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Chapter 13: Q. 29 (page 1079)

In Exercises 27鈥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=uvandy=uvfor1鈮u鈮2and1鈮v鈮3

Short Answer

Expert verified

The Jacobian is equal to2uv.

Step by step solution

01

Given information

The functions are,

x=uvandy=uvfor1鈮u鈮2and1鈮v鈮3

02

Find the Jacobian

The Jacobian is computed as,

鈭(x,y)鈭(u,v)=det鈭x鈭u鈭y鈭u鈭x鈭v鈭y鈭v鈭(x,y)鈭(u,v)=det1vv-uv2u鈭(x,y)鈭(u,v)=uv--uv2v鈭(x,y)鈭(u,v)=uv+uv鈭(x,y)鈭(u,v)=2uv

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