Multivariable Calculus and Vector Analysis (Math 243) for Fall 1998-99

Multivariable Calculus and Vector Analysis (Math 243)

447 Altgeld Hall, 9am, M-Th

Instructor: Mihail Kolountzakis

E-mail: kolount@math.uiuc.edu

Text: J.E. Marsden, A.J. Tromba and A. Weinstein, Basic Multivariable Calculus, Springer Verlag and Freeman.

Material to be covered (approximately): Chapters 1-7. (Not everything will be covered from those chapters.)

Topics include: Vector algebra, partial derivatives, extrema of functions of many variables, vector-valued functions, multiple integrals, curve and surface integrals, Green, Stokes and Gauss' theorems.

Grading policy: There will be two 2-hour-long tests plus the final. Each of the two tests counts for 20% of the grade and the remainder 60% is the final. The homework will not directly contribute to your grade.

Office hours (at 234 Illini Hall): anytime I'm in the office (or anywhere else you find me) or by appointment.

W, Aug 26: Organizational stuff, coordinate systems in 2- and 3-space, R2 and R3 and their identification with 2- and 3-space, addition and multiplication by scalars for n-tuples, vectors in space and their addition.

HWK 1: From 1.1: 13-16, 18, 22, 26, 27, 29, 31, 33, 36

Th, Aug 27: Completion of section 1. The parametrization of a line segment and a straight line.

M, Aug 31: Parametrization of a plane in space. Inner (dot) product, distance, algebraic properties of the dot product, geometric properties (how to find the angle between two vectors).

T, Sep 1: Almost completed section 1.2. Mostly discussed how one computes the projection of a vector onto a line spanned by another vector and how onto a plane spanned by two non-collinear vectors.

HWK 2: (Section 1.2) 3, 5, 9, 13, 17, 20, 21, 25, 26, 27-28, 34.

W, Sep 2: We completed section 1.2 and briefly mentioned the definitions of 2x2 and 3x3 matrices and determinants. Please go over 1.3 yourselves and remember the determinant properties. We also covered some of the hardest homework problems.

Th, Sep 3: We talked about the properties of determinants and introduced the cross product of two vectors. We proved some of its algebraic properties. We also had our first quiz. Most of you did well but those who computed lengths (besides that of vector C on which you need to project) either got it completely wrong or unnecessarily complicated it.

HWK 3: (Section 1.3) 10, 13, 15.

T, Sep 8: Properties of the cross product, the triple product (a x b) . c, the magnitude, direction and orientation of a x b.

HWK 4: (Section 1.4) 3, 4, 12, 20, 21, 24, 27, 29, 31, 32, 33, 37-39, 41, 42, 43, 46.

W, Sep 9: We talked about the interpretation of determinants and cross and triple products as areas and volumes (in two- and three-dimensional space). Also saw how to compute the distance of a point from a given plane. We started talking about n-dimensional space: its vectors, algebraic operations on them, dot product, proved the Cauchy-Schwartz inequality.

Th, Sep 10: We continued Section 1.5 and taled about m x n matrices, algebraic operations on them (especially multiplication), transpose, inverse. Went over some problems in last HWK.

HWK 5: (Section 1.5) 2, 3, 9, 13, 16, 18, 22, 29, 30, 31, 34, 35, 36.

M, Sep 14: We talked a bit about linear transformations from R^m to R^n (vectors with m coord's to vectors with n coord's) and their matrix representations. We also spoke about the parametrization of paths in plane or space and discussed several examples (line segment, circle, parabola, helix).

HWK 6: (Section 1.6) 6, 9, 13, 15, 17.

T, Sep 15: The parametrization of the cycloid. The velocity vector of a moving particle. Chapter 2. Introduction to functions of many variables, graphs and level curves (surfaces).

