \theta$**
$$
\begin{split}
&\frac{1}{r}[\hat \theta A_r \frac{\partial \hat r}{\partial \theta} + 0  + \hat \theta A_\theta \frac{\partial \hat \theta}{\partial \theta} + | \hat \theta | ^2 \frac{\partial A_\theta}{\partial \theta} + 0 + 0] \\[1em]
&= \frac{1}{r}[A_r \hat \theta \cdot \hat \theta + A_\theta \hat \theta \cdot (- \hat r) + 1 \frac{\partial A_\theta}{\partial \theta}] =  \frac{1}{r}[A_r (1) + 0 + \frac{\partial A_\theta}{\partial \theta}]\\[1em]
&= \frac{A_r}{r} + \frac{1}{r}\frac{\partial A_\theta}{\partial \theta}
\end{split}
$$

**Similary For $\hat z$**

$$
\begin{split}
&0 + 0 + 0 + 0 + 0 + \hat z \cdot \hat z \frac{\partial A_z}{\partial z} \\[1em]
&= \frac{\partial A_z}{\partial z}
\end{split}
$$

$\Rightarrow \nabla \cdot A(r) = \frac{\partial A_r}{\partial r} + \frac{A_r}{r} + \frac{1}{r}\frac{\partial A_\theta}{\partial \theta} + \frac{\partial A_z}{\partial z}$

## 8.

:::info
**Proof that**
$$
\nabla \times A(r) = (\frac{1}{r}\frac{\partial A_z}{\partial \theta} - \frac{\partial A_\theta}{\partial z}) \hat r + (\frac{\partial A_r}{\partial z} - \frac{\partial A_z}{\partial r}) \hat \theta + (\frac{\partial A_\theta}{\partial r} - \frac{1}{r}\frac{\partial A_r}{\partial \theta} + \frac{A_\theta}{r}) \hat z
$$
:::

$$
\nabla \times A(r) = (\hat r \frac{\partial}{\partial r} + \hat \theta \frac{1}{r}\frac{\partial}{\partial \theta} + \hat z \frac{\partial}{\partial z}) \times (A_r \hat r + A_\theta \hat \theta + A_z \hat z)
$$

**Similary to divergent**

**For $\hat r$**

$$
\begin{split}
&0 + 0 + \hat r \times \hat \theta \frac{\partial A_\theta}{\partial r} + 0 + \hat r \times \hat z \frac{\partial A_z}{\partial r} + 0 \\[1em]
&= (\hat z)\frac{\partial A_\theta}{\partial r} + (- \hat \theta) \frac{\partial A_z}{\partial r} \\[1em]
&= \frac{\partial A_\theta}{\partial r} \hat z -\frac{\partial A_z}{\partial r}  \hat \theta
\end{split}
$$

**For $\hat \theta$**

$$
\begin{split}
& \frac{1}{r}[\hat \theta A_r \frac{\partial \hat r}{\partial  \theta} + \hat \theta \times \hat r \frac{\partial A_r}{\partial \theta} + \hat \theta A_\theta \frac{\partial \hat \theta}{\partial \theta} + 0 + 0 + \hat \theta \times \hat z \frac{\partial A_z}{\partial \theta}] \\[1em]
&= \frac{1}{r}[A_r \hat \theta \times \hat \theta + (- \hat z)\frac{\partial A_r}{\partial \theta} + \hat \theta \times (- \hat r) A_\theta + (\hat r)\frac{\partial A_z}{\partial \theta}] \\[1em]
&= \frac{1}{r}[0 - \frac{\partial A_r}{\partial \theta} \hat z + A_\theta \hat z + \frac{\partial A_z}{\partial \theta} \hat r]\\[1em]
&= -\frac{1}{r}\frac{\partial A_r}{\partial \theta} \hat z + \frac{A_\theta}{r} \hat z + \frac{1}{r}\frac{\partial A_z}{\partial \theta}\hat r 
\end{split}
$$

**For $\hat z$**

$$
\begin{split}
&0 + \hat z \times \hat r \frac{\partial A_r}{\partial z} + 0 + \hat z \times \hat \theta \frac{\partial A_\theta}{\partial z} + 0 + 0 \\[1em]
&= (\hat \theta) \frac{\partial A_r}{\partial z} + (- \hat r) \frac{\partial A_\theta}{\partial z} \\[1em]
&= \frac{\partial A_r}{\partial z} \hat \theta - \frac{\partial A_\theta}{\partial z} \hat r
\end{split}
$$

