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In Exercises 27–32, functions x = x(u, v) and y = y(u, v) are given that determine transformations from an XY-coordinate system to a UV-coordinate system in R2. Use these functions to determine a region in the XY-plane that has the image specified for the given values of u and v, and find the Jacobian of the transformation.

x=usecvandy=utanvfor0u2and0vπ4

Short Answer

Expert verified

The Jacobian is equal toJ=usecv.

Step by step solution

01

Given information

The functions are,

x=usecvandy=utanvfor0u2and0vπ4

02

Find the Jacobian

The Jacobian is computed as,

(x,y)(u,v)=detxuyuxvyv(x,y)(u,v)=detsecvtanvusecvtanvusec2v(x,y)(u,v)=usec3v-usecvtan2v(x,y)(u,v)=usec3v-usecvsec2v-1(x,y)(u,v)=usec3v-usec3v+usecv(x,y)(u,v)=usecv

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