W, Sep 16: We spoke about how to visualize the graphs and level curves and surfaces of functions in 2 and 3 variables. We skipped many of the examples in 2.1, however do look at them, and pay special attention to the ellipsoid. We started 2.2 and defined partial derivatives of multivariable functions, and showed how to compute them.

HWK 7: (Section 2.1) 9, 13, 17, 21.

Th, Sep 17: We talked a bit more about how to plot the graph of a function in 2 variables (read it, but I am not expecting you to become an expert in drawing such objects), how to tell what a level curve or surface looks like. Finally, we talked more about partial derivatives. We defined also what a directional derivative is (this is further on in your book in 2.5), tha gradient and what it represents, how to get a directional derivative from the gradient, and had a quick preview of the chain rule for differentiating a composite function of two variables. We also saw the definition of a limit of a function of many variables as the arguments approach a point.

M, Sep 21: Limit of a function as (x, y) approaches (x0, y0), several situations that can occur depending on how (x, y) approaches the limit point. Continuity of a function at a point (x0, y0).

HWK 8: (Section 2.2) 3, 4, 7, 8, 10, 11, 13, 15, 18, 20, 22, 27, 30.

T, Sep 22, and W, Sep 23: We derived the equation of the tangent plane to the graph of a function f of two variables at some point P = (x, y, f(x,y)). We also talked about how to approximate our function f in the vicinity of a point (x0, y0) by a linear expression in x, y. This property (of being able to be well approximated) we called "differentiability" of f at (x0, y0) and we defined it for general multi-variable vector-valued functions. We also stated the differentiability criterion which says that if a function f has continuous partial derivatives at (x0, y0) then it is differentiable there.

Th, Sep 24: We saw how to differentiate composite functions (various types of composition) using the chain rule.

HWK 9: (Section 2.3) 8, 12, 16, (Section 2.4) 1, 2, 3, 5, 6, 7, 8.

The first exam is tentatively scheduled for Th, Oct 8, sometime after 5pm. It will be two hours long and the material will be eveything that has been covered until then. More details will follow.

M, Sep 28: We completed our discussion of the chain rule by giving it in matrix form. We talked about directional derivatives and the gradient of a function at a point.

HWK 10: (Section 2.4) 15, 16, 19, 21, 22, 24.

The first exam will be on Thursday, Oct. 8, at 7-9pm in 124 Burrill Hall.

T, Sep 29: We saw how compute the rate of change of a function f(x,y,z) when moving along a curve c(t), using the chain rule. We also saw how to compute the tangent plane on a surface defined by an equation f(x,y,z) = 0.

HWK 11: (Section 2.5) 7, 11, 12, 13, 17, 18, 25, 29, 30, 33, 37, 41, 43.

W, Sep 30: We computed some tangent planes on surfaces. Saw also how to find the derivatives of functions defined implicitly.

Th, Oct 1: (Sectio 3.1) We defined higher order partial derivatives of multivariable functions and saw that the order of differentiation can be arbitrary (in most cases of interest). We also saw several examples of Partial Differential Equations (The Heat Eqn, the Laplace Eqn, the Poisson Eqn and the Wave Eqn). A sample exam was distributed to help you prepare for the coming exam.

M, Oct 5: We talked about the Taylor expansion of order 2 of a function of two variables.

HWK 12: (Section 2.6) 4, 5, 9, 10, 18, 19, 20, (Review Section for Chapter 2) 31, 37, 45, 52, 63, 65, 66, (Section 3.1) 11, 12, 13, 14, 16(a) (Section 3.2) 5, 6, 11, 12.

T, Oct 6: We went over the sample exam as a preparation for the test of Thursday.

W, Oct 7: We discussed how to locate local minima and maxima of functions looking for critical points (points where both partial derivatives are 0), and saw some examples of how to distinguish critical points which are not local extrema and the possible behavior of the function in their vicinity.

HWK 13: (Section 3.3) 6, 7, 11, 13, 14, 15, 17.

