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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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What is Divergence? How to calculate Divergence? What is the physical significance of Divergence?

 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:

$$\vec{\nabla}\cdot \vec{v} = \frac{\partial }{\partial x}v_x+\frac{\partial }{\partial y}v_y+\frac{\partial }{\partial z}v_z$$

 

How to calculate Divergence in various Coordinate Systems?

 

Divergence 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:

$$\vec{\nabla}\cdot \vec{F} = \frac{1}{fgh}\big[\frac{\partial}{\partial u}(ghF_u)+\frac{\partial}{\partial v}(fhF_v)+\frac{\partial}{\partial w}(fgF_w)\big]$$

In Cartesian Coordinate System:


$$\vec{dl} = dx\hat{i}+dy\hat{j}+dz\hat{k}$$

$$f=1, g=1, h=1$$

$$\vec{\nabla}\cdot \vec{F} = \frac{\partial}{\partial x}(F_x)+\frac{\partial}{\partial y}(F_y)+\frac{\partial}{\partial z}(F_z)$$

In Cylindrical Coordinate System:


$$\vec{dl} = ds\hat{s}+sd\phi\hat{\phi}+dz\hat{k}$$

$$f=1, g=s, h=1$$

$$\vec{\nabla}\cdot \vec{F} = \frac{1}{s}\big[\frac{\partial}{\partial s}(sF_s)+\frac{\partial}{\partial \phi}(F_{\phi})+\frac{\partial}{\partial z}(sF_z)\big]$$

In Spherical Coordinate System:


$$\vec{dl} = dr\hat{r}+rd\theta\hat{\theta}+r\sin\theta d\phi\hat{\phi}$$

$$f=1, g=r, h=r\sin\theta$$

$$\vec{\nabla}\cdot \vec{F} = \frac{1}{r^2\sin\theta}\big[\frac{\partial}{\partial r}(r^2\sin\theta F_r)+\frac{\partial}{\partial \theta}(r\sin\theta F_{\theta})+\frac{\partial}{\partial \phi}(rF_{\phi})\big]$$

What is the physical Significance of Divergence? 

Working on it... Will add soon. 


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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