(Math 2401, sections J1 & J2)

**Mihalis Kolountzakis**

School of Mathematics

Georgia Institute of Technology

686 Cherry Street NW

Atlanta, GA 30332

E-mail:`kolount AT gmail.com`

**January 2007**

School of Mathematics

Georgia Institute of Technology

686 Cherry Street NW

Atlanta, GA 30332

E-mail:

- 1 What the course is about
- 2 Schedule
- 3 Grading
- 4 Course Progress
- 4.1 M, 1/8/07: Vector valued functions of a real variable
- 4.2 W, 1/10/07: Differentiation rules for vector-valued functions
- 4.3 F, 1/12/07: Parametrized curves and their tangents
- 4.4 W, 1/17/07: Tangent and normal vector. Length of a curve.
- 4.5 F, 1/19/07: Parametrization by arc-length. Curvature of plane curves.
- 4.6 M, 1/22/07: Curvature of space curves. Decomposition of the acceleration vector.
- 4.7 W, 1/24/07: Applications of vector calculus to Mechanics
- 4.8 F, 1/26/07: Initial Value Problems. Scalar functions of several variables.
- 4.9 M, 1/29/07: Quadric surfaces
- 4.10 W, 1/31/07: Level curves and surfaces; Partial derivatives of functions of many variables
- 4.11 F, 2/2/07: Partial derivatives. Open sets.
- 4.12 M, 2/5/07: Open and closed sets, interior and boundary points
- 4.13 W, 2/7/07: Review
- 4.14 F, 2/9/07: First Test
- 4.15 M, 2/12/07: Continuity of multivariable functions, partial derivatives and mixed second order derivatives
- 4.16 W, 2/14/07: Differentiability of multivariable functions. The gradient of a function
- 4.17 F, 2/16/07: Gradients and directional derivatives
- 4.18 M, 2/19/07: Mean Value Theorem in the multivariable case; Chain rules
- 4.19 W, 2/21/07: The gradient as normal vector to level curves and surfaces
- 4.20 F, 2/23/07: Local extrema
- 4.21 M, 2/26/07: Absolute extrema
- 4.22 W, 2/28/07: Function extremization under side conditions
- 4.23 F, 3/2/07: Function extremization under side conditions, continued
- 4.24 M, 3/5/07: Summation sign
- 4.25 W, 3/7/07: Integral of a continuous function of two variables
- 4.26 F, 3/9/07: Double integrals by repeated integration
- 4.27 M, 3/12/07: Review session
- 4.28 W, 3/14/07: Second Test
- 4.29 F, 3/16/07: Test problems
- 4.30 M, 3/26/07: Double integrals in polar coordinates
- 4.31 W, 3/28/05: Applications of double integrals to mechanics
- 4.32 F, 3/30/07: Triple integrals
- 4.33 M, 4/2/07: Triple integrals in cylindrical coordinates
- 4.34 W, 4/4/07: Spherical coordinates for triple integrals
- 4.35 F, 4/6/07: General change of variables in multiple integrals
- 4.36 M, 4/9/07: Line integrals
- 4.37 W, 4/11/07: The fundamental theorem of calculus for line integrals
- 4.38 F, 4/13/07: Work and conservation of energy
- 4.39 M, 4/16/07: New way of writing line integrals; integrals with respect to arc-length
- 4.40 W, 4/18/07: Green's theorem
- 4.41 F, 4/20/07: Applications of Green's formula to evaluation of some double integrals
- 4.42 M, 4/23/07: Surface area
- 4.43 W, 4/25/07: Review session
- 4.44 F, 4/27/07: Review session
- 4.45 F, 5/4/07: Final Test

Look here for the syllabus and more information.

You can also have a look at a previous semester's (2005) web site of mine for the same course.

*Main lectures*: MWF 14:05 - 14:55, in Skiles 202.

My office hours are: Thu 10-12 in my office (Skiles 133).

You're also welcome to ask me questions any time you see me, anywhere.

