Lecture Summaries and Homework Assignments

Lecture 40, Wednesday, April 30:  Homework: study for the final! Master the material.
Stokes examples and applications. Review of part of the course. "Win one for the Gipper" motivational speech intended to get you "fired up" to study for the final.
Transcript here.
Lecture 39, Monday, April 28:  Do Section 14.8 problems 3, 5, 7, 11, 13, 17, 21, 23, and these problems.
Brief recap of Gauss's Law from last time. If F=curl(G) then div(F)=0. If div(F)=0 and domain is nice enough, a method to find G such that F=curl(G). Example. Statement of Stokes's Theorem. Explanation of the geometric meaning. An example we'll do next time.l
Transcript here.
Lecture 38, Friday, April 25:  Read Section 14.8. Do Section 14.7 problems 1, 3, 5, 7, 17, 21, 25, 29, and these problems.
Statement of Divergence Theorem. Examples. Derivation of Gauss's Law (see pp. 1195-1196 in the text). Another idea for using the Divergence Theorem in a slightly tricky way to compute a flux.
Transcript here.
Lecture 37, Wednesday, April 23:  Read Section 14.7. Do these problems (some answers here).
Definition of surface integrals. Examples. Some technical points about which we should take care. Brief discussion of the Implicit Function Theorem and its relevance. Statement of Divergence Theorem in 3D?
Transcript here.
Lecture 36, Monday, April 21:  Read Sections 14.6 and 14.7. Do Section 14.5 problems 31-36, 45, 55, 57. Do Section 14.6 problems 1-8, 17.
Statement and explanation of the Divergence Theorem in 2D. Parametrization of surfaces. Beginning discussion of surface integrals (preliminaries on what the integrand should look like).
Transcript here.
Lecture 35, Friday, April 18:  Do Section 14.5 problems 1,3, 5, 7, 27 (including Divergence) and these problems (some answers here).
Divergence and its meaning. Examples. Conservative vector fields in 3D. Normal and unit (outwarding-facing) normal to a curve in the plane. Set-up for the Divergence Theorem in 2D.
Transcript here.
Lecture 34, Wednesday, April 16:  Do Section 14.5 problems 1, 3, 5, 7, 27. Do Section 14.4 problems 35, 37, 39.
Proof sketch of Green's Theorem. Basics of curl in 3 dimensions. Relationship to what we discussed in 2 dimensions. Some pictures.
Transcript here.
Lecture 33, Monday, April 14:  Read Section 14.5 and also this web page to get some idea of what curl and divergence are all about. Do Section 14.4 problems 15, 17, 25, 29, 33, and these problems (some answers here).
Statement of Green's Theorem, again. Meaning of the integrand in the double integral: curl. Geometric meaning as ``tendency to spin you.'' Definition in terms of a line integral (limit of rotation per unit area), brief sketch of how that gives the expression dF_2/dx - dF_1/dy. Example. Calculating area using a line integral. Application to computing the area of the Folium of Descartes.
Transcript here.
Lecture 32, Friday, April 11:  Do Section 14.4 problems 1, 3, 5, 9, 11, 13, 19, 21, 27.
Line integral around a circle of a vector field that is not conservative but does satisfy dF_2/dx = dF_1/dy. Explanation of the meaning. Theorem that if the region D is simply-connected then the condition dF_2/dx = dF_1/dy is equivalent to the vector field F = being conservative. Green's Theorem via an example. Statement of theorem. Set-up of the proof in a special case.
Transcript here.
Lecture 31, Wednesday, April 9:  Read Section 14.4. Do Section 14.3 problems 13, 17, 23, 31-36, 37, and these problems (some answers here).
Conservative vector fields and independence of path. Two paths with the same endpoints that give the same line integrals, and a discussion of why (in terms of potential energy). Statement of theorem: independence of path is equivalent to vector field being conservative. Fundamental Theorem of Calculus for Line Integrals. Integrating conservative fields around closed curves gives zero, and a theorem about this. An example in which dF_2/dx = dF_1/dy but a line integral around a circle is not zero.
Transcript here.
Lecture 30, Monday, April 7:  Do these problems (some answers here).
Line integrals with respect to arc length. Examples. Line integrals of F dot dr, where F is a vector field and r is a parametrized curve. Meaning. Application to computing work. Special case and its notation. Examples. Brief discussion of independence of path.
Transcript here.
Lecture 29, Friday, April 4:  Read Sections 14.2 and 14.3. Do the homework posted for Lecture 28. Also, do Section 14.2 problems 1, 3, 7, 13, 21.
