Problems

[1-1]
Prove the identities (A.11).

[1-2]
Using definition (A.14)and the expressions of $\nabla$, $\nabla \cdot$ and $\nabla \times$ in rectangular coordinates, verify the identity (A.12).

[1-3]
Use Stokes' theorem to prove that

$$\oint_S \nabla \times \underline{F} \cdot \hat{n} \, dS=0 \space ,$$ (2.70)

where $S$ is any closed surface with outward normal $\hat{n}$ and $\underline{F}$ is a vector field.

[1-4]
Prove the following relations involving unit vectors in rectangular and cylindrical coordinates:

$$\hat{\rho} =\hat{x} \cos \varphi+ \hat{y} \sin \varphi\space , \quad \hat{\varphi}=-\hat{x} \sin \varphi+ \hat{y} \cos \varphi\space ,$$

$$\hat{x} = \hat{\rho} \cos \varphi-\hat{\varphi} \sin \varphi\space , \quad \hat{y}= \hat{\rho} \sin \varphi+ \hat{\varphi} \cos \varphi\space ,$$

$$\nabla \cdot \hat{\rho} = 1/\rho \space , \quad \nabla \cdot \hat{\varphi} = \nabla \cdot \hat{z} = 0 \space ,$$

$$\nabla \times \hat{\rho} = \nabla \times \hat{z} = 0 \space , \quad \nabla \times \hat{\varphi} = \hat{z} / \rho \space .$$

[1-5]
Prove the following relations involving unit vectors in rectangular and spherical coordinates:

$$\hat{r} =\hat{x} \sin\theta \cos\varphi+ \hat{y} \sin\theta \sin\varphi+ \hat{z} \cos\theta \space ,$$

$$\hat{\theta} = \hat{x} \cos \theta\cos \varphi+ \hat{y} \cos \theta\sin\varphi- \hat{z} \sin \theta \space ,$$

$$\hat{\varphi} =-\hat{x} \sin\varphi+ \hat{y} \cos \varphi\space ,$$

$$\hat{x} =\hat{r} \sin\theta \cos\varphi+\hat{\theta}\cos\theta\cos\varphi-\hat{\varphi}\sin\varphi\space ,$$

$$\hat{y} =\hat{r}\sin\theta\sin\varphi+\hat{\theta}\cos\theta\sin\varphi+\hat{\varphi}\cos\varphi\space ,$$

$$\hat{z} =\hat{r}\cos\theta-\hat{\theta}\sin\theta \space ,$$

$$\nabla \cdot \hat{r} = 2/r \space , \quad \nabla \cdot \varphi=0 \space , \quad \nabla \cdot \hat{\theta}=\frac{1}{r \tan \theta} \space ,$$

$$\nabla \times \hat{r} = 0 \space , \quad \nabla \times \hat{\varphi} = \frac{\hat{r}}{r \tan \theta} -\frac{\hat{\theta}}{r} \space , \quad \nabla \times \hat{\theta}=\frac{\hat{\varphi}}{r}$$

[1-6]
Prove the following relations involving unit vectors in cylindrical and spherical coordinates:

$$\hat{r} =\hat{\rho}\sin\theta+\hat{z}\cos\theta \space , \quad \hat{\theta} =\hat{\rho} \cos \theta-\hat{z}\sin\theta \space ,$$

$$\hat{\rho} =\hat{r} \sin\theta+\hat{\theta}\cos\theta \space , \quad \hat{z} =\hat{r}\cos\theta-\hat{\theta}\sin\theta \space .$$

[1-7]
Find the gradient and the laplacian of the following functions:

$$\cos(\omega t -kz) \space ,$$

$$\sin(\alpha x)\sin(\beta y)\sin(\gamma z) \space ,$$

$$f(\rho,z) \sin ( m \varphi) \space ,$$

$$r^n \quad (n>\geq 0)\space ,$$

$$\frac{e^{-jkr}}{r} \quad (\mathrm{note:} \nabla^2\frac{1}{r}=-4\pi\delta(\underline{r})) \space .$$

[1-8]
Find the divergence and the curl of the following functions:

$$\hat{x} \cos(\omega t -kz) \space , \quad f(\underline{r})\underline{F}(\underline{r}) \space , \quad \underline{F}(\rho,z) \sin(m \varphi) \space ,$$

$$\hat{r} \times \underline{F}(\underline{r}) \space , \quad \hat{\theta}\times\underline{F}(\underline{r}) \space , \quad \hat{\varphi} \times \underline{F}(\underline{r}) \space ,$$

$$\hat{r}f(\underline{r}) \space , \quad \hat{r}\frac{e^{-j k r}}{r} \space , \quad \hat{y}\sin(\pi x)+\hat{z} \cos (\pi x) \space .$$

[1-9]
Derive (A.24) through (A.27) from the definitions of $\nabla$, $\nabla \cdot$, $\nabla \times$ and $\nabla^2$ in rectangular coordinates, using the transformation (A.17) and (A.18) between rectangular and circular cylindrical coordinates.

[1-10]
Derive (A.36) through (A.39) from the definitions of $\nabla$, $\nabla \cdot$, $\nabla \times$ and $\nabla^2$ in rectangular coordinates, using the transformation (A.29) and (A.30) between rectangular and spherical polar coordinates.

[1-11]
Derive the expressions (A.24) through (A.27) and (A.36) through (A.39) as particular cases of the general formulas given in Section A.4.3.

In the following problems, $\underline{a}(t)$ and $\underline{b}(t)$ vary sinusoidally with time at the same frequency $\omega$; their phasors are $\underline{A}$ and $\underline{B}$, respectively.

[1-12]
Prove that the time averaged quantity

$$\langle \underline{a}(t) \times \underline{b}(t) \rangle=\frac{1}{2} \mathrm{Re}\left\{ \underline{A} \times \underline{B}^* \right\} \space .$$ (2.71)

[1-13]
Find the time-domain counterparts of $\underline{A}=x+jy$ and $ul{B}=x-jy$.

[1-14] Find the phasors corresponding to

$$\cos\left(\omega t \pm \frac{\pi}{4}\right) \space , \quad \alpha \sin\omega t + \beta \cos \omega t \space ,$$

$$\sin \left( \omega t \pm \frac{\pi}{2} \right) \space , \quad \sin \left( \omega t + k x - 120 ^{\circ} \right) \space .$$

[1-15]
Find the time-domain counterparts of

$$3\pm4j \space , \quad \exp\left(-j \frac{\pi}{3} \right) \space , \quad 20e^{j (2x+y)} \space ,$$

$$e^{- j \underline{k} \cdot \underline{r} } \mbox{where $\underline{k}$\space is a constant vector}$$