Question

A is a vector field and f is a scalar field. The gradient of f is equal A; that is A=∇f. Evaluate A in spherical coordinates and determine A at the point (R,θ,ϕ)=(5,30∘,90∘) if f=8Rsinθ−15cosϕ

Answer 1

The vector field A is the gradient of the scalar field f in spherical coordinates, which can be calculated by taking the partial derivatives of f with respect to its variables. At point (5, 30°, 90°), with the scalar field f = 8Rsinθ − 15cosϕ, A is found to be 4î_r + 4√3î_θ + 3î_ϕ.

Step-by-step explanation:

The question involves evaluating a vector field A, which is the gradient of a scalar field f in spherical coordinates. The given scalar field is f = 8Rsinθ − 15cosϕ. The gradient in spherical coordinates is expressed as:

• ∇f = ∂f/∂R î_r + (1/R) (∂f/∂θ) î_θ + (1/(R sinθ)) (∂f/∂ϕ) î_ϕ

To calculate the gradient, take the partial derivatives with respect to R, θ, and ϕ. This gives us:

• ∂f/∂R = 8sinθ
• ∂f/∂θ = 8Rcosθ
• ∂f/∂ϕ = 15sinϕ

Therefore, A = ∇f = (8sinθ) î_r + (8Rcosθ/R) î_θ + (15sinϕ/(Rsinθ)) î_ϕ. At the point (5, 30°, 90°) this becomes A = (8sin(30°)) î_r + (8 × 5cos(30°)/5) î_θ + (15sin(90°)/(5sin(30°))) î_ϕ = (4) î_r + (8√3/2) î_θ + (3) î_ϕ. Thus, the vector A at the specified point is 4î_r + 4√3î_θ + 3î_ϕ.