Question

Express V in terms of Spherical coordinates where V = x + y^2z + z^3x.

(a) rsin⁡θ cos⁡φ + r^3sin⁡θ^2  sin⁡φ cos⁡θ + r^4sin⁡θ cos⁡θ^3 cos⁡φ

(b) rsin⁡θ cos⁡φ + r^2sin⁡θ^3  sin⁡φ cos⁡θ + r^4sin⁡θ cos⁡θ^3 cos⁡φ

(c) rsin⁡θ cos⁡φ + r^3sin⁡θ^2  sin⁡φ cos⁡θ + r^4sin⁡θ cos⁡θ^2 cos⁡φ

(d) rsin⁡θ cos⁡φ + r^3sin⁡θ^3  sin⁡φ cos⁡θ + r^4sin⁡θ cos⁡θ^2 cos⁡φ

I have been asked this question in a national level competition.

I'm obligated to ask this question of Conversion from Cartesian, Cylindrical and Spherical Coordinates in portion Vector Differential Calculus of Engineering Mathematics

Answer 1

Right option is (a) rsin⁡θ cos⁡φ + r^3sin⁡θ^2  sin⁡φ cos⁡θ + r^4sin⁡θ cos⁡θ^3 cos⁡φ

Easy explanation: Using the formula x = r sin⁡θ cos⁡ϕ and y = rsin⁡θ sin⁡ϕ and z=rcos⁡θ and substituting the values of x, y, z we get the desired results.

V = x + y^2z + z^3x

V = rsin⁡θ cos⁡φ + r^3sin⁡θ^2  sin⁡φ cos⁡θ + r^4sin⁡θ cos⁡θ^3 cos⁡φ.