Lecture Summaries and Homework Assignments

Lecture 39, Wednesday, May 2:
Lecture 38, Monday, April 30: Section 14.7. Statement of Stokes's Theorem. Geometric meaning. How to use it to do some calculations.
Honework: Section 14.7 problems 1, 5, 7, 17.
Lecture 37, Friday, April 27: Finishing Section 14.5: flux of a vector field across a surface, geometric meaning, how to calculate. Section 14.6. Divergence Theorem in vector form. A more convenient formulation in terms of dydz etc. What it means geometrically. How to use it to do calculations. Example: flux of F=<2x^2, -3y, z+1> over the surface S equal to the part of the plane x-y+z=1 consisting of points (x,y,z) for which x is between -1 and 1 and y is between 1 and 2. I got 52/3 for that surface integral---let me know if you get a different result!
Homework: Section 14.5 problems 19, 21. Section 14.6 problems 1, 7, 9, 13, 15, 16.
Lecture 36, Wednesday, April 25: Section 14.5. Surface integrals. Surface area element dS and what it means. Surface integrals with respect to surface area and surface integrals with respect to coordinate elements (dydz, dzdx and dxdy). How to calculate all of these. Examples.
Homework: Section 14.5 problems 1, 3, 5, 13, 15.
Lecture 35, Monday, April 23: Flux of a vector field in the plane, its geometric significance, relation to the integral of divergence. Start of Section 14.5. The density element dS for surface integrals. The meaning of a basic surface integral and the analogy with line integrals.
Homework: Section 14.4 problems 21, 23; Section 14.5 problems 1, 3, 5.
Lecture 34, Friday, April 20 (guest lecture by Prof. D'Angelo): Section 14.4. Green's Theorem. How to prove it on a rectangle, sketch of the method for proving it in general. Discussion of the hypothesis that the domain be the region enclosed by a simple closed curve.
Homework: Section 14.4 problems 1, 7, 9, 11, 15, 17, 29. If you're feeling ambitious, Section 14.4 problem 41.
Lecture 33, Wednesday, April 18: More on Section 14.3. Two ways to compute the potential of a conservative vector field. Why the word "conservative" appears in this context.
Homework: the remaining problems from Section 14.3 that were assigned on Monday.
Lecture 32, Monday, April 16: Section 14.3. Fundamental theorem and independence of path for line integrals. The line integral of a gradient vector field doesn't depend on the path. This is a general fact. Some sample computations. Definition of a conservative vector field (gradient of a function). How to tell when a vector field on a rectangle in the plane is conservative.
Homework:   Section 14.3 problems 3, 5, 7, 15, 17, 21, 23, 27, 30, 31. Note: problems that ask you to use methods of examples from the text to find potential functions: we will discuss these on Wednesday; you may postpone doing those problems until after Wednesday's class.
Lecture 31, Friday, April 13: Section 14.2. Line integrals with respect to arc length. Line integrals with respect to a coordinate function. Line integrals of general differentials (1-forms). How to compute mass, centroid of a wire. How to compute work when the velocity is nonconstant.
Homework:   Section 14.2 problems 1, 5, 7, 9, 13, 17, 19, 27, 34.
Lecture 30, Monday, April 9: Section 14.1. Vector fields. Geometric meaning. Gradient vector fields of functions (reminder). Divergence of a vector field and its meaning. Curl of a vector field and its meaning.
Homework:   Section 14.1 problems 3, 5, 11-14, 19, 23, 25, 29, 37, 38.
Lecture 29, Friday, April 6: Section 13.8. Parametric surfaces and surface area.
Homework:   Section 13.8 problems 2, 3, 9, 11.
Lecture 28, Wednesday, April 4: More examples from Section 13.9 to get change of variables. Finishing up Section 13.7: change of variables to spherical coordinates.
Homework:   Section 13.9 problems 9, 13. Section 13.7 problems 21, 25, 33, 39.
Lecture 27, Monday, April 2: Section 13.9. Change of variables formula for multiple integrals. An example (Section 13.9, problem 8).
Homework:   Section 13.9 problems 1, 3, 5.
Lecture 26, Friday, March 30: Section 13.7. Triple integrals in cylindrical coordinates. Section 13.9. Discussion of change of variables.
Homework:   None for today!
Lecture 25, Wednesday, March 28: Section 13.6. Triple integrals. Triple integrals in rectangular coordinates. Volume, very brief discussion of mass and centroid in 3 dimensions. Section 13.7 (the beginning). A discussion of triple integrals in cylindrical coordinates.
