Final answer:
To find the Jacobian of the function φ(r,θ) = (5r×cosθ, 4r×sinθ), you calculate the partial derivatives of each component with respect to r and θ to form a 2x2 Jacobian matrix with entries [[5cosθ, -5r×sinθ], [4sinθ, 4r×cosθ]].
Step-by-step explanation:
The student is asking to compute the Jacobian of the function φ(r,θ) = (5r×cosθ, 4r×sinθ). The Jacobian matrix of a function with two variables r and θ is a 2x2 matrix formed by taking the partial derivatives of each component of the vector-valued function with respect to r and θ.
To compute the Jacobian matrix J for the given function, we take the partial derivatives:
- ∂(5r×cosθ)/∂r = 5cosθ
- ∂(5r×cosθ)/∂θ = -5r×sinθ
- ∂(4r×sinθ)/∂r = 4sinθ
- ∂(4r×sinθ)/∂θ = 4r×cosθ
Therefore, the Jacobian matrix J is:
J = [[5cosθ, -5r×sinθ], [4sinθ, 4r×cosθ]]