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Calculus III
(Math 2401, section F3)

Mihalis Kolountzakis

School of Mathematics
Georgia Institute of Technology
686 Cherry Street
Atlanta, GA 30332

E-mail: kolount AT member.ams.org

August 2004-05


Contents

1 What the course is about

Look here for the syllabus and more information.

2 Schedule

MWF 15:05 - 15:55, in Skiles 271.

My office hours are: 9-11 Tuesdays, in my office (Skiles 209). You're also welcome to ask me questions any time you see me, anywhere.

3 Grading Policy

Two midterm exams will be given and a final. If the three grades are $ a$, $ b$ and $ f$ then the final grade for the class will be given by

$\displaystyle 0.6 \max\{0.7 a + 0.3 b, 0.3 a + 0.7 b\} + 0.4 f. $

This allows one to do not so well on one of midterm exams ($ a$ and $ b$) and compensate by doing very well on the other.

Homework will be assigned but will not be normally collected. Do it to be adequately prepared for the tests, as the problems on these will be small variations of those on the homework assignments.

4 Course Progress

4.1 M, 8/16/04: Vector valued functions of a real variable

We went over §13.1, and talked about what is a vector valued function of a real variable. We saw how the properties of limit (as $ t\to t_0$) are defined, the concept of continuity, differentiability and integration. All of these properties and quantities can be defined either with direct reference to the vector values of the function or with reference to the components of the function.

Look at problems §13.1: 39, 40, 43, 46, 51, 53, 55, 57.

4.2 W, 8/18/04: Differentation of vector valued functions and rules

We covered §13.2. We saw that differentiation of vector valued functions obeys the expected rules, very similar or identical to those obeyed by scalar-valued functions. We did problems §13.2: 28, 29, 31, 33.

Look at problems §13.2: 35, 36.

4.3 F, 8/20/04: Tangent and normal vectors to a curve

We covered §13.3. The main things we learned about is the tangent line and the tangent vector on a curve at one of its points and also the normal vector to it. Together the tangent and normal vector to a curve define the osculating plane, which is a plane that goes through a point on the curve and contains the tangent line and the normal vector.

Forgot to mention it in class but do look at problems §13.3: 1, 2, 9, 11, 14, 18, 33, 35, 36, 37, 45.

4.4 M, 8/23/04: Length of a curve

In §13.4 we saw how to define and calculate the length of a curve which is given to us in parametric form. The answer is that we integrate over time (or whatever our paramater is called, but we think about it always as time) the speed of the motion, that is the magnitude of the velocity vector (derivative of the location vector).

We calculated several examples.

Look at problems §13.4: 7, 9, 17, 21, 23 (this last one has to do with the so-called arc-length parametrization of a given curve, so pay special attention to it).

4.5 W, 8/25/04: Curvature of plane curves

We defined the curvature of a plane curve as the rate of change of the angle formed by the tangent and the $ x$-axis as we are moving along the curve at unit speed. We then found how to calculate the curvature for a curve that is the graph of a function and then for a curve which is given to us in parametric form. We also defined the radius of curvature and the center of curvature of a curve at a given point (provided the curvature there does not vanish). Then we proved the formula

$\displaystyle \kappa = {\left\Vert{\frac{d\vec T}{ds}}\right\Vert}, $

where $ \vec T$ is the unit tangent vector.

Look at problems §13.5: 2, 5, 6, 10, 13, 14,21, 22, 41, 42, 58.

4.6 F, 8/27/04: Curvature of space curves; Mechanics

We completed our discussion of curvature for palne curves and evaluated it in some examples. Then we talked about mechanics and we expressed (but not proved) Theorem 13.6.7 in your book and talked about its qualitative consequences for planetary motion.

4.7 M, 8/30/04: Kepler's second law and initial value problems

We proved Kepler's second law (that a particle moving in a central force field is always executing a planar motion and that its location vector sweeps out equal areas in equal times). We also formulated Kepler's first and third law. We solved an initial value problem, where the force field is given and also the initial location and velocity of the particle and the motion of the particle (function $ \vec r(t)$) is to be determined.

Look at problems §13.6: 2, 3, 5, 7, 8, 15.

