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Vector Analysis

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Fourier Series and Transforms

Differential Equations

Partial Differential Equations

Special Co-ordinate System: Cartesian and Cylindrical

Cartesian Co-ordinate system

Fig: Cartesian co-ordinate system
Fig: Cartesian co-ordinate system

In this co-ordinate system we have
$$u_1=x$$
$$u_2=y$$
$$u_3=2$$
The position vector in this co-ordinate system can be written as,
$$\vec r= \vec r(x,y,z)$$
$$\vec r=\vec i x+ \vec j y+ \vec k z$$
a. Scale factor
The scale factor can be written as,
\(h_1=h_x=\biggl|\frac{\delta \vec r}{\delta u_1}\biggr|= \biggl|\frac{\delta \vec r}{\delta x}\biggr|=|\hat i|=1\)
\(h_2=h_y=\biggl|\frac{\delta \vec r}{\delta u_2}\biggr|= \biggl|\frac{\delta \vec r}{\delta y}\biggr|=|\hat j|=1\)
\(h_3=h_z=\biggl|\frac{\delta \vec r}{\delta u_3}\biggr|= \biggl|\frac{\delta \vec r}{\delta z}\biggr|=|\hat k|=1\)

b. Unit Vector

\(\hat e_1=\frac{\frac{\delta \vec r}{\delta u_1}}{|\frac{\delta \vec r}{\delta u_1}|}=\frac{\frac{\delta\vec r}{\delta x}}{|\frac{\delta \vec r}{\delta x}|}=\frac{\hat i}{1}=\hat i\)
\(\hat e_2=\frac{\frac{\delta \vec r}{\delta y}}{|\frac{\delta \vec r}{\delta y}|}=\frac{\hat j}{1}=\hat j\)
\(\hat e_3=\hat k\)
c. Gradient
We know the expression for gradient in orthogonal curvilinear co-ordinate system are 
$$\nabla\phi=\frac{\hat e_1}{h_1}\frac{\delta\phi}{\delta u_1}+\frac{\hat e_2}{h_2}\frac{\delta \phi}{\delta u_2}+\frac{\hat e_3}{h_3}\frac{\delta \phi}{\delta u_3}$$
$$=\frac{\hat i}{1}\frac{\delta\phi}{\delta x}+\frac{\hat j}{1}\frac{\delta \phi}{\delta y}+\frac{\hat k}{1}\frac{\delta \phi}{\delta z}$$
$$=\hat i\frac{\delta \phi}{\delta x}+\hat j\frac{\delta \phi}{\delta \phi}+\hat k\frac{\delta \phi}{\delta z}$$
d. Divergence
We know the expression of divergence is
$$\nabla\cdot\vec F=\frac{1}{h_1h_2h_3}\biggl[\frac{\delta(h_2h_3f_1)}{\delta u_1}+\frac{\delta(h_1h_3f_2)}{\delta u_2}+\frac{\delta(h_2h_1f_3)}{\delta f_3}\biggr]$$
$$\nabla\cdot\vec F=\frac{\delta F_1}{\delta x}+\frac{\delta F_2}{\delta y}+\frac{\delta F_3}{\delta Z}$$

e. Curl
We know the expression for curl is, $$\nabla\times\vec F=\frac{1}{h_1h_2h_3}\begin{vmatrix} \hat e_1 h_1 & \hat e_2 h_2 & \hat e_3 h_3 \\ \frac{\delta}{\delta u_1} & \frac{\delta}{\delta u_2} & \frac{\delta}{\delta u_3}\\ h_1f_1 & h_2f_2 & h_3 f_3\\ \end{vmatrix}$$ $$=\begin{vmatrix} \hat i & \hat j & \hat k\\ \frac{\delta}{\delta x} & \frac{\delta}{\delta y} & \frac{\delta}{\delta z}\\ F_1 & F_2 & F_3\\ \end{vmatrix}$$