Homework:   Section 13.6 problems 5, 9, 13. Section 13.7 problems 1, 7, 8, 9, 19.
Lecture 24, Monday, March 26: Section 13.5. Applications of double integrals. Mass of a thin plate, its centroid (center of mass). Moment of inertia, especially polar moment of inertia, how to calculate it. The interpretation of polar moment of inertia as the rotational analog of mass (kinetic energy due to rotation at angular speed s is one-half of I0 · s² where I0 is the polar moment of inertia).
Homework:   Section 13.5 problems 7, 15, 27, 29 (click here for a solution), 31, 33, 35.
Lecture 23, Friday, March 16: Section 13.3. How to compute area and volume of regions by double integrals.
Homework:   Section 13.3 problems 3, 7, 11, 13, 19, 27, 29.
Lecture 22, Wednesday, March 14: Section 13.4. Double integrals in polar coordinates. How to change variables. If there's time, a brief discussion of how changes of variables get encoded in differentials.
Homework:   Section 13.4 problems 3, 9, 13, 17, 27, 29, 34.
Lecture 21, Monday, March 12: Section 13.2. Double integrals over more general regions. How to do double integrals when the domain is not a rectangle: some cases that aren't too difficult. A brief reminder of polar coordinates in two dimensions.
Homework:   Section 13.2 problems 1, 11, 13, 15, 19, 31, 33.
Lecture 20, Wednesday, March 7: Section 13.1. Double integrals, rigorous definition, intuition behind it, double integrals as iterated integrals.
Homework   Section 13.1 problems 3, 5, 11, 17, 23.
Lecture 19, Monday, March 5:  Conclusion of Section 12.10. Examples of using the two-variable second-derivative test. Introduction to double integrals (Section 13.1).
Lecture 18, Friday, March 2:  Conclusion of Section 12.9, and Section 12.10. An example of the method of Lagrange multipliers in 3 variables, completely worked out. The two variable second-derivative test, its geometric meaning, how to use it to determine whether a function takes a local maximum, minimum, or neither at a given point.
Homework:   Section 12.10 problems 3, 9, 13, 17, 31. Remark: we will do examples on Monday. You may prefer to wait until then to do the homework.
Lecture 17, Wednesday, February 28:  Section 12.9. Geometry behind the idea of Lagrange multipliers. How to find the maximum or minimum, subject to a constraint, by first looking for points where the two gradients are parallel and then checking each of those. An example.
Homework   Section 12.9, problems 1, 7, 13, 19 (we will do an example of this kind of thing at the beginning of Friday's class!), 21.
Lecture 16, Monday, February 26:  Section 12.5. Maxima, minima, optimization. Where do you look for maxima and minima? First partial derivatives, what can happen on the boundary.
Homework:   Section 12.5 problems 3, 9, 13, 19, 29, 39, 47. Also, get caught up on old homework!
Lecture 15, Friday, February 23:  Section 12.8. Directional derivatives. Their meaning. Computing directional derivatives using the gradient vector and the dot product. Geometric interpretation of the gradient vector.
Homework:  Section 12.8 problems 1, 9, 11, 17, 21, 29, 31, 45, 47, 51.
Lecture 14, Wednesday, February 21:  General form of the chain rule. Implicit differentiation. Using implicit differentiation to calculate equations for tangent planes.
Homework: Section 12.7, whatever you didn't finish on Monday. Section 12.4, problems 31, 35.
Lecture 13, Monday, February 19:  Relation of the differential to formulas for tangent planes. Section 12.7. The chain rule. Dependency diagrams for organizing your calculation.
Homework: Section 12.7 problems 3, 5, 9, 13, 19, 23, 29, 33, 40, 49.
Lecture 12, Friday, February 16:  Skip Section 12.5 for now: we'll return to it at a more sensible time. For today, Section 12.6. The differential of a function of several variables. Some discussion of other terminologies and notations for it. Linear approximation and the gradient. Estimating the value of a function given a nearby value.
Homework:   Section 12.6 problems 1, 3, 7, 10, 17, 25, 33, 39.
Wednesday, February 14:  Classes cancelled because of weather: no lecture.
Lecture 11, Friday, February 9:  Finished Section 12.4. Also discussed the midterm exam on Monday. Tangent plane to a surface. Mixed partial derivatives, when they're equal.
Homework:   Section 12.4 problems 31, 35, 37, 71.
