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Chapter 5: Q16P (page 268)

Find the Jacobians鈭�(x,y)/鈭�(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

Expert verified

The Jacobians isJ=u2+v2

Step by step solution

01

Given Information

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)

02

 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.

03

Evaluate Jacobian determinant

The determinant of the Jacobian is equal to:

J=鈭�ux鈭�ux鈭�uy鈭�uy=鈭�x鈭�u鈭�y鈭�v-鈭�y鈭�u鈭�x鈭�v=u脳v-v脳-v=u2+v2

Where 鈭�abis the partial derivative of b with respect to (or鈭�b/鈭�a in other words).

Hence, the Jacobians isJ=u2+v2

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Most popular questions from this chapter

A dielectric lamina with charge density proportional to y covers the area between the parabola y=16路x2 and the x axis. Find the total charge.

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