Chapter 5: Q16P (page 268)

Find the Jacobians $\partial (x, y) / \partial (u, y)$ of the given transformations from the variables x,y to variables u,v :
$x = \frac{1}{2} (u^{2} - v^{2}) y = u v$
( u and v are called parabolic cylinder coordinates)

Short Answer

Expert verified

The Jacobians is $|J| = u^{2} + v^{2}$

Step by step solution

Given Information

The given transformations from the variables x,y to variables u,v

$x = \frac{1}{2} (u^{2} - v^{2}) y = u v$

( u and v are called parabolic cylinder coordinates)

Step 2: Concept of Jacobian

The Jacobian determinant is used when making a change of variables when evaluating a multiple integral of a function over a region within its domain.

Evaluate Jacobian determinant

The determinant of the Jacobian is equal to:

$|J| = |\begin{matrix} \partial_{u} x & \partial_{u} x \\ \partial_{u} y & \partial_{u} y \end{matrix}| = \frac{\partial x}{\partial u} \frac{\partial y}{\partial v} - \frac{\partial y}{\partial u} \frac{\partial x}{\partial v} = u \times v - v \times (- v) = u^{2} + v^{2}$

Where $\partial_{a} b$ is the partial derivative of b with respect to (or $\partial b / \partial a$ in other words).

Hence, the Jacobians is $|J| = u^{2} + v^{2}$

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Find the Jacobians $\partial (x, y) / \partial (u, y)$ of the given transformations from the variables x,y to variables u,v :
$x = \frac{1}{2} (u^{2} - v^{2}) y = u v$
( u and v are called parabolic cylinder coordinates)

Short Answer

Step by step solution

Given Information

Step 2: Concept of Jacobian

Evaluate Jacobian determinant

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Find the Jacobians∂(x,y)/∂(u,y)of the given transformations from the variables x,y to variables u,v :x=12(u2-v2)y=uv( u and v are called parabolic cylinder coordinates)

Short Answer

Step by step solution

Given Information

Step 2: Concept of Jacobian

Evaluate Jacobian determinant

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Electricity

Wave Optics

Electromagnetism

Space Physics

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Engineering Physics

Study anywhere. Anytime. Across all devices.

Find the Jacobians $\partial (x, y) / \partial (u, y)$ of the given transformations from the variables x,y to variables u,v :
$x = \frac{1}{2} (u^{2} - v^{2}) y = u v$
( u and v are called parabolic cylinder coordinates)