Th, Oct 8: We saw how to use the discriminant (an expression made up of second order derivatives of a function) at a critical point in order to tell if this is a local extremum and what kinf if it is (this method is inconclusive sometimes though).

Grades for the first test are here.

M, Oct 12: We talked about constrained optimization. That is, how to find the extrema of a function subject to the variables satisfying some constraint of the type g(x, y) = c. We saw the so-called method of Largange multipliers which reduces this problem (i.e., finding the critical points of f(x, y) subject to g(x, y) = c) to (non-linear in general) algebraic system in three variables:x, y, and lambda (the Largange multiplier).

HWK 14: (Section 3.4) 4, 10, 16, 21, (Section 3.5) 7, 8, 10, 13, 17, 18.

T, Oct 13: We finished the discussion of constrained of constrained optimization by showing the Lagrange multiplier method for a function of 3 variables subject to 1 constraint. We also briefly showed what happens for 2 constraints. Then we did a review of vector valued functions and their derivatives.

HWK 15: (Section 4.1) 1, 5, 6, 7, 8.

W, Oct 14: We continued our discussion about vector valued functions of a single variable, velocity and acceleration. We derived Kepler's law which relates the period of a planet turning about the sun at a circular orbit to the radius of the orbit. We defined arc-length along a curve ans showed that it is independent of the parametrization of the curve chosen.

HWK 16: (Section 4.1) 10, 11, 13, 14, 15, 16, (Section 4.2) 5, 6, 7, 9, 12, 13.

Th, Oct 15: The reparametrization of a curve by arc-length. Introduction to the concept of a vector field.

M, Oct 19: The gravitational and electrostatic force fields. Conservation of energy in a force field that comes from a potential (pay attention to problem 17, p. 249). Flow lines in a vector field. Flow lines as a system of differential equations. The divergence of a vector field and physical interpretation of the divergence of the velocity field in a fluid flow.

HWK 17: (Section 4.3) 3, 4, 6, 8, 11, 14, (Section 4.4) 3, 4, 5, 6, 7, 12.

T, Oct 20: The curl of a vector field and properties.

HWK 18: (Section 4.4) 13, 14, 22, 25, 28, 29, 30, 31.

W, Oct 21: Completion of section 4.4 by proving most of the identities on p. 260.

HWK 19: (Review for Chapter 4) 5, 7, 11, 12, 14, 16, 20, 22, 27, 30, 31, 32, 33.

Th, Oct 22: Multiple integrals defined as volumes. Cavalieri's principle. How to compute multiple integrals as iterated integrals.

HWK 20: (Section 5.1) 3, 4, 5, 7, 8, 9, 10, 12, 15.

M, Oct 26: We saw how to compute integrals over non-rectangular regions (section 5.3 mostly).

HWK 21: (Section 5.3) 5, 6, 7, 8, 9, 10.

T, Oct 27: We did some more examples of double integrals over some regions. We defined the average (mean) value of a function over a region. We started discussion about triple integrals and computed the volume of a tetrahedron as an iterated triple integral.

HWK 22: (Section 5.3) 13, 14, 17, 18, 23, 25, 26, 29, 30.

W, Oct 28: More on triple integrals. The centroid of a region in space.

HWK 23: (Section 5.4) 2, 4, 5, 9, 10, 12, 16, 19, 23, 24, 26, 27.

Th, Oct 29: We went over homework problems for Section 5.3

HWK 24: (Section 5.3) 13, 14, 20, 21.

M, Nov 2: Polar coordinates in the plane, cylindrical coordinates in space.

HWK 25: (Section 5.5) 3, 4, 5, 7, 8, 9, 10, 13, 14, 17.

T, Nov 3: Spherical coordinates in space and how to make a general change of variable in 2- and 3-dimensional space. We computed the volume of a spherical shell and used this to calculate the area of the unit sphere.

HWK 26: (Section 5.5) 27, 28, 30, 32, 33, 36, 37, 39.

