Obtain the curl of the following vector field in spherical polar coordinates:
vector A = x(xi^+yj^+zk^)
I'm sure you have this stuff already,
http://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates
so why not just plug in your function?
Curl curl of vector f
To obtain the curl of a vector field in spherical polar coordinates, we need to express the vector field in terms of the spherical unit vectors (ρ-hat, θ-hat, and φ-hat) and then use the curl formula in spherical coordinates.
In spherical polar coordinates, we have the following unit vectors:
ρ-hat: points radially outward from the origin
θ-hat: points in the direction of increasing θ (polar angle)
φ-hat: points in the direction of increasing φ (azimuthal angle)
To express the vector field A = x(xi^ + yj^ + zk^) in terms of spherical unit vectors, we'll need to convert the cartesian unit vectors (i^, j^, k^) to spherical unit vectors. The conversion formulas are:
i^ = sin(θ)cos(φ)ρ-hat + cos(θ)cos(φ)θ-hat - sin(φ)φ-hat
j^ = sin(θ)sin(φ)ρ-hat + cos(θ)sin(φ)θ-hat + cos(φ)φ-hat
k^ = cos(θ)ρ-hat - sin(θ)θ-hat
Substituting these conversion formulas into the expression for A, we get:
A = x(sin(θ)cos(φ)ρ-hat + cos(θ)cos(φ)θ-hat - sin(φ)φ-hat)
+ y(sin(θ)sin(φ)ρ-hat + cos(θ)sin(φ)θ-hat + cos(φ)φ-hat)
+ z(cos(θ)ρ-hat - sin(θ)θ-hat)
Now, let's compute the curl of A using the curl formula in spherical coordinates:
∇ x A = (1/ρsin(θ))(∂(Az)/∂θ - ∂(Ay)/∂φ) ρ-hat
+ (1/ρ)(∂(Ax)/∂φ - ∂(Az)/∂ρ) θ-hat
+ (1/ρ)(1/sin(θ))(∂(Ay)/∂ρ - ∂(Ax)/∂θ) φ-hat
where Ax, Ay, and Az are the components of A in the ρ, θ, φ directions, respectively.
Taking the partial derivatives of the components of A with respect to ρ, θ, and φ, we can substitute them into the curl formula to obtain the curl of A in spherical polar coordinates.