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Chapter 1: Vector Analysis Problem 1.1: Using the definitions in Eqs. 1.1 and 1.4, and appropriate diagrams, show that the dot product and cross product are distributive, a) when the three vectors are co-planar. b) in the general case.  Solution   Problem 1.2 Is the cross product associative? $$(\vec{A}\times \vec{B}) \times \vec{C} \overset{?}{=} \vec{A}\times (\vec{B} \times \vec{C})$$ If so, prove it; if not, provide a counterexample (the simpler the better). Solution   Problem 1.3 Find the angle between the body diagonals of a cube.  Solution Problem 1.4 Use the cross product to find the components of the unit vector $\hat{n}$ perpendicular to the shaded plane in Fig. 1.11. Solution   Problem 1.5 Prove the BAC-CAB rule by writing out both sides in component form. Solution Problem 1.6 Prove that $$[\vec{A}\times (\vec{B}\times \vec{C})]+[\vec{B}\times (\vec{C}\times \vec{A})]+[\vec{C}\times (\vec{A}\times \vec{B})] = 0$$ Under what conditions does $\vec{A}\times (\vec{B}\times \v
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Chapter 1 Vector Analysis: Problem 1.54

Problem 1.54: Check the divergence theorem for the function
$$v = r^2\cos\theta\hat{r} + r^2\cos \phi \hat{θ} − r^2 \cos\theta \sin \phi \hat{\phi}$$ ,
using as your volume one octant of the sphere of radius $R$ (Fig. 1.48). Make sure
you include the entire surface. 





 

 

Solution: 

Divergence Theorem States That for a verctor $\vec{v}$,

$$\int_V (\vec{\triangledown}\cdot \vec{v})d\tau = \oint_S \vec{v}\cdot d\vec{a}$$

Let's consider the octant where $r$ varies from $0$ to $R$, $\theta$ varies from $0$ to $\pi\over 2$ and $\phi$ varies from $0$ to $\pi\over 2$ 

Given $$\vec{v} = r^2cos\theta \hat{r} + r^2 cos\phi\hat{\theta} - r^2 cos\theta sin\phi \hat{\phi} $$ $$\vec{\triangledown}\cdot \vec{v} = \frac{1}{r^2 sin\theta}\big[\frac{\partial}{\partial r}(r^2sin\theta \cdot r^2 cos\theta) + \frac{\partial}{\partial \theta}(rsin\theta \cdot r^2 cos\phi) + \frac{\partial}{\partial \phi} (r\cdot (-r^2 cos\theta sin\phi))\big]$$ $$\vec{\triangledown}\cdot \vec{v} = \frac{1}{r^2 sin\theta}\big[ sin\theta cos\theta 4r^3+ r^3cos\phi cos\theta -r^3 cos\theta cos\phi\big]$$ $$\vec{\triangledown}\cdot \vec{v} = 4rcos\theta$$ $$\int_{vol.}(\vec{\triangledown}\cdot \vec{v}) d\tau = \int_{vol.} 4rcos\theta \cdot r^2 sin\theta dr d\theta d\phi $$ $$\int_{vol.}(\vec{\triangledown}\cdot \vec{v}) d\tau = 4\times \int_0^R r^3 dr\times \int_0^{\pi \over 2} cos\theta sin\theta d\theta \times \int_0^{\pi \over 2} d\phi$$ $$\int_{vol.}(\vec{\triangledown}\cdot \vec{v}) d\tau = 4\times \frac{R^4}{4}\times \frac{1}{2} \times \frac{\pi}{2} = \frac{1}{4} \pi R^4$$ Divide the surface into 4 parts where $S_1$ represents the part on the on the $xy$ plane. Hence $\theta = \pi/2$ $$\int_{S_1} \vec{v}\cdot \vec{da} = \int_{S_1} (r^2 cos\phi \hat{\theta})\cdot (r dr d\phi \hat{\theta})$$ $$\int_{S_1} \vec{v}\cdot \vec{da} = \int_0^R r^3 dr \int_0^{\pi/2} cos\phi d\phi$$ $$\int_{S_1} \vec{v}\cdot \vec{da} = \frac{R^4}{4} \times 1 = \frac{1}{4} R^4$$ $S_2$ represents the part on the on the $yz$ plane. Hence $\phi = \pi/2$ $$\int_{S_2} \vec{v}\cdot \vec{da} = \int_{S_2} (r^2 cos\theta \hat{r} - r^2 cos\theta \hat{\phi})\cdot (r dr d\theta \hat{\phi})$$ $$\int_{S_2} \vec{v}\cdot \vec{da} = \int_{S_2} - r^3 cos\theta dr d\theta $$ $$\int_{S_2} \vec{v}\cdot \vec{da} = -\int_0^R r^3 dr \int_0^{\pi/2} cos\theta d\theta$$ $$\int_{S_2} \vec{v}\cdot \vec{da} = -\frac{R^4}{4} \times 1 = -\frac{1}{4} R^4$$ $S_3$ represents the part on the on the $xz$ plane. Hence $\phi = 0$ $$\int_{S_3} \vec{v}\cdot \vec{da} = \int_{S_3} (r^2 cos\theta \hat{r} + r^2 cos\phi \hat{\theta})\cdot (-r dr d\theta \hat{\phi}) = 0$$ $S_4$ represents the curved part of the sphere . Hence $r = R$ $$\int_{S_4} \vec{v}\cdot \vec{da} = \int_{S_4} (R^2 cos\theta \hat{r} + R^2 cos\phi \hat{\theta}-R^2 cos\theta sin\phi \hat{\phi})\cdot (R^2 sin\theta d\theta d\phi \hat{r})$$ $$\int_{S_4} \vec{v}\cdot \vec{da} = \int_{S_4} (R^4 cos\theta sin\theta d\theta d\phi)$$ $$\int_{S_4} \vec{v}\cdot \vec{da} = R^4\int_0^{\pi /2} cos\theta sin\theta d\theta \int_0^{\pi/2} d\phi$$ $$\int_{S_4} \vec{v}\cdot \vec{da} = R^4\times \frac{1}{2} \times \frac{\pi}{2} = \frac{1}{4}\pi R^4$$ Hence,$$\oint_S \vec{v}\cdot \vec{da}= \int_{S_1}\vec{v}\cdot \vec{da}+\int_{S_2}\vec{v}\cdot \vec{da}+\int_{S_3}\vec{v}\cdot \vec{da}+\int_{S_4}\vec{v}\cdot \vec{da}$$ $$\oint_S \vec{v}\cdot \vec{da}= \frac{1}{4}\pi R^4$$ Therefore, $$\int_{vol.} (\vec{\triangledown}\cdot \vec{v})d\tau = \oint_S \vec{v}\cdot\vec{da}$$ 

Hence divergence theorem is verified

 

If you have any doubt regarding the solution or you want solution of some problem which is not posted please let me know by commenting. This encourages me to answer more question because sometime it feels like all I am doing is just a waste. If it helps someone I will be happy to do it.

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