$\Rightarrow \nabla \times A(r) = (\frac{1}{r}\frac{\partial A_z}{\partial \theta} - \frac{\partial A_\theta}{\partial z}) \hat r + (\frac{\partial A_r}{\partial z} - \frac{\partial A_z}{\partial r}) \hat \theta + (\frac{\partial A_\theta}{\partial r} - \frac{1}{r}\frac{\partial A_r}{\partial \theta} + \frac{A_\theta}{r}) \hat z$

## 11.

:::info
**Proof that**
$$
\nabla \cdot A(r) = \frac{\partial A_r}{\partial r} + \frac{2 A_r}{r} + \frac{\cos \theta A_\theta}{r \sin \theta} + \frac{1}{r}\frac{\partial A_\theta}{\partial \theta} + \frac{1}{r \sin \theta} \frac{\partial A_\phi}{\partial \phi}
$$
:::

$$
\nabla \cdot A(r) = (\hat r \frac{\partial }{\partial r} + \hat \theta \frac{1}{r} \frac{\partial }{\partial \theta} + \hat \phi \frac{1}{r \sin \theta } \frac{\partial }{\partial \phi}) \cdot (A_r \hat r + A_\theta \hat \theta + A_\phi \hat \phi)
$$

**Similary to cylindrical coordinate system**

**For $\hat r$**

$$
\begin{aligned}
&\hat r \cdot \hat r \frac{\partial A_r}{\partial r} + 0 + 0 + 0 + 0 + 0\\[1em]
&=1 \frac{\partial A_r}{\partial r} = \frac{\partial A_r}{\partial r}
\end{aligned}
$$

**For $\hat \theta$**
$$
\begin{aligned}
&\frac{1}{r}[0 + \hat \theta A_r \frac{\partial \hat r}{\partial \theta} + \hat \theta \cdot \hat \theta \frac{\partial A_\theta}{\partial \theta} + \hat \theta A_\theta \frac{\partial \hat \theta}{\partial \theta} + 0 + 0] \\[1em]
&=\frac{1}{r}[\hat \theta \cdot (\hat \theta) A_r + 1 \frac{\partial A_\theta}{\partial \theta} + \hat \theta \cdot (- \hat r) A_\theta] = \frac{1}{r} [1 A_r + \frac{\partial A_\theta}{\partial \theta}] \\[1em]
&= \frac{A_r}{r} + \frac{1}{r}\frac{\partial A_\theta}{\partial \theta}
\end{aligned}
$$

**For $\hat \phi$**
$$
\begin{aligned}
&\frac{1}{r \sin \theta}[0 + \hat \phi A_r \frac{\partial \hat r}{\partial \phi} + 0 + \hat \phi A_\theta \frac{\partial \hat \theta}{\partial \phi} + \hat \phi \cdot \hat \phi \frac{\partial A_\phi}{\partial \phi} + \hat \phi A_\phi \frac{\partial \hat \phi}{\partial \phi}] \\[1em]
&= \frac{1}{r \sin \theta}[\hat \phi \cdot \sin \theta \hat \phi A_r + \phi \cdot \cos \theta \hat \phi A_\theta + 1 \frac{\partial A_\phi}{\partial \phi} + \hat \phi \cdot (- \sin \theta \hat r - \cos \theta \hat \theta) A_\phi]\\[1em]
&= \frac{1}{r \sin \theta}[\sin \theta A_r + \cos \theta A_\theta + \frac{\partial A_\phi}{\partial \phi} + 0] \\[1em]
&= \frac{1}{r}A_r + \frac{\cos \theta A_\theta}{r \sin \theta} + \frac{1}{r \sin \theta}\frac{\partial A_\phi}{\partial \phi}
\end{aligned}
$$

$\Rightarrow \nabla \cdot A(r) = \frac{\partial A_r}{\partial r} + \frac{2 A_r}{r} + \frac{\cos \theta A_\theta}{r \sin \theta} + \frac{1}{r}\frac{\partial A_\theta}{\partial \theta} + \frac{1}{r \sin \theta} \frac{\partial A_\phi}{\partial \phi}$

## 12.

:::info
**Proof that**
$$
\begin{split}
\nabla \times A(r) &= (\frac{1}{r}\frac{\partial A_\phi}{\partial \theta} - \frac{1}{r \sin \theta} \frac{\partial A_\theta }{\partial \phi} + \frac{\cos \theta A_\phi}{r \sin \theta}) \hat r + (\frac{1}{r \sin \theta} \frac{\partial A_r}{\partial \phi} - \frac{\partial A_\phi}{\partial r} - \frac{A_\phi}{r}) \hat \theta \\[1em]
&+ (\frac{\partial A_\theta}{\partial r} - \frac{1}{r} \frac{\partial A_r}{\partial \phi} + \frac{\partial A_\theta}{\partial r}) \hat \phi
\end{split}
$$
:::

$$
\nabla \times A(r) = (\hat r \frac{\partial }{\partial r} + \hat \theta \frac{1}{r} \frac{\partial }{\partial \theta} + \hat \phi \frac{1}{r \sin \theta } \frac{\partial }{\partial \phi}) \times (A_r \hat r + A_\theta \hat \theta + A_\phi \hat \phi)
$$
  
