1. Region D of the (x, y)-plane is the locus of points satisfying the conditions x > 0, -1-log x < y <- log x and 2 log x < y < 2+2 logx. (a) Sketch and identify region D in the (x, y)-plane. (b) Calculate the Jacobian of the transformation x=u1/3-1/3 y=log(u²/³v1/3), where log indicates a natural logarithm (to base e). You may find it easier to re-write the formula for y before differentiating it. (c) Hence evaluate the integral SS x² dx dy. D Hint: Begin by expressing the boundaries of D in terms of transformed variables.
1. Region D of the (x, y)-plane is the locus of points satisfying the conditions x > 0, -1-log x < y <- log x and 2 log x < y < 2+2 logx. (a) Sketch and identify region D in the (x, y)-plane. (b) Calculate the Jacobian of the transformation x=u1/3-1/3 y=log(u²/³v1/3), where log indicates a natural logarithm (to base e). You may find it easier to re-write the formula for y before differentiating it. (c) Hence evaluate the integral SS x² dx dy. D Hint: Begin by expressing the boundaries of D in terms of transformed variables.
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
Problem 70EQ
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Transcribed Image Text:1.
Region D of the (x, y)-plane is the locus of points satisfying the conditions x > 0,
-1-log x < y <- log x and 2 log x < y < 2+2 logx.
(a) Sketch and identify region D in the (x, y)-plane.
(b) Calculate the Jacobian of the transformation
x=u1/3-1/3
y=log(u²/³v1/3),
where log indicates a natural logarithm (to base e). You may find it easier to re-write the
formula for y before differentiating it.
(c) Hence evaluate the integral
SS
x² dx dy.
D
Hint: Begin by expressing the boundaries of D in terms of transformed variables.
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