Q17P Find the聽Jacobian聽鈭�(x,y)/鈭�... [FREE SOLUTION]

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Chapter 5: Q17P (page 269)

Find theJacobian $鈭� (x, y) / 鈭� (u, y)$ 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

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

Determine the Jacobian

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

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

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

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

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91影视

Find theJacobian $鈭� (x, y) / 鈭� (u, y)$ 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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91影视

Find theJacobian鈭�(x,y)/鈭�(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

Radiation

Oscillations

Turning Points in Physics

Physics of Motion

Engineering Physics

Linear Momentum

Study anywhere. Anytime. Across all devices.

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