Jacobian Calculator

Calculate Jacobian step by step

The calculator will find the Jacobian matrix of the set of functions and the Jacobian determinant (if possible), with steps shown.

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Calculate the Jacobian of {x=rcos(θ),y=rsin(θ)}\left\{x = r \cos{\left(\theta \right)}, y = r \sin{\left(\theta \right)}\right\}.

Solution

The Jacobian matrix is defined as follows: J(x,y)(r,θ)=[xrxθyryθ].J{\left(x,y \right)}\left(r, \theta\right) = \left[\begin{array}{cc}\frac{\partial x}{\partial r} & \frac{\partial x}{\partial \theta}\\\frac{\partial y}{\partial r} & \frac{\partial y}{\partial \theta}\end{array}\right].

In our case, J(x,y)(r,θ)=[r(rcos(θ))θ(rcos(θ))r(rsin(θ))θ(rsin(θ))].J{\left(x,y \right)}\left(r, \theta\right) = \left[\begin{array}{cc}\frac{\partial}{\partial r} \left(r \cos{\left(\theta \right)}\right) & \frac{\partial}{\partial \theta} \left(r \cos{\left(\theta \right)}\right)\\\frac{\partial}{\partial r} \left(r \sin{\left(\theta \right)}\right) & \frac{\partial}{\partial \theta} \left(r \sin{\left(\theta \right)}\right)\end{array}\right].

Find the derivatives (for steps, see derivative calculator): J(x,y)(r,θ)=[cos(θ)rsin(θ)sin(θ)rcos(θ)].J{\left(x,y \right)}\left(r, \theta\right) = \left[\begin{array}{cc}\cos{\left(\theta \right)} & - r \sin{\left(\theta \right)}\\\sin{\left(\theta \right)} & r \cos{\left(\theta \right)}\end{array}\right].

The Jacobian determinant is the determinant of the Jacobian matrix: cos(θ)rsin(θ)sin(θ)rcos(θ)=r\left|\begin{array}{cc}\cos{\left(\theta \right)} & - r \sin{\left(\theta \right)}\\\sin{\left(\theta \right)} & r \cos{\left(\theta \right)}\end{array}\right| = r (for steps, see determinant calculator).

Answer

The Jacobian matrix is [cos(θ)rsin(θ)sin(θ)rcos(θ)]\left[\begin{array}{cc}\cos{\left(\theta \right)} & - r \sin{\left(\theta \right)}\\\sin{\left(\theta \right)} & r \cos{\left(\theta \right)}\end{array}\right]A.

The Jacobian determinant is rrA.