If x = uv and y = (u - v) / (u + v), find the Jacobian ∂(u,v)/∂(x,y).
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
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Identify the transformation equations We have the transformations given as: $$ x = uv $$ $$ y = \frac{u - v}{u + v} $$
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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.
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Using the chain rule for Jacobian To find the inverse Jacobian, we use: $$ J = \begin{vmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \end{vmatrix} $$
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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} ).
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Calculate ( \frac{\partial x}{\partial u} ), ( \frac{\partial x}{\partial v} ), ( \frac{\partial y}{\partial u} ), and ( \frac{\partial y}{\partial v} )
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For ( x ): $$ \frac{\partial x}{\partial u} = v $$ $$ \frac{\partial x}{\partial v} = u $$
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For ( y ): $$ \frac{\partial y}{\partial u} = \frac{(u+v)(1) - (u - v)(1)}{(u + v)^2} = \frac{2v}{(u + v)^2} $$ $$ \frac{\partial y}{\partial v} = \frac{(u + v)(-1) - (u - v)(1)}{(u + v)^2} = \frac{-2u}{(u + v)^2} $$
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Construct the Jacobian matrix Now put these into the Jacobian matrix: $$ J = \begin{vmatrix} v & u \ \frac{2v}{(u + v)^2} & \frac{-2u}{(u + v)^2} \end{vmatrix} $$
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Calculate the determinant Compute ( J ): $$ J = v \cdot \frac{-2u}{(u + v)^2} - u \cdot \frac{2v}{(u + v)^2} $$ Simplifying, we get: $$ J = \frac{-2uv - 2uv}{(u + v)^2} = \frac{-4uv}{(u + v)^2} $$
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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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