(a) Express the field \(\mathbf{D}=\left(x^{2}+y^{2}\right)^{-1}\left(x \mathbf{a}_{x}+y \mathbf{a}_{y}\right)\) in cylindrical components and cylindrical variables. ( \(b\) ) Evaluate \(\mathbf{D}\) at the point where \(\rho=2, \phi=0.2 \pi\), and \(z=5\), expressing the result in cylindrical and rectangular components.

The field D in rectangular components is given by: D = (x^2 + y^2)^{-1} (x * ax + y * ay) Using the transformation equations, we can rewrite this in cylindrical components. For the x and y components, we will use the chain rule: dx/dρ = cos(φ), dx/dφ = -ρ * sin(φ) dy/dρ = sin(φ), dy/dφ = ρ * cos(φ) Now, substitute these into the expression for D and compute the cylindrical components: Dρ = (ρ^2)^{-1} *(ρ * cos(φ) * aρ + ρ * sin(φ) * aφ) Dφ = (ρ^2)^{-1} *(-ρ * sin(φ) * aρ + ρ * cos(φ) * aφ) Since D has no z-component, Dz will be 0. Simplifying, we find that the field D in cylindrical coordinates is: D = (1/ρ^2) (ρ * cos(φ) * aρ + ρ * sin(φ) * aφ) #Step 3: Substitute the given cylindrical variables to find D at the specified point#

Using the transformation equations, compute the corresponding rectangular coordinates at the specified point: x = 2 * cos(0.2π) y = 2 * sin(0.2π) z = 5 The field D in rectangular components at the specified point is: D = (1/4) (x * ax + y * ay) Thus, we have found the field D in both cylindrical and rectangular components at the specified point.