The calculator will find the Jacobian matrix of the set of functions and the Jacobian determinant (if possible), with steps shown.
Solution
The Jacobian matrix is defined as follows: J(x,y)(r,θ)=[∂r∂x∂r∂y∂θ∂x∂θ∂y].
In our case, J(x,y)(r,θ)=[∂r∂(rcos(θ))∂r∂(rsin(θ))∂θ∂(rcos(θ))∂θ∂(rsin(θ))].
Find the derivatives (for steps, see derivative calculator): J(x,y)(r,θ)=[cos(θ)sin(θ)−rsin(θ)rcos(θ)].
The Jacobian determinant is the determinant of the Jacobian matrix: ∣∣cos(θ)sin(θ)−rsin(θ)rcos(θ)∣∣=r (for steps, see determinant calculator).
Answer
The Jacobian matrix is [cos(θ)sin(θ)−rsin(θ)rcos(θ)]A.
The Jacobian determinant is rA.