It was also announced that your first test will come after we complete Chapter 14 and the material covered will be Chapters 13 and 14.

4.8 W, 9/1/04: Proof that the planets' trajectories are ellipses

We proved Kepler's 1st law, namely that the planets are moving in elliptical orbits with the sun at a focus. We carried out the proof to the point where the equation of an ellipse was determined in polcar coordinates.

No problems for homework from this session.

4.8.1 Practice one hour exam for Chapter 13

Do the following problems in an hour with closed books (no calculators will be needed). In any test you write for this class you'll have to show all your work.

  1. If $ \vec r(t) = (t, t^2, 2t^2)$ find the unit tangent vector $ \vec T(t)$, the principal normal vector $ \vec N(t)$ and an equation in $ x, y, z$ satisfied by the osculating plane at $ t=1$.
  2. Find the length of the curve described by $ \vec r(t) = (t^3/3 - t, t^2, 0)$, $ 0\le t \le 2$.
  3. A particle moves according to $ \vec r(t) = (e^t \cos t, e^t \sin t, e^t)$. Find the curvature of its trajectory at time $ t$ and determine the tangential and normal components of its acceleration.
  4. If a particle moves with constant velocity show that its angular momentum is also constant. Give an example to show that this is not true of we only assume that the speed is constant.

4.9 F, 9/3/04: Functions of several variables. The ellipsoid.

We covered §14.1, and we talked about functions which depend on several variables (typically two or three, but that's not necessary) and how to find their domain (given a formula for them) and their range. We also talked about the general quadric surface in $ {\mathbf R}^3$ which is a surface defined by a polynomial equation in three variables $ x, y, z$ such that the degree of each monomial is at most two (the general quadratic). We described the ellipsoid, which is one of them.

Look at problems §14.1: 1-10, 35-37, 39.

4.10 W, 9/8/04: Quadric surfaces

After a brief reminder of the several different kinds of curves in the plane that have a quadratic equation describing them (ellipses, hyperbolas and parabolas) we described some of the different types of surfaces that a quadratic equation in $ x, y$ and $ z$ can represent. We did not give an exhaustive list but worked with 3-4 of those kinds and tried to figure out how they look like by studying their intersection with some specific planes. It is this skill that I want you to learn from §14.2, not a list of surfaces.

Do problems §14.2: 1-5, 10, 22, 24, 26, 39, 40, 43.


4.10.1 ANNOUNCEMENT:

The first test will be held on W 9/22/04, during the regular meeting time. You should know the material in §13 and §14. Books will be closed. (But see §4.12.1.)

4.11 F, 9/10/04: Graphs of functions, level curves and level surfaces; partial derivatives

We talked about graphs of functions and how to visualize them by drawing their level curves (§14.3).

Do problems §14.3: 4, 5, 8, 14, 20, 21, 25, 28.

We also defined the partial derivatives of functions of several variables and calculated soem examples.

Please let me know in class Monday what your preferences are regarding when the first test will be administered.

4.12 M, 9/13/04: Partial derivatives

We went again over the concept of partial derivatives, how to calculate them, and talked about their geometrical interpretation, mainly through problem 46 (§14.4) of your book.

Do problems §14.4: 1-10, 23, 24, 41, 53.


4.12.1 ANNOUNCEMENT: First test

The first test will be held on F 9/24/04, during the regular meeting time. You should know the material in §13 and §14. Books will be closed. Please disregard the previous Announcement of §4.10.1.

The change has been asked by the class.

4.13 W, 9/15/04: Elements of the topology of Euclidean spaces

We covered the material in §14.5: what is a neighborhood of a point, which points of a set are called interior points, which are boundary points, which sets are called open and which closed. We gave several examples.

Do problems §14.5: 1-20.

4.14 F, 9/17/04: Limits of multivariable functions

We gave some more examples and general statements about open and closed sets.

We defined what it means for a function $ f(x,y)$ to have a limit as $ (x,y) \to (x_0, y_0)$, and we observed, through examples, that the situation is much more subtle that with functions of one variable.

4.15 M, 9/20/04: Continuity of multivariable functions, partial derivatives and mixed second order derivatives

We saw again 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).