Lecture 10, Wednesday, February 7:  A little bit of Section 12.3, and Section 12.4 through p. 873. Definition and explanation of the limit of a function of several variables. An example, its graph, some values of the limits, and where the limit is undefined. Partial derivatives, their definition and geometric interpretation. How to compute partial derivatives (e.g.: to compute df/dx, think of y and z as constants, and do what you normally would to compute the derivative of a function of x).
Homework:  Make sure you have read Sections 12.1 - 12.4. Section 12.4 problems 1,3, 5, 9, 55, 62, 63, 65.
Lecture 9, Monday, February 5:  Sections 12.1 - 12.3 (we'll do more on 12.3 next time, I expect). Brief motivation for functions of several variables. Definition, discussion of graphs and level curves. Definition of limits for functions of several variables, and, even more generally, for vector-valued functions of several variables. Some discussion of the properties of limits in this setting, why we might care.
Homework:  Section 12.2, problems 21, 23, 27, 31, 33, 35, 53-58 (note: once you understand what's going on, doing just the odd-numbered ones takes only slightly less long than doing all of them!); Section 12.3 problems 3, 5, 17.
Lecture 8, Friday, February 2:  Some graphs of surfaces in 3 dimensional space; surfaces of revolution; some special types of quadric surfaces, their graphs. A brief discussion of cylindrical and spherical coordinates.
The Mathematica notebook I used (in HTML).
The web site on quadrics I mentioned.
Homework:  Section 11.2 problems 20, 25. Section 11.7 problems 3, 9, 17, 23, 25, 32; Section 11.8 problems 1, 7, 23, 25, 40.
Lecture 7, Wednesday, January 31:  A quick explanation of the "easier" way to compute curvature; review of what we did last time; a couple of examples; why curvature in 3 dimensions is exactly the same as curvature in 2 dimensions; the principal normal vector and its meaning; tangential and normal components of acceleration, the geometric meaning, some useful formulas for computing them, and an example.
Homework: Reread Section 11.6; read Sections 11.7, 11.8; do Section 11.6, problems 23, 33, 38, 43, 46.
Lecture 6, Monday, January 29:  Integrals of vector-valued functions. An example of integrating to find the position function from the acceleration function plus initial position and velocity. Arc-length of curves in the plane, parametrizing by arclength. Curvature of curves in the plane, how to understand it geometrically. The example of the curvature of a circle.
Homework:  Section 11.6, problems 1, 9, 11, 21, 29.
Lecture 5, Friday, January 26:  A little more on planes. Parametric curves, limits, derivatives, the example of a helix, tangent vector. Velocity, acceleration, speed, scalar acceleration. Rules for derivatives of vector-valued functions.
Homework: Section 11.5, problems 5, 8, 15, 17, 23, 27, 35, 42, 55. Note: we did not quite discuss integrals of vector-valued functions today, so you may want to wait to do problems 17 and 27 until after Monday's lecture!
Lecture 4, Wednesday, January 24:  Lines and planes in space. Vector, scalar (implicit), and parametric equations for lines and planes, with examples. After this lecture and doing the homework, you should be able to: Homework: Section 11.4, problems 3, 7, 9, 17, 23, 31, 33, 35, 37, 41. Also, read Section 11.5 for Friday.
Lecture 3, Monday, January 22:  All about the cross product! How to use it to get a perpendicular vector, how to remember the formula for the cross product, how to interpret as a determinant of a 3x3 matrix, geometric interpretation, how to use it to tell whether two vectors are parallel. Also, scalar triple products and their geometric interpretation in terms of volume of a parallelepiped.
Homework: Section 11.3, problems 1, 7, 13, 15, 17, 19. Also, read Sections 11.3 and 11.4.
Lecture 2, Friday, January 19:  Vectors in 3-space. Formula for the distance between two points. Equality, addition, scalar multiplication. Length of vectors. Dot product, what is it good for? Direction angles. Component of one vector in the direction of another, and how to use it to compute work (i.e. force times distance) when the direction in which force is applied is not identical to the direction in which movement takes place.
Homework: Section 11.2, problems 39, 41, 43, 45, 49, 53, 57.
Lecture 1, Wednesday, January 17:  Vectors in the plane. The position vector of a point. Length of vectors. Equality, addition, scalar multiplication. Some standard properties of addition and scalar multiplication. What is the difference between a vector and a point in the plane?
Homework: Section 11.1, problems 1, 5, 9, 17, 21, 33, 51, 55.
Main course page.