MATH 252: NOTES FOR QUIZ 4
TABLE OF CONTENTS (under construction)
BIG FILE: Chapter 17 notes (handwritten); see links and comments
below
SMALLER FILES (if you cant get the big file):
CHAPTER 17: MULTIPLE INTEGRALS
Web links (look under Chapter 16):
http://www.grossmont.edu/jennyvandeneynden/archive/M281_links.htm
Section 17.1 / 17.2 of
Swokowski : Double Integrals, Area, and Volume
Animation: Riemann Sums for Volume Converging
to the Integral:
http://math.bu.edu/people/paul/225/two_var_Riemann_sum.mov
Section 17.3 of Swokowski
: Double Integrals in Polar Coordinates
Example from 17.3: Math252SinSquaredTheta.pdf
Two Animations by Professor Lou Talman,
Metropolitan State College of Denver
Area in Polar Coordinates:
http://math.bu.edu/people/paul/225/SimplePolarArea.mov
Area Between Two Polar Curves:
http://math.bu.edu/people/paul/225/StandardPolarArea.mov
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):
http://math.bu.edu/people/paul/225/one_spherical_rec.html
40 Cubes in Spherical Coordinates (you can
rotate this):
http://math.bu.edu/people/paul/225/many_spherical_recs.html
Section 17.9 of Swokowski
: Change of Variables and Jacobians (last revised
12/1/07)
More
on TSPs: Math252JumblingTSP.pdf
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: 87 pages total (Ch.17 including additional notes) (4.9 MB)