We saw (without proof) conditions that guarantee that the mixed partial derivatives of a function are equal.

Do problems §14.6: 1-5, 21, 23, 24, 26, 27.

On Wednesday we will review chapters 13 and 14. Please come armed with questions you want to ask.

4.15.1 Practice one hour exam for Chapter 14

Do the following problems in an hour with closed books (no calculators will be needed). In any test you write for this class you'll have to show all your work.

  1. Which are the boundary points of the set $ {\left\{{(x,y): x>0, y\ge 0}\right\}}$, and why?
  2. Let $ C$ be the curve which is the intersection of the plane $ y=3$ with the surface $ z=x^2+y^2$. Find a parametrization for the line tangent to $ C$ at point $ P(1,3,10)$.
  3. Show that the function $ f(x,y) = \frac{x^2-y^2}{x^2+y^2}$ is not continuous at $ (0,0)$.
  4. Let $ g$ have partial derivatives of second order and set $ h(x,y) = g(x+y)-g(x-y)$. Show that $ h$ satisfies the equation $ h_{xx} - h_{yy} = 0$.

4.16 W, 9/22/04: Review session for Chapter 13 and 14

We had a review session for Chapters 13 and 14. On Friday we write our first test on these two chapters.

4.17 F, 9/24/04: First Test

Today we had our first test. The problems were the following:

  1. Two particles move in the plane and at time $ t$ their locations are given by $ \vec{r_1}(t) = (t, t^2)$ and $ \vec{r_2}(t) = (2t, t^3)$.
    (a) Where do the trajectories of the two particles meet?
    (b) At what angle?
    (c) Do the particles collide after time 0?
  2. (a) Find all first and second order partial derivatives of the function $ f(x,y) = 3x^2 y^5 + 2ye^{5x}$.
    (b) Find the length of the curve $ \vec{r}(t) = (4\sin{3t}, 5t, 4 \cos{3t})$, $ 2 \le t \le 8$.
    (c) Find the unit tangent vector of the curve in (b) at $ t=2$.
  3. Consider the curve $ \vec{r}(t) = (t, t^2/2, t^3/3)$. Find the curvature at time $ t$ and determine the tangential and normal components of the accelaration. Show all intermediate work.
  4. (a) Which are the boundary points of the set $ {\left\{{(x,y): x+y\le 5}\right\}}$. Explain why each such point is a boundary point and why the rest are not.
    (b) The subset $ A$ of the real line consists of all intervals $ (\frac{1}{n}-\frac{1}{10n^2}, \frac{1}{n}+\frac{1}{10n^2})$, for $ n\ge 100$. (You may take it for granted that these are nonoverlapping.) Is this an open set?

4.17.1 The results of the first test

You can find the results of the first test here.

The most important comment I have to make is the almost complete and universal failure to do reasonably on the last problem.

If your score is less than 20 you should start working much harder on this class.

4.18 M, 9/27/04: Differentiability of multivariate functions. The gradient of a function

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

$\displaystyle f(\vec x + \vec h) - f(\vec x) = \vec y \cdot \vec h + o(\vec h). $

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.

Do problems §15.1: 12-16, 33-37, 39, 40.

4.18.1 ANNOUNCEMENT: No office hours tomorrow

Most likely I will not be able to be in my office tomorrow T, 9/28/04. Please schedule an appointment with me if you'd like to see me, or drop by at any other time.

This announcement is cancelled. Office hours wil be held as usual.

4.19 W, 9/29/04: Gradients and directional derivatives

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.

Do problems §15.2: 11-14, 23-26, 40, 41.

4.20 F, 10/1/04: Mean Value Theorems; Chain Rules

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 connect set then they differ by a constant.

Next we reviewed the chain rule for functions of one variable and saw teh form it takes for the composition of a scalar function of a vector variable with a vector valued function of one variable.

4.21 M, 10/4/04: Chain Rules, implicit differentiation

We completed our discussion of the various chain rules and saw how to differentiate functions defined implicitly.

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

4.22 W, 10/6/04: The gradient as normal vector to level curves and surfaces

We pointed out that $ \nabla f(\vec r)$ is a normal vector to the curve (or surface) $ f(\vec r) = C$, $ C$ a constant. We used this to compute normal and tangent vectors to curves and surfaces.

