Introduction to Electrodynamics by David J Griffiths Solutions

  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} & ...

 What is Divergence? How it looks? Divergence is a vector operator that is it acts on a vector to output a scalar just like a normal dot/scalar product and in simple terms it tells how much the vector divergences i.e whether the vector flux increases or decreases at a point(infinitesimal volume). First del operator in Cartesian coordinate is given by: $$\vec{\nabla} = \hat{i}\frac{\partial }{\partial x}+\hat{j}\frac{\partial }{\partial y}+\hat{k}\frac{\partial }{\partial z}$$  Since, divergence resembles the dot product in the way that it takes a vector and outputs a scalar. Divergence acting on an vector resembles dot product. Consider two vectors $\vec{A} = A_x\hat{i}+A_y\hat{j}+A_z\hat{k}$ and $\vec{B}=B_x\hat{i}+B_y\hat{j}+B_z\hat{k}$ . Their dot product is given by: $$\vec{A}\cdot \vec{B} = A_xB_x+A_yB_y+A_zB_z$$  Consider A vector function $\vec{v}=v_x\hat{i}+v_y\hat{j}+v_z\hat{k}$. The divergence of vector $\vec{v}$ is given be: $$\vec{\nabla}\cdot \vec{v} = \frac...