The 2nd test will take place on Nov 19, 7-9pm, at 112 Chem Annex. The material to be examined is everything that I shall have tought by the previous Monday.

W, Nov 4: We talked (again) about the center of mass of regions in space with a (non-uniform) mass distribution in them, and computed some examples.

HWK 27: (Section 5.6) 6, 7, 11, 13, 14, 15, 16.

F, Nov 6: We did several problems regarding multiple integrals in class.

NO CLASS for the week of the 9th of November. We resume on Monday the 16th.

***** And a little postcard for you kids from California. *****

M, Nov 16: We started talking about line integrals. The work done by a particle moving in a force field. Conservative force fields come from gradients of scalar functions.

HWK 28: (Section 6.1) 1, 3, 4, 6, 7, 9, 11.

EXTRA CLASS on T, Nov 17, and W, Nov 18, at 6pm in 145 Altgeld. Very important to attend given that we have a test on Thursday.

T, Nov 17: Continuation of line integrals. Independence on the choice of the parametrization. Conservative vector fields and dependence of the line integral only on the endpoints of the curve and not on the path. The integral of a scalar function along a curve.

HWK 29: (Section 6.1) 13, 15, 17, 20, 22, 27, 28.

W, Nov 18: Parametrized surfaces. How to find the parametrization. How to find the normal vector at a given point. How to find the area of a surface.

HWK 30: (Section 6.2) 5, 6, 9.

Th, Nov 19: The area element dS of a parametrized surface. Graphs of two-variable functions as parametrized surfaces and how to find their area. Surfaces of revolution.

HWK 31: (Section 6.3) 3, 5, 7, 9, 12, 16.

The 2nd test will take place on Nov 19, 7-9pm, at 112 Chem Annex. The material to be examined is everything that I shall have tought by the previous Monday.

NO CLASS on Monday, Nov 30, and on Tuesday, Dec 1 (after the Thanksgiving holiday). We do have class this week of the 23rd of November on Monday and Tuesday.

2nd test grades are here. Unfortunately there has been a noticeable decline in scores.

W, Dec 1: We discussed examples of surface integrals representing flux, such as heat flow and fluid flow.

Th, Dec 3: Finished discussion of surface integrals talking about Gauss' Law in electrostatics and how to derive Coulomb's law from it. We also described how to compute surface integrals when the surface is the graph of a function. We then moved to section 7.1 and discussed Green's theorem. I showed how to computed the area of a polygon using it.

HWK 32: (Section 6.4) 6, 8(b), 9, 11, 14, 15, (Section 7.1) 1, 2.

We shall have a problem session next Thursday, Dec 10, at 7pm in 145 Altgeld Hall. Please come prepared to ask questions.

The Final Exam will take place on Monday, December 14, 8-11am, in 217 Noyes Lab.

M, Dec 7: Applications of Green's theorem in computing areas. The divergence theorem in two and three dimensions. We computed the volume of the unit sphere using the divergence theorem in three dim.

HWK 33: (Section 7.1) 5, 7, 12, 18, 20, 22, 26, 27.

T, Dec 8: Green's Theorem in vector (curl) form. Stokes theorem for surfaces in space. Applications to electromagnetics.

HWK 34: (Section 7.2) 6, 9, 14, 15, 16, 23, 24.

W, Dec 9: We did several of the problems of Section 7.2.

Th, Dec 10: We discussed the divergence theorem and its applications.

HWK 35: (Section 7.3) 2, 3, 6, 7, 8, 10, 13.

We shall have a problem session tonight, Thursday, Dec 10, at 7pm in 145 Altgeld Hall. Please come prepared to ask questions.

The Final Exam will take place on Monday, December 14, 8-11am, in 217 Noyes Lab.

Your final grades are here. I will consider appeals, complaints, etc, only until Tuesday, 12/15/98, 6pm.