*Recitation sessions*: TTh 14:05-14:55 in Skiles 202 (instructor **Daniel Tiefa**)

Dan's office hours are TBA.

There will be quizzes each Thursday (except for the first one) for the last 15 minutes of the session. One problem will be asked among those (or very much resembling one of those) that were assigned as homework.

The quizzes are worth 20% of the grade (the two smallest quiz scores will be discounted for each of you),
each of the two midterms (**Feb 9 and March 14**) is worth another 20% and the final is worth 40%.

We covered §13.1.
We saw examples of functions of a real variable whose values are 2D or 3D vectors.
We defined the limit

and we saw that the limit can be computed by evaluating the limits of the coordinate functions. We also defined the derivative and the integral of such a function and spoke briefly about how to draw the curve that such a function defines.

HW problems: §13.1: 1, 2, 15, 16, 21, 22, 27, 28, 39, 41.

For the first week only, my office hours will be held on Friday, 3-5.

We saw how to compute the derivatives of vector valued functions which have been
made by piecing together, in various ways, other functions. For example, we saw that
if is a scalar function and is a vector function then the derivative
of the function which is defined by

is given by

We saw various such rules and proved a few.

HW problems: §13.2: 1, 2, 4, 6, 7, 9, 11, 17, 18, 21, 22, 27, 29, 32.

The first midterm exam will take place on Fri, Feb 9, during our regular meeting and in the same place.

The second midterm will take place on Wed, Mar 14, also during our regular meeting and in the same place.

We saw a few examples of vector-valued functions viewed as parametrizations of curves in space. We saw that when traverses a curve then is a tangent vector of at point , and solved some problems which asked geometric questions about curves that had been described in parametric form. We also saw how to find the intersection points of two curves that have been given to us in parametrized form. One must be careful in this problem not to use the same name of the parameter for both curves.

Some of the problems listed below you cannot solve without reading the rest of §13.3 (we will cover this on Wednesday).

HW problems: §13.3: 1-4, 9, 14, 15, 18, 35-38.

We finished §13.3: talked about the unit tangent vector to a curve at point and the principal normal vector as well as the osculating plane they define. We computed an example.

Then we saw how to calculate the length of a curve parametrized by the vector function , for . We computed some examples and also saw by example that the formula we gave for arc-length does not depend on the parametrization of the curve. That is we took two different paraetrizations of a specific curve (a circle, actually) and used our length formula with both of these parametrizations and we got the same result, as we should.

HW problems: §13.4: 1-4, 8, 9, 17, 23.

My office hours tommorrow, Thursday Jan 18, will be held from 9-11 (not 10-12) as I want to go to a lecture at 11 (this change may become permanent).

If any of your planned to see me from 11-12, please send me a message to set up an appointment (but not on Thursday).

We covered the part of §13.5 that concerns plane curves and their curvature. We saw how to
reparametrize a curve which has been given to us in parametric form in such a way that
the parameter is now the arc-length along the curve. This parametrization has the property,
which characterizes it, that the velocity vector has constant magnitude equal to .
The curvature of a plane curve is defined as the derivative with respect to of the angle formed
by the tangent line to the curve and the -axis, in absolute value. (This definition will have to
be amended next time in order to define the curvature of *space* curves.) We then saw formulas
that allow us to calculate the curvature given the parametrization of the curve. We applied these
formulas to several examples. Finally we defined the radius of curvature and the center of curvature.

Of the following problems, do now those that concern plane curves, and leave those about space curves for after having finished §13.5.

HW Problems: §13.5: 1, 2, 5, 6, 10, 13, 14, 21, 22, 41, 42, 58.

We saw how one defines the curvature of space curves (curves which are not known to be contained in a plane) and computed some examples.

We also saw that acceleration of a particle moving along a curve , that is
the vector

is a linear combination of the two vectors and , i.e. there are numbers and , such that

We then saw how to computed these coefficients and and did so in some examples.

Do the problems of §13.5, listed above, which you could not do before today's lecture.

