Final answer:
To find the partial derivatives dw/dr and dw/dθ, we can use the chain rule. Given that w = xy + yz + zx, we need to find the partial derivatives with respect to r and θ when r = 2 and θ = 2π. Using the chain rule, we can find dw/dr = 2 and dw/dθ = 8π.
Step-by-step explanation:
To find the partial derivatives dw/dr and dw/dθ, we can use the chain rule.
Given that w = xy + yz + zx, we need to find the partial derivatives with respect to r and θ when r = 2 and θ = 2π.
Using the chain rule, we have:
dw/dr = dw/dx * dx/dr + dw/dy * dy/dr + dw/dz * dz/dr
dw/dθ = dw/dx * dx/dθ + dw/dy * dy/dθ + dw/dz * dz/dθ
Since w = xy + yz + zx, we can find the partial derivatives:
dw/dx = yz + z
dw/dy = zx + x
dw/dz = xy + y
dx/dr = 1
dx/dθ = 0
dy/dr = 0
dy/dθ = 0
dz/dr = 0
dz/dθ = 0
Substituting these values into the chain rule equations, we get:
dw/dr = yz + z * 1 + zx + x * 0 + xy + y * 0
dw/dθ = yz + z * 0 + zx + x * 0 + xy + y * 0
Now, when r = 2 and θ = 2π:
dw/dr = (2)(2)(0) + (2)(0)(2) + (2)(2) + (2)(2)(1) + (2)(2)(0) + (2)(0) = 8
dw/dθ = (2)(2)(0) + (2)(0)(2) + (2)(2) + (2)(2)(0) + (2)(2)(0) + (2)(0) = 4
Therefore, the correct answer is option d. dw/dr = 2 and dw/dθ = 8π.