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

4.23 F, 10/8/04: Local extrema

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.

Do problem §15.5: 1, 2, 5-8, 25, 26.

4.24 M, 10/11/04: Absolute extreme values in a given domain

We saw how to find the absolute maximum and minmum for a function $ f$ of two variables in a given domain $ D$ (§15.6).

Do problems §15.6: 1-6, 19-22, 27.

4.24.1 ANNOUNCEMENT: No office hours tomorrow

I will not be in my office in the morning tomorrow T, 10/12/04. Please schedule an appointment with me if you'd like to see me, or drop by at any other time.

4.25 W, 10/13/04: Function extremization under side conditions

We talked about the problem of finding the minimum or maximum of a function $ f(\vec x)$ when $ \vec x$ is not free to take any values in the domain of definition of $ f$ but it has to satisfy some condition, which is usually given in the form $ g(\vec x) = 0$. Sometimes one can solve $ g(\vec x)$ for one of the variables in $ \vec x$ and substitute the resulting expression in $ f(\vec x)$ 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 $ g(\vec x) = 0$ for one of the variables, especially if $ \vec x$ is a three-component vector and $ g$ is non-linear. Even if possible the resulting expression for $ f$ 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 $ \vec x$ and $ \lambda$, the latter being a auxiliary variable which is not used except to help find the values of $ \vec x$. The system of equations is

$\displaystyle g(\vec x)$ $\displaystyle =$ 0  
$\displaystyle \nabla f (\vec x)$ $\displaystyle =$ $\displaystyle \lambda \nabla g(\vec x).$  

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

Some examples were discussed, and we will continue with this Friday.

4.26 F, 10/15/04: Function extremization under side conditions, continued

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.

Do problems §15.7: 1-4, 13, 15, 18-21, 23, 26

4.27 W, 10/20/04: Summation sign

We remembered a few basic things about one variable integrals and how they are defined via sums corresponding to partitions of the intervals of integration. We then introduced the summation sign for both single and double sums and evaluated several examples.

Do problems §16.1: 1-4, 13-17.

4.28 F, 10/22/04: Integral of a continuous function of two variables

We defined the integral of a function $ f(x,y)$ over a rectangle $ R$ via lower and upper sums corresponding to partitions of $ R$. We evaluated, using the definition, only some simple integrals, and we then saw how to extend this definition to arbitrary domains of integration. Finally we saw some properties of the operation of integration which carry over from the case of one-variable integration.

Do problems §16.2: 1, 2, 6, 7, 10, 11.

4.29 M, 10/25/04: Double integrals by repeated integration

When a domain is such that all its intersections with lines parallel to the $ y$-axis are intervals, and the set of $ x$-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 $ x$ and $ y$ 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.

Do problems §16.3: 1-6, 13, 14, 33, 34, 43, 46.

4.30 W, 10/27/04: Evaluating double integrals using polar coordinates

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

$\displaystyle 0 \le r \le 1, 0\le \theta < 2\pi. $

Next we transcribe the function to be integrated, from the cartesian variables into the polar variables. This is achieved by just substituting $ r\cos\theta$ for $ x$ and $ r\sin\theta$ for $ y$. Last, we replace the ``area element'' $ dx dy$ by the expression $ rdr d\theta$.

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

4.30.1 ANNOUNCEMENT: Second Test

The second test will be held after we finish Chapter 16. You'll be tested on chapters 15 and 16.

4.31 F, 10/29/04: Applications of double integrals to mechanics

We covered 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 how to compute the moment of inertia of a solid body (with variable density) rotating around a line in space. We specialized this to a rotating plate and evaluated some relevant double integrals. Last, we mentioned tha Parallel axis theorem. We did not have time to prove this (the proof is very simple and is in your book) but talked about what it means.

Do problems §16.5: 1-4, 11, 12, 14, 17, 25.

4.32 M, 11/1/04: Triple Integrals

We saw briefly how triple integrals are defined (in a completely analogous way to oduble integrals, so we did not insist much on §16.6) and proceeded to evaluate some triple integrals by repeated integration. We also talked about the average fo a function $ f(x,y,z)$ over a domain $ \Omega$ on which a density function $ \lambda(x,y,z)$ is defined, and how this applies to the center of mass of a domain with variable mass density.

