MATH 252: NOTES FOR QUIZ 4
TABLE OF CONTENTS (under construction)
CHAPTER 17:
MULTIPLE INTEGRALS
Web links (look under Chapter 16):
http://www.grossmont.edu/jennyvandeneynden/archive/M281_links.htm
Notes (pdf format):
Section 17.1 / 17.2 of Swokowski : Double Integrals, Area, and Volume
Animation: Riemann Sums for Volume Converging to the Integral:
Section 17.3 of Swokowski : Double Integrals in Polar Coordinates
Two Animations by Professor Lou Talman, Metropolitan State College of Denver
Area in Polar Coordinates:
Area Between Two Polar Curves:
Section 17.4 of Swokowski : Surface Area
Section 17.5 of Swokowski : Triple Integrals
Section 17.6 of Swokowski : Moments and Centers of Mass (last revised 8/8/07)
• WIKIPEDIA LINKS ON MOMENTS:
http://en.wikipedia.org/wiki/First_moment_of_inertia (First moment)
http://en.wikipedia.org/wiki/Shear_Stress (Shear stress)
http://en.wikipedia.org/wiki/Second_moment_of_inertia (Second moment)
http://en.wikipedia.org/wiki/Polar_moment_of_inertia (Polar moment of inertia)
http://en.wikipedia.org/wiki/Torsion_%28mechanics%29 (Torsion)
http://en.wikipedia.org/wiki/Moment_of_inertia (Mass moment of inertia)
Section 17.7 of Swokowski : Cylindrical Coordinates (last revised 8/8/07)
• CORRECTION TO 17.7.10:
I forgot to put the integral sign inside the innermost brackets of the bottom triple integral.
Section 17.8 of Swokowski : Spherical Coordinates (last revised 8/9/07)
A Cube in Spherical Coordinates (you can rotate this):
40 Cubes in Spherical Coordinates (you can rotate this):
Section 17.9 of Swokowski : Change of Variables and Jacobians (last revised 12/1/07)
IMPORTANT COMMENT ON JACOBIANS (SECTION 17.9)
• Read my reciprocal comment on 17.9.1. How does this idea extend to higher dimensions?
• Look at Step 1 on my Notes 17.9.9. Notice that the corresponding coefficient matrix (A) has determinant 3. Look at Step 4 on Notes 17.9.11. The Jacobian we want is 1/3, the reciprocal of 3. This is not a coincidence!
• The Jacobian of x and y with respect to u and v is essentially the reciprocal of the Jacobian of u and v with respect to x and y. This latter Jacobian is easy to find! You just use the A matrix from 17.9.10, which is easily obtained by writing down coefficients from your substitution statements.
• It helps that our last example in 17.9 dealt with a linear transformation; the relationships between (x,y) and (u,v) are linear, so our Jacobians are constants. For nonlinear transformations, however, you may well need to solve for x and y in terms of u and v, anyway.
Jacobian Applet (I don't know if this works):
http://cs.jsu.edu/mcis/faculty/leathrum/Mathlets/jacobian.html
Quiz 4 Notes: 80 pages total (Ch.17) (4.3+ MB)