**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)