Math 241 Lectures and Homework

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Notation: 1.2: 3, 4 means Chapter 1 Section 1.2 Problems 3 and 4.

Weekly homework will be collected on Tuesday discussion sessions. For practice please do all problems marked in red even if they are not assigned because they come with online Homework Hints. You are required to staple your homework and to draw a box around each answer.

Lecture 1, Mon Aug 24. 12.1-12.2. Coordinate system and vectors. Notes
Homework 1 due Sep 1:
12.1: 8, 9ab (use scalar multiplication of vectors; do not use the distance formula as in the solution manual), 14, 16, 26, 32;
12.2: 20, 22, 24, 25.

Lecture 2, Wed Aug 26. 12.3. The dot product. Notes
Homework 1 due Sep 1:
12.3: 1, 10, 12, 22, 24, 34, 38, 46, 52;
Problem not from textbook:
1*. Given the vectors a=<8,8,-6> and b=<1,-1,-2>, express a as a sum of a vector parallel to b and a vector perpendicular to b.

Lecture 3, Fri Aug 28. 12.4. The cross product. Notes
Homework 2 due Sep 8:
12.4: 12, 13, 17, 20, 25, 30, 36, 38, 42, 43.

Lecture 4, Mon Aug 31. 12.5. Lines and planes. Notes
Homework 2 due Sep 8:
12.5: 1a-e, 8, 16ab, 20, 26, 28, 34, 44, 52, 58, 70.

Lecture 5, Wed Sep 2. 12.6. Surfaces. Notes
Homework 2 due Sep 8:
12.6: 4, 8, 12, 14, 16, 24, 26, 28, 34, 36.

Lecture 6, Fri Sep 4. 13.1-13.2. Vector functions and curves. Notes
Homework 3 due Sep 15:
13.1: 4, 9, 16, 22, 24, 26, 38;
13.2: 4, 16, 20, 26, 34, 50.
Problem not from textbook:
2*. Describe and sketch the parametric curve r(t)=< cos t, (sin t)^2>, 0 < t < pi. Mark the direction on the curve. (The charachter ^ means raising to a power. I hope you get what I mean by "pi" !)

Lecture 7, Wed Sep 9. 13.3-13.4. Arc length, curvature, acceleration. Notes
(The procedure for computing the unit normal vector is not clearly explained in the textbook. Use your Lecture 7-8 notes.)
Homework 3 due Sep 15:
13.3: 2, 14, 18, 24, 30, 46;
13.4: 12, 16, 22, 24, 26, 36(find also the normal vector N).
Problem not from textbook:
3*. Let v=<1,2,-1> and a=<8,6,8> be the velocity and acceleration vectors of a moving particle at a certain moment of time. Find the unit tangent and normal vectors and the curvature of the trajectory of the particle; find the tangent and normal components of the acceleration at that moment of time.

Lecture 8, Fri Sep 11. 13.3-13.4. Arc length, curvature, acceleration. Notes
No additional homework

Lecture 9, Mon Sep 14. Exam 1 review. Notes

Lecture 10, Wed Sep 16. 14.1-14.2 Functions of several variables. Limits and continuity. Notes
Homework 4 due Sep 22:
14.1: 14, 28, 30ac, 34, 46, 63;
14.2: 6, 10, 12, 14, 38.

Midterm Exam 1, Fri Sep 18.

Lecture 11, Mon Sep 21. 14.3 Partial derivatives. Notes
Homework 5 due Sep 29:
14.3: 15, 24, 39, 46, 54, 57, 70ab, 73, 74ab.

Lecture 12, Wed Sep 23. 14.4 Tangent planes and linear approximation. Notes
Homework 5 due Sep 29:
14.4: 1, 6, 14, 18, 19, 36.

Lecture 13, Fri Sep 25. 14.5 The Chain Rule. Notes
Homework 6 due Oct 6:
14.5: 10, 14, 17, 22, 28, 32, 40, 50, 52.

Lecture 14, Mon Sep 28. 14.6 The gradient and directional derivatives. Notes
Homework 6 due Oct 6:
14.6: 4, 16, 20, 24, 38, 40, 48.

Lecture 15, Wed Sep 30. 14.7 Extrema of functions of several variables. Notes
Homework 7 due Oct 13:
14.7: 2, 4, 10, 12, 16, 32, 34, 42, 44, 50.

