Chapter 5: Q17P (page 269)

Find theJacobian $\partial (x, y) / \partial (u, y)$ of the given transformations from variables x,yto variables u,v:
$_{y = a s i n h u s i n v,}^{x = a c o s h u c o s v,} (u and v are called elliptic cylinder coordinates)$

Short Answer

Expert verified

The determinant of the Jacobian is $|J| = a^{2} \sin h^{2} u \cos^{2} v + a^{2} \cos h^{2} u \sin^{2} v$ .

Step by step solution

Given Condition

The giventransformation is

$x = a \cos h u \cos v y = a \sin h u \sin v$ .

Concept of the Jacobian

The Jacobian of x,ywith respect to s,t is

$J = J (\frac{x, y}{s, t}) = \frac{\partial (x, y)}{\partial (s, t)} = | \begin{matrix} \frac{\partial x}{\partial s} & \frac{\partial x}{\partial t} \\ \frac{\partial y}{\partial s} & \frac{\partial y}{\partial t} \end{matrix} |$

Determine the Jacobian

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

$= |\begin{matrix} a \sin h u \cos v & - a \cos h u \sin v \\ a \cos h u \sin v & a \sin h u \cos v \end{matrix}|$

$= (a \sin h u \cos v) (a \sin h u \cos v) + (a \cos h u \sin v) (a \cos h u \sin v) = a^{2} \sin h^{2} u \cos^{2} = a^{2} \cos h^{2} u \sin^{2} v$

The determinant of the Jacobian is $|J| = a^{2} \sinh^{2} {ucos}^{2} = a^{2} \cosh^{2} {usin}^{2} v$ .

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Find theJacobian $\partial (x, y) / \partial (u, y)$ of the given transformations from variables x,yto variables u,v:
$_{y = a s i n h u s i n v,}^{x = a c o s h u c o s v,} (u and v are called elliptic cylinder coordinates)$

Short Answer

Step by step solution

Given Condition

Concept of the Jacobian

Determine the Jacobian

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Find theJacobian∂(x,y)/∂(u,y)of the given transformations from variables x,yto variables u,v:y=asinhusinv,x=acoshucosv,(uandvarecalledellipticcylindercoordinates)

Short Answer

Step by step solution

Given Condition

Concept of the Jacobian

Determine the Jacobian

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Linear Momentum

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Study anywhere. Anytime. Across all devices.

Find theJacobian $\partial (x, y) / \partial (u, y)$ of the given transformations from variables x,yto variables u,v:
$_{y = a s i n h u s i n v,}^{x = a c o s h u c o s v,} (u and v are called elliptic cylinder coordinates)$