Chapter 5: Q19P (page 269)

In the integral
$I = \int_{0}^{\infty} \int_{0}^{\infty} \frac{x^{2} + y^{2}}{{(x^{2} + y^{2})}^{2}} e^{- 2 xy} dxdy$ .
Make the change of variables
$u = x^{2} - y^{2} v = 2 xy$
And evaluate I. Hint: Use (4.8) and the accompanying discussion.

Short Answer

Expert verified

The evaluation of the given integral upon changing the variables is as follows,

$I = \int_{0}^{\infty} \int_{0}^{\infty} \frac{x^{2} + y^{2}}{{(x^{2} + y^{2})}^{2}} e^{- 2 xy} dxdy$

$= \frac{ττ}{4}$ .

Step by step solution

Given the condition

Theintegral given is as follows,

$I = \int_{0}^{\infty} \int_{0}^{\infty} \frac{x^{2} + y^{2}}{{(x^{2} + y^{2})}^{2}} e^{- 2 xy} dxdy$

The change of variables are given to be the following,

$u = x^{2} - y^{2} v = 2 xy$

Concept of the Jacobian

The Jacobian of x,ywith respect to s,tis

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

Determination of the Integral

The substitution of variables in the given integral is as follows,

$u = x^{2} - y^{2}, v = 2 xy$

$\frac{\partial (u, v)}{\partial (x, v)} = |\begin{matrix} 2 x & - 2 y \\ 2 y & 2 y \end{matrix}| = 4 x^{2} = 4 y^{2} = 4 (x^{2} + y^{2})$

$\begin{array}{l} dxdy = J dudv \\ dxdy = \frac{1}{4 (x^{2} + y^{2})} dudv \\ (x^{2} + y^{2}) dudv = \frac{1}{4} dudv \end{array}$

$I = \int_{v = 0}^{\infty} \int_{u = 0}^{\infty} \frac{e^{- v}}{4 (1 + u^{2})} dudv = \frac{1}{4} \int_{v = 0}^{\infty} e^{- v} dv \int_{u = 0}^{\infty} \frac{1}{1 + u^{2}} du = \frac{1}{4} {(e^{- v})}^{\infty} {(\tan^{- 1} u)}_{- \infty}^{\infty} = \frac{1}{4} (\frac{ττ}{2} + \frac{ττ}{2}) = \frac{ττ}{4}$

The integral

$I = \int_{0}^{\infty} \int_{0}^{\infty} \frac{x^{2} + y^{2}}{{(x^{2} + y^{2})}^{2}} e^{- 2 xy} dxdy = \frac{ττ}{4}$

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In the integral
$I = \int_{0}^{\infty} \int_{0}^{\infty} \frac{x^{2} + y^{2}}{{(x^{2} + y^{2})}^{2}} e^{- 2 xy} dxdy$ .
Make the change of variables
$u = x^{2} - y^{2} v = 2 xy$
And evaluate I. Hint: Use (4.8) and the accompanying discussion.

Short Answer

Step by step solution

Given the condition

Concept of the Jacobian

Determination of the Integral

One App. One Place for Learning.

Most popular questions from this chapter

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In the integralI=∫0∞∫0∞x2+y2(x2+y2)2e-2xydxdy. Make the change of variablesu=x2-y2v=2xyAnd evaluate I. Hint: Use (4.8) and the accompanying discussion.

Short Answer

Step by step solution

Given the condition

Concept of the Jacobian

Determination of the Integral

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Translational Dynamics

Magnetism

Measurements

Electrostatics

Electric Charge Field and Potential

Space Physics

Study anywhere. Anytime. Across all devices.

In the integral
$I = \int_{0}^{\infty} \int_{0}^{\infty} \frac{x^{2} + y^{2}}{{(x^{2} + y^{2})}^{2}} e^{- 2 xy} dxdy$ .
Make the change of variables
$u = x^{2} - y^{2} v = 2 xy$
And evaluate I. Hint: Use (4.8) and the accompanying discussion.