Answer to Question #185023 in Physics for Satyam

Question

Express the field D = (x2 + y2)−1(xax + yay) in cylindrical components and cylindrical

variables.

Answer 1

$$x = ρ \cos φ, y = ρ \sin φ\\\bold{D}=\frac{1}{\rho}(\cos φ\bold{a_x}+ \sin φ\bold{a_y})$$

Then

$$D_\rho=\frac{1}{\rho}(\cos φ\bold{a_x\cdot a_\rho}+ \sin φ\bold{a_y\cdot a_\rho})\\=\frac{1}{\rho}(\cos^2 φ+ \sin^2 φ)=\frac{1}{\rho}$$

$$D_\phi=\frac{1}{\rho}(\cos φ\bold{a_x\cdot a_\phi}+ \sin φ\bold{a_y\cdot a_\phi})\\=\frac{1}{\rho}(-\sin φ\cos φ+ \sin φ\cos φ)=0$$

So,

$$\bold{D}=\frac{1}{\rho}\bold{a_\rho}$$