Homework problems for Chapter 15

15.9: 1,5,7,10,13,17,19,23

Topics: changes of variables in multiple integrals, the Jacobian of a transformation

15.8: 1,3,9,17,19,24,33,35

Topics: triple integrals in cylindrical and spherical coordinates

15.7: 3,5,7,9,13,17,19,29,31,33,38,43

Topics: triple integrals over parallelpipeds and general regions, Fubini's theorem, applications of triple integrals

15.6: 1,3,5,7,9,21

Topics: surface area

15.5: 1,3,5,7,11,15,23,25

Topics: moments and centers of mass for lamina of variable density, probability density functions for pairs of random variables, expected values, normal distributions

15.4: 7,9,11,12,15,17,20,26,27,29

Topics: double integrals in polar coordinates, converting integrals in rectangular coordinates to polar coordinates and vice versa

15.3: 1,3,5,7,11,21,35,37,39,45

Topics: double integrals over general regions, regions of type I and type II, switching limits of integration, properties of double integrals

15.2: 1,4,7,11,15,18,21,24,33

Topics: Iterated integrals, Fubini's theorem

15.1: 3,4,5,12

Topics: Double integrals over rectangles, Riemann sums, Midpoint Rule, average value

Homework problems for Chapter 14

14.8: 3,5,7,17,19,31

Topics: Lagrange multipliers

14.7: 3,5,7,14,29,33,43,49

Topics: relative and absolute maximum and minimum values, saddle points for functions of several variables

14.6: 3,4,9,13,22,24,39,47,48

Topics: directional derivatives, the gradient vector, tangent planes to level surfaces

14.5: 1,5,7,10,13,21,30,31,36,40

Topics: the Chain Rule and the Implicit Function Theorem for functions of several variables

14.4: 1,5,17,19,23,28,29,33,37

We covered the material related to questions 23,28,29,33 and 37 in class on Thursday (10/10) and will cover the material related to questions 1,5,17 and 19 in class on Tuesday (10/15).

Topics: tangent planes (explicit and parametric form), linear approximations, differentials, error computations

14.3: 1,6,11,19,25,41,46,66(abcf),69

For problem 46, verify the conclusion of Clairaut's Theorem.

Topics: partial derivatives, partial differential equations

14.2: 7,11,12,28,33,37

Topics: limits and continuity in higher dimensions

14.1: 2,4,8,15,30,31,37

MATHEMATICA will generate plots of the graphs and level sets for functions of two variables.

Topics: functions of several variables, various representations (graphical, tabular, descriptive), domain and range

Homework problems for Chapter 13

13.4: 10,14,15,25,29

Applied Project (p. 867): 4

For problem #4 in the applied project, use Kepler's Third Law T^2=(4 pi^2 a^3)/(GM) from problem 2(c).
You will need to know:

• the gravitational constant G=6.67 x 10^(-11) N m^2/kg^2,
• the mass of the Earth M=5.98 x 10^24 kg,
• the radius of the Earth = 6.37 x 10^6 m.

Topics: motion in space, velocity, acceleration, tangential and normal components of acceleration, Kepler's Laws

13.3: 9,13,15,18,30,33

Topics: arc length, parametrizations by arc length, curvature, normal and binormal vectors, osculating planes and circles

13.2: 1,3,7,13,17,21,25,33

Topics: derivatives and integrals of vector functions, tangent vectors

13.1: 1,5,14,16,17,21,30

Again, try using MATHEMATICA to generate pictures of some of these space curves.

Topics: vector functions, space curves, intersections of surfaces

Homework problems for Chapter 12

12.7: 7,9,17,21,37,39,42,55

Topics: cylindrical and spherical coordinates

12.6: 5,15,19,32

Try using MATHEMATICA to generate 3D plots of the given surfaces (a handout describing how to do this will be provided in class).

Topics: cylindrical surfaces, quadric surfaces, graphing surfaces in 3D by the method of traces

12.5: 3,7,12,21,23,29,47

For problems 21,23 and 29 you should also make sure that you know how to write parametric equations (using two parameters) for the plane.

Topics: vector (parametric) and symmetric equations of lines and planes, angles and distance between lines and planes

12.4: 1,7,11,13,31,35,45

Topics: the cross product, determinants, area and volume in terms of cross products, torque

12.3: 3,7,14,15,25,51

Topics: the dot product, angles between vectors, projections

12.2: 3,5,12,13,17,24,36

Topics: vectors, geometric and algebraic descriptions of vectors

12.1: 2,6,8,11,15,21

Topics: 3D coordinate system, locating and plotting points in 3D, equations of planes and spheres, the 3D distance formula