Do problems: §16.7: 3-6, 11, 14-16, 21-22.

4.33 W, 11/3/04: More on triple integrals

We computed some volumes and saw the following principle. Suppose we take a domain $ \Omega$ in $ {\mathbf R}^3$ and stretch it along the $ z$-axis by a factor of $ \lambda$ to get a domain which we call $ \Omega_\lambda$. This means that a point $ (x, y, z) \in \Omega$ if and only if $ (x, y, z/lambda) \in \Omega_\lambda$. Then

$\displaystyle {\rm vol }\Omega_\lambda = \lambda \cdot {\rm vol }\Omega. $

One can apply this principle to show, for example, that the volume of the ellipsoid

$\displaystyle \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} \le 1, $

is equal to $ abc$ times the volume of the unit ball.

4.33.1 ANNOUNCEMENT: Second Test

The second test will be held during the regular class hours on Monday, November 15, 2004. A review hour will be held on the previous Friday in class. You'll be tested on chapters 15 and 16 and the format of the test will be similar to the first one.

4.34 F, 11/5/04: Triple integrals in cylindrical coordinates

We showed how to compute a triple integral after first describing its domain of integration in cartesian coordinates.

Do problems: §16.8: 1-8, 11, 12, 17, 18, 25, 26.

4.34.1 Practice one hour exam for Chapter 15

Do the following problems in an hour with closed books (no calculators will be needed). In any test you write for this class you'll have to show all your work.

  1. Let $ u = u(x,y)$, where $ x=x(t)$ and $ y=y(t)$ and assume that these functions have continuous second derivatives. Show that

    $\displaystyle u_{tt} = u_{xx}(x_t)^2 + 2 u_{xy} x_t y_t + u_{yy}(y_t)^2 + u_x x_{tt} + u_y y_{tt}. $

  2. Show that the sphere $ x^2+y^2+z^2-4y-2z+2 = 0$ is perpendicular to the paraboloid $ 3x^2+2y^2-2z=1$ at the point $ (1,1,2)$.
  3. Find the absolute extreme values taken by the function $ f(x,y) = \frac{-2y}{x^2+y^2+1}$ on the set $ D = {\left\{{(x,y): x^2+y^2 \le 4}\right\}}$.
  4. Minimize $ x+2y+4z$ on the sphere $ x^2+y^2+z^2 = 7$.

4.35 M, 22/8/05: Spherical coordinates for triple integrals

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 $ dx dy dz$ becomes now $ \rho^2 \sin\phi d\rho d\phi d\theta$). In the last example we did (Example 3 in p. 1009 of your book) we got the wrong answer (0) because teh range for $ \phi$ is not 0 to $ \pi$, as we took it to be, but 0 to $ \pi/2$ (remember that $ z\ge 0$ is obvious from the equation of the surface).

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

4.35.1 Practice one hour exam for Chapter 16

Do the following problems in an hour with closed books (no calculators will be needed). In any test you write for this class you'll have to show all your work.

  1. Calculate the average value of the function $ xy$ in the region defined by

    $\displaystyle 0\le x,y \le 1 $   and$\displaystyle x^2+y^2 \le 4. $

  2. Find the volume of the solid bounded below by the $ xy$ plane, above by the surface $ x^2+y^2+z^2 = 4$ and on the sides by the surface $ x^2+y^2 = 1$.
  3. Evaluate the integral $ \int\int\int_T (x^2+y^2+z^2)^{-1} dx dy,dz$, where $ T$ is the region defined by $ 0\le x \le 1$, $ 0\le y \le (1-x^2)^{1/2}$ and $ 0\le z \le (1-x^2-y^2)^{1/2}$.
  4. Find the area of the region enclosed by the curves

    $\displaystyle x^2-2xy+y^2+x+y = 0 $   and$\displaystyle x+y+4 = 0. $

    You may use the change of variable $ u=x-y, v=x+y$.