Cylindrical co-ordinate system
In this co-ordinate system,
$$u_1=\rho$$
$$u_2=\phi$$
$$u_3=z$$
The transformation equation relating cartesian and cylindrical co-ordinate system are,
$$x= \rho cos\phi$$
$$y=\rho sin \phi$$
$$z=z$$
The displacement vector in cylindrical co-ordinate system are
$$\vec r=\vec r(u_1, u_2, u_3)$$
$$=\vec r(x,y,z)$$
$$=\hat ix+\hat j y+\hat k z$$
$$\rho cos\phi\hat i+\rho sin\phi\hat j+z\hat k$$

a. Scale Factor

$$h_1=h_\rho=|\frac{\delta \vec r}{\delta u_1}|=|\frac{\delta \vec r}{\delta \rho}|=|cos\phi \hat i+sin\phi\hat j|=\sqrt{cos^2\phi+sin^2\phi}=1$$
$$ h_2= h_{\phi}=|\frac{\delta \vec r}{\delta u_2}|=|\frac{\delta\vec r}{\delta \pi}|=|-\rho sin\phi\hat i+\rho cos\phi \hat j|=\sqrt{\rho^2 sin^2\phi+\rho^2 cos^2\phi}=\rho$$
$$h_3=h_z=|\frac{\delta \vec r}{\delta u_3}|=|\frac{\delta \vec r}{\delta z}|= \hat k=1$$

b. Unit vector

$$\hat e_1=\hat e_{\rho}=\frac{\frac{\delta\vec r}{\delta u_1}}{|\frac{\delta\vec r}{\delta u_1}|}=\frac{\delta \vec r}{\delta \rho}=cos\phi\hat i+sin\phi\hat j$$
$$\hat e_2=\hat e_{\phi}=\frac{\frac{\delta\vec r}{\delta u_2}}{|\frac{\delta\vec r}{\delta u_2}|}=\frac{1}{\rho}(\frac{\delta\vec r}{\delta \phi})$$
$$=\frac{1}{\rho}(-\rho sin\phi \hat i+\rho cos\phi \hat j)$$
$$=-sin\phi\hat i+cos\phi\hat j$$
$$\hat e_3= e_z=\frac{\frac{\delta\vec r}{\delta u_3}}{|\frac{\delta\vec r}{\delta u_3}|}=\frac{\hat k}{1}=\hat k$$

c. Gradient
The expression for gradient can be written as, $$\nabla\phi=\frac{\hat e_1}{h_1}\frac{\delta \phi}{\delta u_1}+\frac{\hat e_2}{h_2} \frac{\delta \phi}{\delta u_2}+\frac{\hat e_3}{h_3}\frac{\delta \phi}{\delta u_3}$$ $$=\hat e_{\rho} \frac{\delta\phi}{\delta u_1}+\frac{\hat e_{\phi}}{\rho}\frac{\delta\phi}{\delta \phi}+\hat e_z\frac{\delta \phi}{\delta z}$$

d. Divergence

$$\nabla\cdot\vec F=\frac{1}{h_1h_2h_3}\biggl[\frac{\delta(h_2h_3f_1)}{\delta u_1}+\frac{\delta(h_1h_3f_2)}{\delta u_2}+\frac{\delta(h_2h_1f_3)}{\delta f_3}\biggr]$$ $$=\frac{1}{\rho}\biggl[\frac{\delta(\rho f_1)}{\delta \rho}+\frac{\delta F_z}{\delta \phi}+\frac{\delta(\rho F_3)}{\delta z}\biggr]$$

e. Curl
$$\nabla\times\vec F=\frac{1}{h_1h_2h_3}\begin{vmatrix} \hat e_1 h_1 & \hat e_2 h_2 & \hat e_3 h_3 \\ \frac{\delta}{\delta u_1} & \frac{\delta}{\delta u_2} & \frac{\delta}{\delta u_3}\\ h_1f_1 & h_2f_2 & h_3 f_3\\ \end{vmatrix}$$

 

Reference

  1. Adhikari, Pitri Bhakta, and Dya Nidhi Chhatkuli.A Textbook of Physics. Third Revised Edition ed. Vol. III. Kathmandu: SUKUNDA PUSTAK BHAWAN, 2072.
  2. Vaughn, Michael T.Introduction to Mathematical Physics. Weinheim: Wiley-VCH, 2007
  3. Butkov, Eugene.Mathematical Physics. Reading, MA: Addison-Wesley Pub., 1968.
  4. Carroll, Robert W.Mathematical Physics. Amsterdam: North-Holland, 1988.
  5. Dass, H. K.Mathematical Physics. New Delhi: S. Chand, 2005.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


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