Reminder of vector fields. Conservative vector fields. A necessary condition in terms of partial derivatives for a vector field to be conservative. How to integrate to find a potential function for a conservative vector field. C^1 paths in space. Definition of line integrals with respect to arc length.
Transcript here.
Lecture 28, Monday, March 31:  Read Section 14.1. Do Section 14.1 problems 1,3,5, 9, 11, 12, 13, 17, 23, 25, 29, 37, 39, and these problems (some answers here).
Vector fields. Definitions. Examples, with graphs. Meaning of a vector field. Sources: velocity fields, force fields. Force fields from potential energy functions. Example of the gravitational potential. A necessary condition for a vector field in the plane to be conservative.
Transcript here.
Lecture 27, Friday, March 28:  Do all of the following homework before Wednesday's midterm: Do Section 13.7 problems 1, 3, 7, 9, 11, 19, 25, 27, 33, 37, 41, 43, 45, 49, 51, 53, 59. Do Section 13.8 problems 23, 25, 27, 29, 31, 32. Note: this looks like a huge number of problems, but many of them are routine and will go quickly.
One more example in cylindrical coordinates. Spherical coordinates. Jacobian determinant of the spherical coordinate transformation. Triple integrals in spherical coordinates. Examples.
Transcript here.
Lecture 26, Wednesday, March 26:  Read Sections 13.5, 13.6, 13.7. Do these problems.
Basics of triple integrals. A couple of examples. Change of variables to cylindrical coordinates. Examples. The general change of variables formula for triple integrals.
Transcript here.
Lecture 25, Monday, March 24:  Do these problems.
Reminder of double integrals and change-of-variables. Why you should take the absolute value of the Jacobian determinant. Why you need to be careful about whether the transformation is one-to-one. (Almost) none of our standard examples will be one-to-one, but that's ok, because the subsets of the domain on which they fail to be one-to-one have no area, so they don't affect the integral.
Transcript here.
Lecture 24, Friday, March 14:  Read Section 13.4. Do Section 13.4 problems 1, 3, 7, 9, 29, 31.
Surface integrals.
Lecture 23, Wednesday, March 12:  Do Section 13.3 problems 17, 19, 21, 23, 27. Also do these problems (some answers here).
More on double integrals in polar and elliptical coordinates. Computation of a Gaussian integral using polar coordinates.
Lecture 22, Monday, March 10:  Do these problems (some answers here).
Reminder of transformations and basic examples. How does a parallelepiped transformation change area? Linear approximation tells us a general rescaling factor. Change of variables formula in general. What is the change of variables factor for our examples. Elliptical coordinates. Examples.
Lecture 21, Friday, March 7:  Read Section 13.8. Do Section 13.8 problems 1, 3, 5, 7, 9, 11. Also do these problems (some answers here; note: I believe the x- and y-components of the centroid in problem 1 have been switched in the written answer!).
Computing center of mass of a plane lamina using double integrals. An example. Start of change-of-variables in multiple integrals: (coordinate) transformations, two basic examples (polar coordinates and ``parallelepiped coordinates'' (i.e. T(u,v) = ua + vb where a and b are vectors in the plane). What does such a thing do to volume?
Lecture 20, Wednesday, March 5:  Do this homework (some answers here).
Reminder of last time. What if R is a general region in the plane? Definition of the integral in such a case. Some good cases in which we don't need to revert to the definition because the region can be handled by Fubini-esque methods. An example in which switching the order of integration helps a lot. Computing area, volume using double integrals.
Lecture 19, Monday, March 3:  Read Sections 13.1-13.3. Do these problems (some answers here).
Quick review of integration in one variable. An example "by hand" using Riemann sums and a limit. Discussion of the definition of the integral of a function f(x,y) of two variables over a rectangle R. Fubini's theorem. How we actually use Fubini's theorem to calculate an integral. What about if R is a more general region in the plane?
Lecture 18, Wednesday, February 27:  Review of quadratic approximation in one variable (i.e. the quadratic part of the Taylor series). Why quadratic approximation makes the second derivative test work. Discussion of quadratic approximation in two variables in terms of the Hessian. Discussion of the model geometries. How the second derivative test can fail to provide information and why.
Lecture 17, Monday, February 25:  Do Section 12.7 problems 21, 25. Also do these problems (some answers here).
Linear regression via an example. Lagrange multipliers: geometry via an example. Formal statement of the method of Lagrange multipliers. Examples.
A nice article about the real-world uses of constrained optimization can be found here.
Lecture 16, Friday, February 22:  Read Section 13.1 for Monday. Catch up on old homeworks! There is a midterm next Friday.
Review of 2nd derivative test. Some discussion of examples, "model geometries," and what the test is really testing for. Constrained optimization and Lagrange multipliers: motivation via "the boundary," basic geometric picture via an example.
