Obtain the curl of the following vector field in spherical polar coordinates :
vector A = x( xi^+yj^+zk^)
To obtain the curl of the vector field in spherical polar coordinates, we need to express the vector field in terms of the spherical polar coordinates system. The vector field, A, is given as:
A = x(xi^ + yj^ + zk^)
In spherical coordinates, x, y, and z are expressed in terms of the radial distance (r), the polar angle (θ), and the azimuthal angle (φ). The unit vectors in spherical coordinates are denoted as ȩᵣ, ȩθ, and ȩφ, respectively.
So, we need to express the vector field A, in terms of the spherical coordinate unit vectors. To do this, we can use the following relationship:
xi^ = rȩᵣ
yj^ = r*sin(θ)ȩθ
zk^ = r*cos(θ)ȩᵣ
Substituting these expressions into A, we have:
A = x(xi^ + yj^ + zk^)
= x(rȩᵣ) + y(r*sin(θ)ȩθ) + z(r*cos(θ)ȩφ)
= r*xȩᵣ + r*sin(θ)*yȩθ + r*cos(θ)*zȩφ
Now that we have expressed A in terms of the spherical coordinate unit vectors, we can find the curl of A in spherical polar coordinates. The curl of a vector field in spherical coordinates is given by:
∇ × A = (1/r)*[ (1/sin(θ))* ∂(Asin(θ))/∂φ - ∂Aφ/∂θ ]ȩᵣ
+ (1/r)*[ ∂Ar/∂θ - ∂(Asin(θ))/∂r ]ȩθ
+ (1/r)*[ (1/sin(θ))* ∂(rAφ)/∂r - ∂Ar/∂φ ]ȩφ
Now, let's compute the curl of A using these formulas.