Math 2500 Summer 2009, Smith, lectures: Boyd 304, 10:30A-11:30A MTWThFr

We will schedule a group problem session, tentatively 2:15-4:30, daily in Room 322.

Book: University Calculus, Hass/Weir/Thomas, chaps 10-14, see the dept. syllabus at: http://www.math.uga.edu/undergraduate/departmental_syllabi.html

My Office: Boyd 448; phone: 542-2595; rcsmith99@gmail.com, roy@math.uga.edu,

use both edresses for quickest response, (not just the one in webwork).

Official course theme song: “We’ve got a long way to go and a short time to get there”.

Prerequisites: Working knowledge of algebra, geometry, trig, one variable calculus, and basic vector algebra. Know definitions, geometric and physical interpretations, be able to compute accurately, state (and prove some) theorems, set up and solve problems.

Universal homework assignment: Come to every class, ask questions, review/rewrite your notes daily. Read every assigned section in the book, work out the examples, then do as many problems as possible (I did them all the first time I taught the course), at least the ones on the departmental syllabus and those assigned on webwork. Ask questions about everything you don’t understand as soon as possible.

Webwork: In addition to hw problems on the departmental syllabus, there will be regular webwork assignments at https://webwork.math.uga.edu/webwork2/

Your email handle is your login name, and your 810-xx-xxxx number your password.

Tests:(tentative) I: 6/22; II: 7/06; III: 7/20; Final(firm): Friday, July 31, noon-3, in our room.

The final cannot be moved, nor missed tests made up, but number of tests and dates may change.

Verify your exam schedule: http://www.reg.uga.edu/or.nsf/html/Summer_Exam_Schedule_2009

Grades: Your grade will not be lower than that given by the formula:

50% tests+30% final+20% hw, quizzes, classroom performance.

0-59:F,60-69:D,70-72:C-,73-76:C,77-79:C+,80-82:B-,83-86:B,87-89:B+,90-92:A-,93-100:A.

Excessive absence can result in a WF. If you withdraw, do so officially and promptly. The semester midpoint is July 1, and the last day to withdraw is July 9. (Verify this.)

Academic Honesty: In all work for credit, do your own research, thinking, computations and writeup. You may brainstorm with others and you should. Notes, books, calculators are not allowed on tests. Read the policy at http://www.uga.edu/honesty/ahpd/culture_honesty.htm

Disclaimer: This syllabus provides a general plan for the course; deviations may be necessary.

2500 Overview: We will study the mathematics needed to analyze such problems as the motion of a particle or fluid under the influence of forces. A moving particle in space is represented by a function f:RàR^3, and its velocity by a family of vectors tangent to the trajectory of this curve.

We also want to understand such questions as the propagation of heat in space, measured by a temperature function g:R^3àR, or the motion of waves and electric force fields represented by functions R^3àR^3. Thus we also study derivatives and integrals of such functions.

This course covers calculus, both differential and integral, for functions of several variables. The derivative of a function f:RàR^3 representing a path in space, is the velocity vector tangent to the path of motion, and the derivative of a “potential” function g:R^3àR is a vector perpendicular to the surface where the potential is constant. There is also a concept of differentiation of vector fields so that the integral of a vector field over a path or surface bounding a region, equals the integral of its derivative over the region bounded. These analogs of the fundamental theorem of calculus: Green’s theorem, Stokes‘ theorem, and divergence theorem, all particular cases of one general statement, form a final highlight of the course.

It is useful to introduce the algebra of vectors for dealing with the geometry of higher dimensional spaces, so we begin with a review of vector algebra.

Review of vector algebra

That is, addition of vectors by the parallelogram law, dot products of vectors to measure lengths and angles, and (in three dimensional space only) the cross product of vectors to measure area. We also study linear real valued functions of several variables, and vector valued functions of one variable, and how to represent them by planes or lines in space.

Differential calculus

Next we consider non linear functions and the technique of finding linear approximations to them. First we consider derivatives of vector valued functions of a single variable. By representing such a function as a sequence of real valued functions the derivative becomes just a sequence of ordinary derivatives of one variable functions as in 2250, so there is no new difficulty here. The first derivative is the vector tangent to the path of motion. Second and third derivatives measure the curvature and twisting of that path.

Then we study the differential calculus of functions of two and three variables, which means the theory of defining and calculating linear and quadratic approximations to functions of several variables. Polynomials provide our model, since for those functions the approximations are determined simply by the monomial terms of degrees one and two. The technique of partial derivatives allows us to simplify the computation of these approximations for polynomials, and also to generalize them to functions more general than polynomials. This leads to the “Taylor” polynomials approximating smooth functions and in good cases to the “Taylor series” representing an analytic function. For smooth functions, we find equations for the approximating tangent plane to the graph at each point. For parametrized curves, we give equations for the velocity vector at each point.

The basic idea is to identify those properties of a function which are determined just by the linear, or by the linear plus quadratic terms. E.g. the possibility of having a local extremum is reflected by the linear term, which must be zero, although this is not sufficient. In case this term is zero, we then introduce the quadratic terms which in the general case can give a sufficient condition for an extremum, and also a necessary condition. I.e. it is necessary that the quadratic term be “semi definite” for an extremum to exist, and “definiteness” (either positive or negative) is sufficient. In order to determine definiteness of the quadratic terms, we show how to complete the square and give a simple computational criterion reminiscent of the discriminant of a quadratic equation in one variable. I have no idea what the third degree term tells you.

Integral calculus

Integration in one variable can be thought of finding areas under the curve forming the graph of a function, or as averaging the values, i.e. the height of the graph at various points, of that function. For functions of two variables, we can compute volumes under the graph (which is a surface), and also think of it as averaging the heights or values of the function. For functions of more variables, we can still think of averaging the values, and define an integral formally analogous to that in lower dimensions, thus giving a plausible concept of higher dimensional “volume”. Indeed the concept of computing work done to raise a solid mass through a given distance has already led us to an integral over a three dimensional region that can be thought of as four dimensional volume.

We will recall the fundamental principle of computing areas and volumes from earlier courses, the fact that two plane regions whose vertical slices are all the same length have the same area (i.e. area between the graphs of f ≥ g is the integral of f-g) and the fact that two solids whose horizontal plane slices are all the same area have the same volume, Cavalieri’s principle. This method of volumes by slicing, which allowed us to compute volumes by integrals in one variable, generalizes to the theorem of Fubini, which reduces an integral of a function of two variables to two repeated integrals of functions of one variable. This can also be taken further to any finite number of dimensions, with appropriate hypotheses of continuity.

To deal with problems having a certain symmetry, spherical or cylindrical, we find it helpful to introduce appropriate coordinates, and we learn the corresponding transformations of volume integrals for these important special cases. We may give the general formula for transformation of volumes as well.

Finally, the integral of a vector field can be used to compute the work done by a force on a moving particle, or the total amount of fluid flowing across a given boundary surface or curve in a given time. Generalizations of the fundamental theorem of calculus then have physical interpretations such as the fact that the total amount of an incompressible fluid flowing across a closed curve or surface equals the amount of fluid being created or produced inside the region it bounds, or that the tendency of a fluid to rotate around a closed curve is caused by the presence of “whirlpools” inside the curve. One can even deduce consequences such as the fundamental theorem of calculus, that every polynomial has at least one complex root, and that the wind cannot be blowing at every point of the earth simultaneously (assuming the earth were a perfect sphere, and some other ideal assumptions).