If x = uv and y = (u - v) / (u + v), find the Jacobian ∂(u,v)/∂(x,y).

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Understand the Problem

The question is asking to find the Jacobian determinant of the variables u and v with respect to the variables x and y, given a specific transformation. This involves using partial derivatives and is a fundamental concept in multivariable calculus.

Answer

The Jacobian is \( \frac{(u + v)^2}{-4uv} \).
Answer for screen readers

The Jacobian ( \frac{\partial(u,v)}{\partial(x,y)} ) is given by: $$ \frac{(u + v)^2}{-4uv} $$

Steps to Solve

  1. Identify the transformation equations We have the transformations given as: $$ x = uv $$ $$ y = \frac{u - v}{u + v} $$

  2. Calculate partial derivatives Next, we need to find the Jacobian determinant ( J = \frac{\partial(u,v)}{\partial(x,y)} ). This involves finding the necessary partial derivatives.

  3. Using the chain rule for Jacobian To find the inverse Jacobian, we use: $$ J = \begin{vmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \end{vmatrix} $$

  4. Find derivatives of ( u ) and ( v ) Using implicit differentiation, we first find ( \frac{\partial u}{\partial x} ) and ( \frac{\partial u}{\partial y} ), and then ( \frac{\partial v}{\partial x} ) and ( \frac{\partial v}{\partial y} ).

  5. Calculate ( \frac{\partial x}{\partial u} ), ( \frac{\partial x}{\partial v} ), ( \frac{\partial y}{\partial u} ), and ( \frac{\partial y}{\partial v} )

    • For ( x ): $$ \frac{\partial x}{\partial u} = v $$ $$ \frac{\partial x}{\partial v} = u $$

    • For ( y ): $$ \frac{\partial y}{\partial u} = \frac{(u+v)(1) - (u - v)(1)}{(u + v)^2} = \frac{2v}{(u + v)^2} $$ $$ \frac{\partial y}{\partial v} = \frac{(u + v)(-1) - (u - v)(1)}{(u + v)^2} = \frac{-2u}{(u + v)^2} $$

  6. Construct the Jacobian matrix Now put these into the Jacobian matrix: $$ J = \begin{vmatrix} v & u \ \frac{2v}{(u + v)^2} & \frac{-2u}{(u + v)^2} \end{vmatrix} $$

  7. Calculate the determinant Compute ( J ): $$ J = v \cdot \frac{-2u}{(u + v)^2} - u \cdot \frac{2v}{(u + v)^2} $$ Simplifying, we get: $$ J = \frac{-2uv - 2uv}{(u + v)^2} = \frac{-4uv}{(u + v)^2} $$

  8. Find the inverse of the Jacobian To find ( \frac{\partial(u,v)}{\partial(x,y)} ), we take the reciprocal of ( J ): $$ \frac{\partial(u,v)}{\partial(x,y)} = \frac{(u + v)^2}{-4uv} $$

The Jacobian ( \frac{\partial(u,v)}{\partial(x,y)} ) is given by: $$ \frac{(u + v)^2}{-4uv} $$

More Information

The Jacobian determinant provides essential information about how the area changes under a transformation. In this case, it reveals how the variables ( u ) and ( v ) scale when transformed into ( x ) and ( y ).

Tips

  • Confusing the order of differentiation can lead to incorrect results. Be sure to correctly identify which variable you are differentiating with respect to.
  • Neglecting to simplify the determinant properly can result in mistakes.

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