Question #297117

Let (u,v)=u^2-v^2,y(u,v)=2uv.


Find the the Jacobian determinant,J(u,v).

1
Expert's answer
2022-02-15T13:31:37-0500

Let us first write the Jacobian matrix :

J=(uxvxuyvy)J = \begin{pmatrix} \partial_u x & \partial_v x \\ \partial _u y & \partial_v y \end{pmatrix}

Now let us calculate every term in this matrix :

{ux=2uvx=2vuy=2vvy=2u\begin{cases} \partial_ u x = 2u \\ \partial_v x = -2v \\ \partial _u y = 2v \\ \partial_v y = 2u \end{cases}

And now we will calculate the determinant :

detJ=2u2u2v(2v)=4(u2+v2)\det J = 2u \cdot 2u - 2v\cdot (-2v) = 4(u^2+v^2)


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