Question

Derive the unit vectors ρ, θ, and φ as functions of the spherical coordinates <ρ, θ, φ> and the Cartesian unit vectors i, j, and k.

1. Suppose that ρ, θ, and φ depend on time t. Compute dρ/dt , dθ/dt , and dφ/dt , leaving results in terms of the spherical coordinates and the Cartesian unit vectors.

2. Express the derivatives dρ/dt , dθ/dt , and dφ/dt in terms of the spherical coordinates and spherical unit vectors.

3. Compute the second derivatives d^2ρ/dt^2 , d^2θ/dt^2 , and d^2φ/dt^2 in terms of the spherical coordinates and spherical unit vectors.

Answer 1