{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

LSweek30 - MATH1211/LSweek30/TNK/10-11(2 THE UNIVERSITY OF...

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
1 MATH1211/LSweek30/TNK/10-11(2) THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS MATH1211: Multivariable Calculus Lecture Summary – Week 30 March 21, 2011 We have the second class test. March 23, 2011 We recalled the method of change of variables for integration of single-variable functions: Theorem . Let f : [ A; B ] ! R be an integrable function over the closed interval [ A; B ] . Suppose x = x ( u ) is a differentiable function in variable u such that x maps an interval with end points a and b to the closed interval [ A; B ] in a one-to-one manner, with x ( a ) = A and x ( b ) = B . Then Z B A f ( x ) dx = Z b a f ( x ( u )) dx du du: (1) Note that on the LHS of (1), the variable of integration is x , the function f is written in terms of x , and the limits A; B of integration are limits on x ; on the RHS, the variable of integration is u , the function f is written in terms of u , and the limits a; b of integration are limits on u , but it also has an “extra” term which is dx du (which can be written as x 0 ( u ) ). “Justification” of this extra term (Note: this is not a rigorous proof; but a rigorous proof can be constructed based on this idea): We partition [ A; B ] into A = x 0 < x 1 < ¢ ¢ ¢ < x n = B , and form Riemann sums P n i =1 f ( c i x i where ¢ x i = x i ¡ x i ¡ 1 and c i 2 [ x i ¡ 1 ; x i ] . By writing the integral on the LHS of (1) as the limit of Riemann sums, we have: R B A f ( x ) dx = lim P f ( c i x i = lim P f ( x ( d i ))¢ x i , where d i are the values of u such that x ( d i ) = c i . Note that, by using differentials, ¢ x i ¼ dx du ¢ u i . Hence P f ( x ( d i ))¢ x i ¼ P f ( x ( d i ))
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern