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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}...
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Chapter 1 Vector Analysis: Problem 1.54

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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...
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Chapter 1 Vector Analysis: Problem 1.27

Problem 1.27: Prove that the divergence of a curl is always zero. Check it for function $v_a$ in Prob. 1.15. ($v_a = x^2\hat{x} + 3xz^2\hat{y} - 2xz\hat{z}$) Answer: Consider a vector function $$\vec{F} = F_x \hat{i} + F_y \hat{j} + F_z \hat{k}$$  To show $$\vec{\triangledown}\cdot (\vec{\triangledown} \times \vec{F}) = 0$$  Points To Remember: $$\vec{\triangledown}\cdot \vec{F} = \frac{\partial}{\partial x}(F_x) + \frac{\partial}{\partial y}(F_y) + \frac{\partial}{\partial z}(F_z) $$ And $$\vec{\triangledown} \times \vec{F} = \begin{vmatrix} \hat{i} && \hat{j} && \hat{k} \\ {\partial \over \partial x} && {\partial \over \partial y} && {\partial \over \partial z}\\ F_x && F_y && F_z \end{vmatrix}$$  Consider, $$\vec{\triangledown} \times \vec{F} = \begin{vmatrix} \hat{i} && \hat{j} && \hat{k} \\ {\partial \over \partial x} && {\partial \over \partial y} && {\partial \over ...
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