MATH 252: NOTES FOR QUIZ 4

SMALLER FILES (if you can’t get the big file):

CHAPTER 17: MULTIPLE INTEGRALS

Web links (look under Chapter 16):

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

Example from 17.3: Math252SinSquaredTheta.pdf

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)

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/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)

More on TSPs:

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):

Quiz 4 Notes: 87 pages total (Ch.17 including additional notes) (4.9 MB)