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List of Questions and link to solutions

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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Chapter1 Vector Analysis: Problem 1.18

  Problem 1.18 Calculate the CURL of the following vector functions: (a) $v_a = x^2\hat{i}+3xz^2\hat{j}-2xz\hat{k}$ (b) $v_b = xy\hat{i}+2yz\hat{j}+3zx\hat{k}$ (c) $v_c = y^2\hat{i}+(2xy+z^2)\hat{j}+2yz\hat{k}$ Solution: Here's the link on "How to calculate curl in different coordinate system?" (a) $v_a = x^2\hat{i}+3xz^2\hat{j}-2xz\hat{k}$ $$\vec{\nabla}\times \vec{v_a} = \begin{bmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ x^2 & 3xz^2 & (-2xz) \end{bmatrix}$$ $\vec{\nabla}\times \vec{v_a} = \big[\frac{\partial }{\partial y}(-2xz) - \frac{\partial}{\partial z}(3xz^2)\big]\hat{i}- \big[\frac{\partial }{\partial x}(-2xz) - \frac{\partial}{\partial z}(x^2)\big]\hat{j}+ \big[\frac{\partial }{\partial x}(3xz^2) - \frac{\partial}{\partial y}(x^2)\big]\hat{k}$ $$\vec{\nabla}\times \vec{v_a} = 6xz\hat{i}+2z\hat{j}+ 3z^2\hat{k}$$ (b) $v_b = xy\hat{i}+2yz\hat{j}+3zx\hat{k}
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Calculate CURL in different Coordinate Systems.

  How to calculate CURL in various Coordinate Systems?   Curl of a vector function $\vec{F} = F_u\hat{u}+F_v\hat{v}+F_w\hat{w}$ in a coordinate system where $\vec{dl} = f du\hat{u}+gdv\hat{v}+hdw\hat{w}$   is given by: $$\vec{\nabla}\times \vec{F} = \frac{1}{fgh} \begin{bmatrix} f\hat{u} & g\hat{v} & h\hat{w} \\ \frac{\partial}{\partial u} & \frac{\partial}{\partial v} & \frac{\partial}{\partial w} \\ fF_u & gF_v & hF_w \end{bmatrix}$$ In Cartesian Coordinate System: $$\vec{dl} = dx\hat{i}+dy\hat{j}+dz\hat{k}$$ $$f=1, g=1, h=1$$ $$\vec{\nabla}\times \vec{F} = \begin{bmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ F_x & F_y & F_z \end{bmatrix}$$ In Cylindrical Coordinate System: $$\vec{dl} = ds\hat{s}+sd\phi\hat{\phi}+dz\hat{k}$$ $$f=1, g=s, h=1$$ $$\vec{\nabla}\times \vec{F} = \frac{1}{s} \begin{bmatrix} \hat{s} & s\hat{\phi} & \hat{z} \\ \f
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Chapter 1 Vector Analysis: Problem 1.12

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 Problem 1.12 The height of a certain hill (in feet) is given by $$h(x, y) = 10(2xy − 3x2 − 4y2 − 18x + 28y + 12),$$ where $y$ is the distance (in miles) north, $x$ the distance east of South Hadley. (a) Where is the top of the hill located? (b) How high is the hill? (c) How steep is the slope (in feet per mile) at a point 1 mile north and one mile east of South Hadley? In what direction is the slope steepest, at that point? Here's is the (c) part. https://introductiontoelectrodynamics.blogspot.com/2021/09/problem-1_3.html 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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Chapter 1 Vector Analysis: Problem 1.8

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 Problem 1.8 (a) Prove that the two-dimensional rotation matrix ($\bf Eq. 1.29$) preserves dot products. (That is, show that $\overline{A_y B_y} + \overline{A_z B_z} = A_y B_y + A_z B_z$ .) (b) What constraints must the elements ($R_{i j}$ ) of the three-dimensional rotation matrix ($\bf Eq. 1.30$) satisfy, in order to preserve the length of $\bf{A}$ (for all vectors $\vec{A}$)?   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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