Lecture 15, Wednesday, February 20:  Do these problems (some answers here).
Global max and min. Local max and min. Pictures. Critical points. How to identify critical points, and thus look for maxima/minima. Model geometries for second derivative test in two variables. Statement of second derivative test.
Lecture 14, Monday, February 18:  Read Sections 12.7 (again) and 12.8 for Wednesday (although we very likely won't start 12.8 until Friday). Do Section 12.5 problems 21, 23, 25, 27, and these problems (some answers here).
Reminder of the Chain Rule. Differentiation rules of 1-variable calculus via the multivariable chain rule. Implicit differentiation. Brief discussion of optimization in general. Local max and min, global max and min. Some pictures.
Lecture 13, Friday, February 15:  Read Section 12.7 for Monday. Do Section 12.6 problems 27, 31, 45. Do Section 12.5 problems 3, 5, 7, 17. Also do these problems (some answers here).
Reminder of directional derivatives. Directional derivative as the component of the gradient in the given direction. Gradient as direction of fastest increase. Statement of the chain rule and derivation using linear approximation. Examples. Justification for the claims about the gradient as normal vector to the tangent plane.
Lecture 12, Wednesday, February 13:  Read Section 12.5 for Friday. Do Section 12.6 problems 11, 13, 17, 23, 41, 43. Also do these problems (some answers here).
Tentative summary: Reminder of the gradient and linear approximation. Gradient as normal vector to a level surface. Examples. First steps in directional derivatives: discussion of definition and its meaning.
Lecture 11, Monday, February 11:  Read Section 12.6. Do Section 12.6 problems 1, 3, 5, 7, 9, 39. Do Section 12.4 problem 42. Also do these problems (some answers here).
Parametrization of traces ("x-curves" and "y-curves") in general. Tangent lines to these curves at (a,b,f(a,b)). How this gives the formula in Theorem 4.1, p. 963, for the tangent plane. Brief discussion: you could do the same thing in general, if you knew a parametrization of the surface that interests you. Linear approximation and its geometry. Definition of the gradient as the thing that allows linear approximation. Examples.
Lecture 10, Friday, February 8:  Reread Sections 12.2-12.4 for Monday. Do the following:
  • Find both implicit and parametric equations for the tangent lines to:
    • The ellipse x^2 + 4y^2 = 4 at the point (0, 1) and the point (square root of 3, 1/2).
    • The graph of f(x) = x^3 - 4x + 3 at the point (1, 0).
  • Let F(x,y) = x^3 -2xy + 3y^2.
    • Give parametrizations of the "x-curve" and "y-curve" (also known as traces) through the point (1,2,8) on the graph z=F(x,y).
    • Compute parametric equations for the tangent lines to these two curves at the point (1,2,8).
    • Compute a vector n perpendicular to these two tangent lines.
    • Use this to give an equation of the tangent plane to the graph at the point (1,2,8).
    • Using your solution to problem 47 in Section 10.6 (see below), carry out the same procedure to compute a tangent plane to the surface defined by x^2 + y^2/4 + z^2 = 1 at the point (1/2, 1, 1/(square root of 2)).
  • Do Section 12.3 problem 37. Do Section 10.6 problems 47 and 49.
A bit more on Clairault's Theorem and what the hypotheses mean. Review of tangent lines to graphs y=f(x). Parametrizing a graph y=f(x) in the plane. Finding a tangent line to a parametrized curve in the plane. An example worked both ways. Traces (i.e. "x-curve" and "y-curve") on a surface z=f(x,y) in 3-dimensional space. Parametrizing them and finding tangent lines. How this determines a tangent plane in an example (we'll discuss the general theory on Monday).
Lecture 9, Wednesday, February 6:  Read Sections 12.3 and 12.4. Do Section 12.3 problems 3, 5, 7, 9, 15, 17, 19, 29, 43, 44. [In connection with problems 43 and 44, you may find it interesting to look here.] Also, do these problems (some answers here).
Partial derivatives: motivation from one variable. Derivative as linear approximation (Taylor series in one variable). Derivative as rate of change. Rates of change in different variables when f(x,y) is a function of more than one variable. Definition of partial derivatives. Interpretation as slopes of tangent lines to curves on the graph in the x- and y-directions. Why the definition leads to an easy principle for calculating ("think of y as a constant, compute the derivative, etc."). Examples. Higher partial derivatives, notation. A calculation of mixed partials. Brief discussion of an example in which mixed partials are different. Theorem of Clairault-Schwarz (Theorem 3.1 in Section 12.3, p. 954 in the text).