We covered §13.6. We saw how to apply some of the things we've seen so far to problems of Mechanics (motion of particles under certain forces). In particular, we proved Kepler's 2nd law which states that if a particle is moving under the influence of central force (force parallel to the location vector) then its location vector sweeps out equal areas in equal times.

We did not have time to cover the subsection on initial value problems. Please go over Examples 3 and 4 on p. 809 and 810.

HW Problems: §13.6: 1-4, 8, 9, 11, 14, 15.

We talked a little about how to solve initial value problems of the second order such as those that arise from Newton's equation in kinematics (read §13.6).

Then we moved on to §14.1 and talked about real-valued functions of two or three variables

We talked about the

HW Problems: §14.1: 1-10, 35, 37, 39.

We first remembered what kind of curves in the plane have an equation which is
a polynomial in and of degree at most , a quadratic equation. These are the so-called
*conic sections*: ellipse, hyperbola, parabola, straight line, a pair of straight lines or
a single point. Next we wrote down the general quadratic equation in three variables , that is
an arbitrary polynomial in these variables whose degree is at most . The case now are many more
and I am not asking that you memorize the entire list in your book (§14.2) but that you learn,
as we did in class, how to figure out the shape of a given surface, whose equation you're given, by
cleverly fixing some of the variable to appropriate values (in your book these are called *intercepts*
and *traces*), and also by taking advantage of the symmetries of the equation. Do learn the few examples
we did in class.

HW Problems: §14.2: 2, 4, 10, 12, 20, 22, 26, 40, 43.

A student note taker is needed in this course to take notes for a student with a disability. The note taker will be paid a stipend for this assignment. Skills needed are the ability to take accurate, legible, and organized notes and a commitment to attend every lecture. If interested, please contact Tina Allen via office phone at 404-894-2563 or via email at notetaker@vpss.gatech.edu as soon as possible. Be sure to indicate the Professor's name, time, day and course number/ section in the subject line of the announcement.

We saw how to present geometric information about a function or (or its graph, representable by the equation or ) by drawing its level curves for two-variable functions, i.e. the curves whose equation is given by for various values of , or its level surfaces (three-variable case) which are the surfaces representable by the equation , again for various values of .

HW Problems: §14.3: 4, 5, 8, 14, 19, 20, 21, 25, 28.

Next we saw how one defines the *partial* derivatives (with respect to each of the
variables) of functions of more than one variable, and saw how to compute some examples
by treating all variables but the one we're differentiating with as constants.

My apologies to those of you (if any) who came to my office hours Thursday. I entirely forgot.

I will be in my office from 4-5 tomorrow Friday.

Mihalis

We finished §14.4, by giving more examples of how to compute the partial derivatives of functions of several variables and by talking about the geometric significance of the partial derivatives of a function .

HW Problems: §14.4: 1-10, 23, 41, 53.

We then started speaking about §14.5. We defined the -neighborhood (and deleted neighborhood)
of a point
, where . We also defined the concept
of an open set
and saw, as an example, that the *upper half plane*
:

is indeed an open set.

You can find a practice exam (for 50 min) here in PDF.

We defined what closed sets are (the complements of open) sets as well as what the interior points of a set are and its boundary points. We saw several examples, both in dimension 2 as well as in dimension 1.

HW Problems: §14.5: 1-20.

For the first test (this Friday, in class) you are expected to know all that has been taught to you by today (Monday). Please come early so as to start on 14:05 sharp.

There will be a quiz on Thursday as usual.

We went over the practice exam and aswered some questions.

We had our first test. Here are the problems.

- A particle of mass moves according to the law
.
Find, as a function of , its velocity, its speed and the force which is applied on
the particle.
**Solution:**Velocity: .

Speed: .

Force: . - For the curve parametrized by
, , find

(a) the velocity vector , and

(b) a parametrization of the tangent line to the curve at the earliest (smallest ) moment that line becomes parallel to the -plane.**Solution:**(a) .

