This answer is section 15.10 number 13 in stewarts multivariable calculus book.

Question

This answer is section 15.10 number 13 in stewarts multivariable calculus book. I don't get their explanation at all from the solution guide. I need someone to preferably show me on paper how to do this problem, and why they let U and V equal to what they are.

QUESTION :

A region R in the xy plane is given. Find equations for a transformation T that maps a rectangular region S in the uv plane onto R, where the sides of S are parallel to the u and v axes.

R lies between the circles x^2 + y^2 = 1 and x^2 + y^2 = 2 in the first quadrant.

The solution manual lost me when they let v = inverse tangent y / x

Explanation / Answer

(u,v) ------>(r,theta)

knowing that x = r cos(theta) y=r sin(theta) ==>tan(theta) =y/x ==>theta=arctan(y/x)

x^2+y^2 = 1 ====> x^2 + y^2 = 1 ====> r^2 =1 ====> r = 1

x^2+y^2=2 ====> r^2 = 2 ====> r = sqrt(2)

first quadrant =====> pi/2

pi/2....sqrt(2)
?........? r*dr d? = pi/2
0.......0

pi/2....1
?.........? r *dr d? = pi/4
0.......0

pi/2 - pi/4 = pi/4

Related Questions

Navigate

• Browse (All)
• Browse T
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.