Lecture 8, Friday, February 1:  Read Sections 12.1 and 12.2. Do Section 11.4 problems 1, 3. Do Section 12.1 problems 1, 3, 7, 11, 17 (see page 837 for the definition of a "trace"), 19, 35, 47, 49, 67.
Scalar-valued functions of 2 and 3 variables. Examples. The domain may not be all of the plane or space. How to graph scalar-valued functions of 2 variables. Examples. How to tell whether a surface in space is the graph of a function. The utility of drawing level curves and traces. Level surfaces of a function of 3 variables. Vector-valued functions of several variables and how they arise.
Lecture 7, Wednesday, January 30:  Read Section 11.4 up to Example 4.2. Do Section 11.3 problems 1, 5, 7, 11, 13, 39. Do Section 11.4 problems 7, 9, 11. Also, compute the arc length of r(t) = < 3 sin(2t), 3 cos(2t), 8t> from t=0 to t = pi, and compute the arc length function as in Problem 19 here.
Theorem that speed is constant if and only if velocity is orthogonal to acceleration for all times t. Example of a helix. Acceleration does not determine position, but acceleration plus initial velocity plus initial position does. Examples. Arc length function. Reparametrization by arc length. Examples.
Lecture 6, Monday, January 28:  Read Sections 11.1-11.3. Do Section 11.1 problems 5, 11, 17, 31, 49, 50. Do Section 11.2 problems 13, 17, 35, 39.
Vector-valued functions of one variable, i.e. parametrized curves. Particles moving in space. Example of a particle moving around a circle in two different ways. How to compute velocity as a derivative (i.e. tangent vector). How to compute speed. Example of a helix. How to compute distance travelled/arc length as an integral. Examples. Arc length formula in space reproduces arc length formula for a graph of a scalar-valued function of one variable. Some basic properties of the derivative of a vector-valued function of one variable. Vector-valued function has "constant length" if and only if velocity is perpendicular to position.
Lecture 5, Friday, January 25:  Look at this web page and play around with it a bit. I'll give some homework related to it shortly. Also, read Section 10.6 from the text. Do these problems. [Solutions: (1) x-y+2z=8. (2) x-1 = y-2 and y-2 = 2z-6. (3) y-x+3z=2. (4) A=-4/3. (5) r(t) = <2t+5, -3t, 5t+6>.] Also, read this and do the problem contained there.
Implicit vs. parametric description of subsets of space. Implicit and parametric equations for a line. Implicit equation for a plane via dot product and the normal vector.
Lecture 4, Wednesday, January 23:  Do Section 10.4 problems 29, 33, 51, 61. Read Section 10.5 and do problem 3.
Volume of a parallelepiped in terms of scalar triple product/determinant. The geometric meaning of the determinant. Magnus forces (see also Wikipedia on this subject). Lines and planes in space. Implicit vs. parametric representation of subsets of the plane or of space.
Lecture 3, Friday, January 18:  Read Section 10.4 about torque. Read Section 10.5. Do Section 10.4 problems 1, 7, 11, 19, 21, 25, 41, 53, 65.
2x2 and 3x3 determinants and how to calculate them. Definition of the cross product. Examples. Some basic properties of cross product. a x b is orthogonal to a and b. Right-hand rule for determining in which direction the cross product points. Length of the cross product as a function of lengths of the vectors and sine of the angle between. Area of a parallelogram in terms of cross product.

A student reminded me of another (maybe nicer) yoga for computing 3x3 determinants, which is summarized here (PDF file). You may want to read it and see which you prefer!
Lecture 2, Wednesday, January 16:  Read Sections 10.3 and 10.4. Do Section 10.3 problems 3, 5, 9, 13, 17, 21, 25, 26, 27, 29, 31, 32, 50. Optional: do problem 39 after reading about the Cauchy-Schwartz inequality.
Dot product of vectors. Basic properties of dot product. A few examples. Formula for the dot product in terms of cosine of the angle between vectors. Using the formula to find the angle between vectors. Orthogonal vectors. Standard basis vectors. Component and projection; how to compute them, what they mean geometrically. How to compute work in two- or three-dimensional space.
Lecture 1, Monday, January 14:  Read Sections 10.1 and 10.2. Do Section 10.1 problems 1, 5, 25, 35; Section 10.2 problems 11, 13, 21, 23, 27, 43, 47. You could also try Section 10.2 problems 63-65 but they are optional.
Definition of a vector in 2 and 3 dimensions. Position vector of a point. Vector addition and scalar multiplication; geometric interpretation of them. Length of a vector. When are two vectors parallel? Unit vectors and how to find them. In Tuesday discussion section: right-handed coordinate systems. Equations of spheres.
Main course page.