Lecture 16, Fri Oct 2. 14.7 Extrema of functions of several variables. Notes

Lecture 17, Mon Oct 5. 14.8 Constrained optimization and Lagrange multipliers. Notes
Homework 7 due Oct 13:
14.8: 4, 12, 16, 30, 38, 42b.
Problems not from textbook:
4*. Find the maximum and minimum values of the function f(x,y)=y^2-4y-3x^2 on the circle x^2+y^2=1. Sketch the level curves of f passing through the points in which the constrained extrema are attained.
5*. Find the maximum and minimum values of the function f(x,y)=y^2-4y-3x^2 on the disk bounded by the circle x^2+y^2=1. 6*. Find the maximum and minimum values of the function f(x,y)=x^3+y on the ellipse x^2+4y^2=4. Sketch the level curves of f passing through the points in which the constrained extrema are attained.

Lecture 18, Wed Oct 7. 15.1-15.3 Double integrals. Notes
Homework 8 due Oct 20:
15.1: 8, 12;
15.2: 20, 26.

Lecture 19, Fri Oct 9. 15.3 Double integrals. Notes
Homework 8 due Oct 20:
15.3: 14, 18, 20, 26, 40, 46, 50, 54, 56.

Lecture 20, Mon Oct 12. 15.4 Double integrals in polar coordinates. Notes
Homework 8 due Oct 20:
15.4: 8, 14, 18, 22, 26, 30, 32.

Lecture 21, Wed Oct 14. Exam 2 review. Notes

Midterm Exam 2, Fri Oct 16.

Lecture 22, Mon Oct 19. 15.5 Applications of double integrals. Notes
Homework 9 due Oct 27:
15.5: 10, 16, 18, 26, 28, 32.

Lecture 23, Wed Oct 21. 15.6 Triple integrals. Notes
Homework 9 due Oct 27:
15.6: 12, 18, 20, 28, 34, 42, 46abc, 50abc.

Lecture 24, Fri Oct 23. 15.7 Triple integrals in cylindrical coordinates. Notes
Homework 10 due Nov 3:
15.7: 4, 8, 10ab, 18, 22, 28;
Problem not from textbook:
7*. Set up (but don't evaluate) the iterated integral in cylindrical coordinates for the moment of inertia about the y-axis of the solid bounded by the surfaces z=0, z=sqrt(x^2+y^2), and (x-1)^2+y^2=1, if the density at (x,y,z) is equal to the distance of (x,y,z) to the origin. ("sqrt" means the square root.)

Lecture 25, Mon Oct 26. 15.8 Triple integrals in spherical coordinates. Notes
Homework 10 due Nov 3:
15.8: 4ab, 8, 10ab, 22, 30, 32ab, 40.
Problems not form textbook:
8* Let Q be the solid defined by x^2+y^2>1 (outside the cylinder), x^2+y^2+z^2<4 (inside the sphere). The solid has density x^2+y^2 at (x,y,z). Set up the iterated integrals for the mass of Q in both cylindrical and spherical coordinates. Don't evaluate the integrals.
9* Let Q be the solid defined by x^2+y^2>z^2 (below the cone), x^2+y^2+(z-1)^2< 1 (inside the sphere), x>0, y>0. The solid is homogeneous with density 1. Set up the iterated integrals for the moment of inertia of Q about the z-axis in all types of coordinates: cartesian, cylindrical and spherical coordinates. Don't evaluate the integrals.

Lecture 26, Wed Oct 28. 15.9 Change of variables in multiple integrals. Notes
Homework 10 due Nov 3:
15.9: 4, 8, 12, 14, 16, 20.

Fri Oct 30. Class cancelled due to power failure in the building.

Lecture 27, Mon Nov 2. 16.1-16.2 Vector fields. Line integrals. Notes
Homework 11 due Nov 10:
16.1: 6, 14, 23, 30;
16.2: 2, 4, 8, 14, 18, 22, 33, 40.

Lecture 28, Wed Nov 4. 16.3 The fundamental theorem for line integrals. Notes
Homework 11 due Nov 10:
16.3: 2, 3, 6, 16, 20, 23, 28, 30, 32, 33.

Lecture 29, Fri Nov 6. 16.4 Green's Theorem. Notes
Homework 12 due Nov 17:
16.4: 2, 8, 12, 18, 22, 28.

Lecture 30, Mon Nov 9. 16.5 Curl and divergence. Notes
Homework 12 due Nov 17:
16.5: 2, 10, 12a-f, 20, 25, 32, 33, 36.

Lecture 31, Wed Nov 11. Exam 3 review. Notes

Midterm Exam 3, Fri Nov 13.

Lecture 32, Mon Nov 16. 16.6 Parametric surfaces and their areas. Notes
Homework 13 due Dec 1:
16.6: 4, 6, 14, 18, 24, 34, 38, 44, 54abc, 56abc.

Lecture 33, Wed Nov 18. 16.7 Surface integrals. Notes
Homework 13 due Dec 1:
16.7: 4, 6, 10, 14, 18.

Lecture 34, Fri Nov 20. 16.7 Surface integrals.
Homework 14 due Dec 8:
16.7: TBA
Problems not from textbook