Let (u,v)=u^2-v^2,y(u,v)=2uv.
Find the the Jacobian determinant,J(u,v).
Let us first write the Jacobian matrix :
J=(∂ux∂vx∂uy∂vy)J = \begin{pmatrix} \partial_u x & \partial_v x \\ \partial _u y & \partial_v y \end{pmatrix}J=(∂ux∂uy∂vx∂vy)
Now let us calculate every term in this matrix :
{∂ux=2u∂vx=−2v∂uy=2v∂vy=2u\begin{cases} \partial_ u x = 2u \\ \partial_v x = -2v \\ \partial _u y = 2v \\ \partial_v y = 2u \end{cases}⎩⎨⎧∂ux=2u∂vx=−2v∂uy=2v∂vy=2u
And now we will calculate the determinant :
detJ=2u⋅2u−2v⋅(−2v)=4(u2+v2)\det J = 2u \cdot 2u - 2v\cdot (-2v) = 4(u^2+v^2)detJ=2u⋅2u−2v⋅(−2v)=4(u2+v2)
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