(b) For a vector to be parallel to the -plane means that its -coordinate is . This happens for the first time at . At that moment the velocity vector is . A parametrization of the tangent line (which goes through the point and has the direction of the vector is

- Let
.

(a) Find the maximum domain where is defined and describe it geometrically.

(b) Compute all three partial derivatives of .**Solution:**(a) The argument of the function must be positive, hence the domain is

This is the outside of the sphere centered at the origin with radius 2,*without*the surface of the sphere.

(b) , , . - (a) Describe in geometric language the region of the plane

Find its interior and boundary points.

(b) Same question for the region

**Solution:**(a) This is the region where and have the same sign or some of them is zero. This is the first and third quadrant of the plane,*together*with the - and -axes.Any point of that set which is not on the axes is an interior point as one can draw a disk centered at it which is entirely in the set. Any point on the axes is not an interior point as any disk around such a point will contain points outside the first and third quadrant.

For this reason all points of the axes are boundary points. There are no other boundary points as any point inside a quadrant and not on an axis will have a small disk around it which contains either only points of the set or of its complement and not from both.

(b) The only difference of this shape from that of (a) is that the - and -axes are not part of the set now. However the set of interior points and the set of the boundary points remains the same as that of (a). (In this case the set of boundary points, namely the two axes, is not part of the set itself.)

We defined formally what the limit of a two-variable function is when , and when a function is continuous at a point . We calculated the limits using the definition for some very simple functions.

We also saw examples of functions of two variables which exhibit strange behaviour, by one variable standards. For example we saw a function which is everywhere continuous with respect to each of the variables and has everywhere partial derivatives, yet is not continuous at (0,0).

HW Problems: §14.6: 1-5, 21, 23, 24, 26, 27.

You can find your grade here in PDF.

We defined what it means for a function of many variables to be differentiable at a point ,and also defined the gradient of a differentiable function at , as the only
vector that makes the following true:

We then saw that if a function has continuous partial derivatives at a point then it is differentiable there and the components of its gradient are just its partial derivatives with respect to the corresponding variables.

See, for example, this page for the big-O and little-o notation that we talked about today.

HW Problems §15.1: 12-16, 33-37, 39, 40.

We talked about the concept of directional derivatives and how to compute them using the gradient of a function. Also saw that the direction of the gradient is the direction of maximum rate of increase of a function. Saw several examples.

HW Problems §15.2: 11-14, 23-26, 40, 41.

We went over the Mean Value Theorem for scalar functions of one variable, and used it to
prove the mean value theorem for scalar functions of a vector variable. We remarked
that the Mean Value Theorem is not true for vector *valued* functions.
We saw some consequences of the MVT: if two functions have the same gradient in a connected open
set then they differ by a constant.

Next we reviewed the chain rule for functions of one variable and saw the form it takes for the composition of a scalar function of a vector variable with a vector valued function of one variable. We saw how to construct the dependency diagram when we have many quantities that depend on others, and use it to derive the appropriate chain rule.

Please read from §15.3 about implicit differentiation.

HW Problems §15.3: 1, 3, 4, 6-8, 17, 18, 25, 27, 29, 30, 36, 58.

We pointed out that
is a normal vector to the curve (or surface)
, a constant. We used this to compute normal and tangent vectors
to curves and surfaces amd also angles between curves and surfaces. These angles
are *defined* to be the corresponding angles between their tangent objects at
the intersection. For example, if we are seeking the angle between a curve and a surface
which intersect at a point
we must measure the angle between a tangent vector of the curve at and the
tangent plane to the surface at . This is most easily measured by first finding
the angle between the tangent to the curve and the normal to the tangent plane, and then
subtracting that angle from .

HW Problems §15.4: 1, 2, 10, 11, 19, 20, 26, 27, 28, 34, 36.

We saw what is the analogue in two variables of the criteria, involving first and second derivatives, that we know for deciding where a function's local maxima and minima are. The first stage of the method is to locate the points where the function's gradient vanishes. Each of those points is then checked using higher order partial derivatives of the function at that point in order to decide if there is a local extremum at that point or if it is a saddle point.

