`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, R

**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.