Question

To express the partial derivatives with respect to rho, θ, and ϕ in terms of the partial derivatives with respect to x, y, and z provide the function f(x,y,z), and find the expressions for ∂rho/∂f, ∂θ/∂f, and ∂ϕ /∂f in terms of ∂x/∂f, ∂y / ∂f, and ∂z/∂f.

Answer 1

To express the partial derivatives with respect to rho, theta, and phi in terms of the partial derivatives with respect to x, y, and z, we need to relate the Cartesian coordinates (x, y, z) to the spherical coordinates (rho, theta, phi). The conversion equations between the two coordinate systems are x = rho*sin(theta)*cos(phi), y = rho*sin(theta)*sin(phi), and z = rho*cos(theta). Using the chain rule and these conversion equations, we can differentiate and solve for the partial derivatives.

Step-by-step explanation:

To express the partial derivatives with respect to rho (ρ), theta (θ), and phi (φ) in terms of the partial derivatives with respect to x, y, and z, we need to relate the Cartesian coordinates (x, y, z) to the spherical coordinates (ρ, θ, φ). The conversion equations between the two coordinate systems are:

• x = ρsin(θ)cos(φ)
• y = ρsin(θ)sin(φ)
• z = ρcos(θ)

We can differentiate these equations with respect to x, y, and z to find the expressions for ∂ρ/∂f, ∂θ/∂f, and ∂φ/∂f in terms of ∂x/∂f, ∂y/∂f, and ∂z/∂f. Applying the chain rule and solving for the partial derivatives, we get:

• ∂ρ/∂f = (∂x/∂f)sin(θ)cos(φ) + (∂y/∂f)sin(θ)sin(φ) + (∂z/∂f)cos(θ)
• ∂θ/∂f = (∂x/∂f)ρcos(θ)cos(φ) + (∂y/∂f)ρcos(θ)sin(φ) - (∂z/∂f)ρsin(θ)
• ∂φ/∂f = -(∂x/∂f)ρsin(θ)sin(φ) + (∂y/∂f)ρsin(θ)cos(φ)