HW Problems §15.5: 1, 2, 5-8, 25, 26.

We saw how to find the (global) minima or maxima of a function of two variables defined in a closed and bounded domain. We have to find the critical points in the interior of the domain and then parametrize our boundary, which gives rise to a function of one variable corresponding to our function being evaluated on the boundary. This one-variable function we extremize (find its extrema) separately. The extrema of this one-variable function along with the critical points in the interior are then compared in the value of the function at them to find the global extrema.

HW Problems §15.6: 1-6, 19-22, 27.

We talked about the problem of finding the minimum or maximum of a function when is not free to take any values in the domain of definition of but it has to satisfy some condition, which is usually given in the form . Sometimes one can solve for one of the variables in and substitute the resulting expression in thus getting a problem of ordinary function extremization without side conditions and in one variable less. We saw two such examples but this method is most often inapplicable as it is not easy to solve for one of the variables,especially if is a three-component vector and is non-linear. Even if possible the resulting expression for may be too complicated to work with.

We then saw the method of Lagrange (or *Lagrange multipliers* as it is commonly known),
which allows one to solve the above problem by solving a, generally non-linear,
system of equations in the unknowns and , the latter being a auxiliary
variable which is not used except to help find the values of .
The system of equations is

The last equation is actually many, as many as the number of components in (two or three in our cases).

HW Problems §15.7: 1-4, 13, 15, 18, 19, 21, 23.

We gave some more examples of the method of Lagrange multipliers and saw how it applies to the case of a function of three variable subject to two conditions.

We also saw that there is a way to avoid introducing the Lagrange multiplier (the
variable, which we throw away after solving the system of equations). This can
be done by checking the parallelism between the two fectors and
(the gradients of the function to be optimized and the constraint function) by checking the
equation

The answer is that the vector equation is really only two equations. If one writes it out (using the determinant form) one notices immediately that any of the resulting three equations can be obtained from the other two, so one can throw away any one of them (but only one) without changing the solutions to our system. Now it is clear that we have three equations in three unknowns.

We saw how to use the summation sign and iterated (multiple) summation sign via several examples. We also evaluated several simple sums.

HW Problems §16.1: 1-4, 13-17.

You can find a practice exam (for the 50 min 2nd midterm) here in PDF.

Note Taker Announcement:

A student note taker is needed in this course to take notes for a student with a disability. The note taker will be paid a stipend for this assignment. Skills needed are the ability to take accurate, legible, and organized notes and a commitment to attend every lecture. If interested, please contact Christina Bibbs atnotetaker@vpss.gatech.edu, and be sure to indicate the Professors name, time, day and course number in the subject line of your email. Thank you so much for your willingness to assist our office in this process. Please contact our office if we can be of any assistance.

We defined the integral of a function over a rectangle via lower and upper sums corresponding to partitions of . We evaluated, using the definition, only some simple integrals.

HW Problems §16.2: 1, 2, 6, 7, 10, 11.

We stated the Mean Value Theorem for double integrals over connected domains (§16.2).

When a domain is such that all its intersections with lines parallel to the -axis are intervals, and the set of -values used in the domain consititute an interval the domain is called of Type I (and of Type II if the same properties hold with and reversed). For such a domain we saw how to evaluate a double integral of a function as a single integral whose function to be integrated is an integral itself.

HW Pproblems §16.3: 1-6, 13, 14, 33, 34, 43, 46.

We did not cover any new material. Instead we went over the practice exam, in preparation for Wednesday's exam.