**Similary to Divergent**

**For $\hat r$**

$$
\begin{split}
&0 + 0 + \hat r \times \hat \theta \frac{\partial A_\theta}{\partial r} + 0 + \hat r \times \hat \phi \frac{\partial A_\phi}{\partial r} + 0 \\[1em]
&= \frac{\partial A_\theta}{\partial r} \hat \phi - \frac{\partial A_\phi}{\partial r} \hat \theta
\end{split}
$$

**For $\hat \theta$**

$$
\begin{split}
& \frac{1}{r}[\hat \theta \times \hat r \frac{\partial A_r}{\partial \theta} + \hat \theta \times A_r \frac{\partial \hat r}{\partial \theta} + 0 + \hat \theta \times A_\theta \frac{\partial \hat \theta}{\partial \theta} + \hat \theta \times \hat \phi \frac{\partial A_\phi}{\partial \theta} + 0] \\[1em]
&= \frac{1}{r}[(- \hat \phi) \frac{\partial A_r}{\partial \theta} + \hat \theta \times (\hat \theta) A_r + \hat \theta \times (- \hat r) A_\theta + \hat r \frac{\partial A_\phi}{\partial \theta}] = \frac{1}{r}[- \hat \phi \frac{\partial A_r}{\partial \theta} + 0 + \hat \phi A_\theta + \hat r \frac{\partial A_\phi}{\partial \theta}] \\[1em]
&= -\frac{1}{r}\frac{\partial A_r}{\partial \theta} \hat \phi + \frac{A_\theta}{r} \hat \phi + \frac{1}{r}\frac{\partial A_\phi}{\partial \phi} \hat r
\end{split}
$$

**For $\hat \phi$**

$$
\begin{split}
& \frac{1}{r \sin \theta}[\hat \phi \times \hat r \frac{\partial A_r}{\partial \phi} + \hat \phi \times A_r \frac{\partial \hat r}{\partial \phi} + \hat \phi \times \hat \theta \frac{\partial A_\theta}{\partial \phi} + \hat \phi \times A_\theta \frac{\partial \hat \theta}{\partial \phi} + 0 + \hat \phi \times A_\phi \frac{\partial \hat \phi}{\partial \phi}] \\[1em]
&= \frac{1}{r \sin \theta}[\frac{\partial A_r}{\partial \phi} \hat \theta + A_r \hat \phi \times (\sin \theta \hat \phi) + (- \hat r) \frac{\partial A_\theta}{\partial \phi} + A_\theta \hat \phi \times (\cos \theta \hat \phi) + A_\phi \hat \phi \times (- \sin \theta \hat r - \cos \theta \hat \theta)] \\[1em]
&= \frac{1}{r \sin \theta}[\frac{\partial A_\theta}{\partial \phi} \hat \theta + 0 - \frac{\partial A_\theta}{\partial \phi} \hat r + 0 - A_\phi \sin \theta \hat \theta + A_\phi \cos \theta \hat r] \\[1em]
&= \frac{1}{r \sin \theta}\frac{\partial A_r}{\partial \phi} \hat \theta - \frac{1}{r \sin \theta}\frac{\partial A_\theta}{\partial r} \hat r - \frac{A_\phi}{r} \hat \theta + \frac{A_\phi \cos \theta}{r \sin \theta} \hat r
\end{split}
$$

$\Rightarrow \begin{split}
\nabla \times A(r) &= (\frac{1}{r}\frac{\partial A_\phi}{\partial \theta} - \frac{1}{r \sin \theta} \frac{\partial A_\theta }{\partial \phi} + \frac{\cos \theta A_\phi}{r \sin \theta}) \hat r + (\frac{1}{r \sin \theta} \frac{\partial A_r}{\partial \phi} - \frac{\partial A_\phi}{\partial r} - \frac{A_\phi}{r}) \hat \theta \\[1em]
&+ (\frac{\partial A_\theta}{\partial r} - \frac{1}{r} \frac{\partial A_r}{\partial \phi} + \frac{A_\theta}{r}) \hat \phi
\end{split}$
