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Chapter 13: Q. 31 (page 1079)

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=usinvandy=ucosvfor0≤u≤2and0≤v≤π

Short Answer

Expert verified

The Jacobian is equal toJ=-u.

Step by step solution

01

Given information

The functions are,

x=usinvandy=ucosvfor0≤u≤2and0≤v≤π

02

Find the Jacobian

The Jacobian is computed as,

∂(x,y)∂(u,v)=det∂x∂u∂y∂u∂x∂v∂y∂v∂(x,y)∂(u,v)=detsinvcosvucosv-usinv∂(x,y)∂(u,v)=-usin2v-ucos2v∂(x,y)∂(u,v)=-usin2v+cos2v∂(x,y)∂(u,v)=-u×1∂(x,y)∂(u,v)=-u

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