We had our second test. Here are the problems:

- A rectangular box is inscribed in the sphere
with its center at the origin
and its sides parallel to the coordinate axes.
(The corners of the box are all on that sphere.)
Find the maximum volume of such a box.
**Solution:**Suppose the vertex of the box lying in the first octant has coordinates , with . It follows that the volume of the box is

We have therefore to maximize the function subject to the condition

We easily calculate

Following the method of Lagrange multipliers we have to solve the system (in )

Expanding it we get the four equations

Since we can divide the first and second equations to get . Similary we get and the last equation now gives (the box is a*cube*). It follows that the maximum volume is

- Find the absolute minimum and maximum of the function
in the closed
region bounded by the ellipse
.
**Solution:**We have

and setting we get the single solution

This critical point of is inside the ellipse as it satisfies

We now turn our attention to the boundary of the region, namely on the ellipse itself, which can be parametrized as

We thus have to find the maximum of the one-variable function

Differentiating with respect to we get

which vanishes precisely when , that is for and . These two values of correspond to the two points

on the ellipse, on which the function takes the (positive) values

It follows that (the value at the critical point inside the ellipse) and (the largest of the two values we found on the boundary). - Find the points on the surface
at which the tangent plane to the surface
is horizontal. Write the equations of the corresponding tangent planes.
**Solution:**The surface is a level surface of the function

therefore the normal vector to the surface at point is just

and this vector must have its first two coordinates equal to in order for the tangent plane to the surface at , which has that vector as normal vector, to be horizontal. This gives rise to the system in :

which we easily solve (start by subtracting the second equation from the first) to get

as its only solution. The equation of the (horizontal) tangent plane at that point is simply . - Let be a function of and . Let also
, .
Calculate the second order partial derivative
in terms of: , and the partial derivatives of with respect to the variables and .
Your formula for may not involve partial derivatives of with respect to or .

**Solution:**We have and . By the chain rule we get

We now differentiate this equation with respect to to get

To get and we apply the chain rule again to get

Putting this together we get

You may find them here in PDF.

We went through the problems that were on the last test and their solutions.

We saw how to evaluate a double integral using polar coordinates. The first task is to find
the domain in the -plane which corresponds to the given domain
(in the -plane). For example, if is the unit disk in the cartesian plane
(the -plane) then is a rectangle in the -plane defined by

Next we transcribe the function to be integrated, from the cartesian variables into the polar variables. This is achieved by just substituting for and for . Last, we replace the ``area element'' by the expression .

Please read by yourselves the evaluation of from the end of §16.4. This is a very important integral in all mathematics and especially its applications to science and engineering.

HW Problems §16.4: 1, 2, 5, 6, 9, 10, 17-20, 23, 24.

We covered a couple more examples from §16.4 and proved the identity

using polar integration.

We covered some of the examples in §16.5. We saw how to compute the mass of a two-dimensional domain (a ``plate'') with variable density, and also how to compute its center of mass. We also saw the definition of the moment of inertia of a plate (with variable density) rotating around a line or point. We evaluated some relevant double integrals.

HW Problems §16.5: 1-4, 11, 12, 14, 17, 25.

We saw briefly how triple integrals are defined (in a completely analogous way to double integrals, so we did not insist much on §16.6) and proceeded to evaluate some triple integrals by repeated integration.

HW Problems §16.7: 3-6, 11, 14-16, 21-22.

We showed how to compute a triple integral in cylindrical coordinates.

HW Problems §16.8: 1-8, 11, 12, 17, 18, 25, 26.

We introduced the spherical coordinate system and how to use it for evaluation of triple integrals. We saw several examples of how to transform the domain of integration from cartesian to spherical coordinates and carry out the integration in spherical coordinates (the form becomes now ).

HW Problems §16.9: 1-4, 9-14, 16, 19, 20, 24, 26, 27.

We saw the general procedure for evaluating a multiple (double or triple) integral over a domain after first doing a change of variables. This is essentially a way of parametrizing using two or three variables (depending on the whether the domain is in the plane or space) which run over a more convenient domain . We say how to do this when is a parallelogram (and we got a parametrization with paramaters and running through the rectangle ) and also in some other cases. We also saw that the form transforms into the form (and similarly in three dimensions), where is the so-called Jacobian determinant, which can be computed from the functions and .

HW Problems §16.10: 1-2, 8-10, 12-14, 19, 20, 27, 29.

We defined the line integrals of a vector field
along a curve given parametrically by , ,
as the expression

We proved that the line integral so defined is independent of which parametrization is being used for the curve , as long as we do not change the orientation. We computed some examples.

HW Problems §17.1: 1-4, 7, 15, 16, 20, 21, 23, 25, 28-30.

This says that if a vector field is a gradient field, then a line integral of that
along a curve equals the value of (where
) equals
where and are the endpoints of . This is true in two and three dimensions,
and, in two dimensions, in order to decide if is a gradient field we need to verify
that when the domain , where is defined, is *simply connected* (i.e. connected
and with no ``holes'').

HW Problems §17.2: 1-4, 12, 13, 22, 24-26, 28.

My office hours tomorrow, April 12, will last only until 11am.

We discussed kinetic energy and why its change is due to the work is done by the force field on the particle. Also we talked about conservative (gradient) fields and the potential function.

HW Problems §17.3: 1-3, 6, 7, 8.

We saw an alternative way of writing the line integral as , where . We also saw the line integral w.r.t. arc-length, denoted by , where is a scalar function.

HW Problems §17.4: 2, 3, 5, 17, 18, 26, 27, 29, 30, 32, 36.

We stated Green's theorem. This expresses a line integral along a closed curve as a double integral over the interior of that curve. We proved this in the case when the domain if of Type I and of Type II and showed how one proves this if the domain is more general by cutting up the domain into a finite number of non-overlapping parts each of which is of both Type I and II. We also explained how to parametrize the boundary of a domain if that is not simply connected or even not connected (walk along the boundary in such a way that the domain is always on your left).

HW Problems §17.5: 1, 2, 5, 6, 18-20, 26, 28, 30, 31.

We saw how to apply Green's formula to derive a formula for the area of a polygonal region that has been described to us by the coordinates of its vertices. We also saw how to find an analogous formula for the centroid of the polygon and for the volume of the solid that arises if we rotate a polygonal region (which is part of the right half plane ) about the -axis.

We saw how to calculate the area of a surface which has been given to us in parametric form.

HW Problems §17.6: 1, 2, 5, 6, 15, 16, 17, 22, 23, 35, 36.

You may find it here in PDF.

I answered various questions of the students.

We went through the practice final exam that was distributed on Wednesday.

It will take place in Skiles 202 (lecture room), on Friday May 4, from 11:30 to 14:20.

We had our final exam. Here are the problems:

- Let the surface be defined by the equation (a paraboloid) and the surface
be defined by the equation (also a paraboloid).
The intersection of the surfaces and is a curve and the point
belongs to
as its coordinates satisfy the equations of both and .
Find a
*unit*tangent vector to the curve at .*Hint:*Use the normal vectors to and at . - Find the point of the surface which is closest to the point .
- Let and
be a quarter-disk in the first quadrant.
Using polar coordinates compute the double integral

- Compute the line integral

where is the circle of radius centered at oriented counterclockwise.*Hint:*Do not use Green's theorem. - A simple closed curve in the -plane encloses a domain .
Show that the moment of inertia of with respect to the -axis
is equal to the line integral

- Compute the volume of the solid
defined by the three inequalities

You may find them here.

I will be in my office on Sunday May 6, from 6 to 7pm, in case somebody wants to see an exam paper. After that the grades will be finalized.

**Mon May 7 20:15:18 EDT 2007**: There had been an error in the software that computed your numerical
scores, which had as an effect that the quizzes were not counted (and possibly other effects). This error
has now been corrected (I hope) and your numerical grades are now correctly computed (please, check yours, and
let me know if you believe there is a numerical error in your posted numerical grade).

Some letter grades have inevitably been changed as the landscape changed, most of them for the better I believe, but not all.

I will enter the corrected grades into the system tomorrow but the process is slow and the changes may not show up online for several days.

I apologize for this mistake. I wish one of you had checked the numbers concerning him/her but, amazingly, no one did (at least, no one notified me of the discrepancy). Neither did I until today when I realized the error.

Mihalis Kolountzakis 2007-05-07