4.36 W, 11/10/04: General change of variables in multiple integrals

We saw the general procedure for evaluating a multiple (double or triple) integral over a domain $ \Omega$ after first doing a change of variables. This is essentially a way of parametrizing $ \Omega$ using two or three variables (depending on the whether the domain $ \Omega$ is in the plane or space) which run over a more convenient domain $ \Gamma$. We say how to do this when $ \Omega$ is a parallelogram (and we got a parametrization with paramaters $ u$ and $ v$ running through the rectangle $ 0 \le u,v \le 1$) and also in some other cases. We also saw that the form $ dx dy$ transforms into the form $ {\left\vert{J(u,v)}\right\vert} du dv$ (and similarly in three dimensions), where $ J(u,v)$ is the so-called Jacobian determinant, which can be computed from the functions $ x=x(u,v)$ and $ y=y(u,v)$.

Do problems §16.10: 1-4, 8-10, 12-14, 19, 20, 25, 27, 29.

Please come to the review session on Friday prepared to ask questions.

4.37 F, 11/12/04: Review before test

Today we talked about some problems from chapters 15 and 16, as a preparation for Monday's test. The questions were chosen by the students.

4.38 M, 11/15/04: Second test

Today we had our second test, during the usual class meeting. Here are the problems:

  1. The surfaces $ x^2y^2+2x+z^3=16$ and $ 3x^2+y^2-2z=9$ intersect in a curve that passes through the point $ (2,1,2)$.
    (a) What are the equations of the respective tangent planes for the two surfaces at this point?
    (b) Describe the intersection of these two planes in parametric form.
  2. (a) Determine the maximum of the function $ f(x,y,z)=xyz$ given that $ x+y+z=k$, where $ k$ is a constant and $ x,y,z \ge 0$.
    (b) Using (a) show that for any three nonnegative numbers $ x, y, z$ we have

    $\displaystyle (xyz)^{1/3} \le \frac{x+y+z}{3}. $

    (This inequality is called the geometric-arithmetic mean inequality.)
  3. Find the volume of the solid bounded above by the surface $ z=xy$ and below by the half-disk defined by $ z=0$, $ x\ge 0$ and $ (y-1)^2+x^2\le 1$.
  4. Evaluate the integral $ \int\int\int_T \sqrt{x^2+y^2} dx dy dz$ using cylindrical coordinates, where $ T$ is the domain defined by the inequalities

    $\displaystyle -1 \le x \le 1, -\sqrt{1-x^2}\le y\le \sqrt{1-x^2}, x^2+y^2 \le z \le 2-(x^2+y^2). $

Someone forgot his calculator in class. It's in my office.

4.38.1 ANNOUNCEMENT: Message from School administration 

Date: Mon, 15 Nov 2004 13:30:34 -0500
From: Rhonda Mozingo <rmozingo@math.gatech.edu>
Subject: CIOS Survey Times

Please inform your students of the following CIOS schedule for the fall 2004 semester:

From Monday, November 22 -- 12:00 AM  until Friday, December 10 --  12:00 midnight course surveys will be on-line, 24/7
excluding Tuesdays Thursdays, and Saturdays from midnight to 3 AM when the system is down for maintenance.

Students may access the surveys:   www.coursesurvey.gatech.edu

If you or your students have any questions, please let me know.

Thank you,
Rhonda

4.38.2 The results of the second test

You can find the results of the second test here.

4.39 W, 11/17/04: Line integrals

We defined the line integrals of a vector field $ \vec f = \vec f (x,y,z)$ along a curve $ C$ given parametrically by $ \vec r(t)$, $ a\le t\le b$, as the expression

$\displaystyle \int_C \vec f \cdot d\vec r = \int_a^b \vec f(\vec r(t))\cdot \vec r'(t) dt. $

We pointed out, but did not have time to prove it, that the line integral so defined is independent of which parametrization is being used for the curve $ C$, as long as we do not change the orientation. We computed some examples.

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

4.40 F, 11/19/04: The fundamental theorem of calculus for line integrals

This says that if a vector field $ \vec F$ is a gradient field, then a line integral of that along a curve $ C$ equals the value of $ f$ (where $ \nabla f = \vec F$) equals $ f(\vec b) - f(\vec a)$ where $ \vec a$ and $ \vec b$ are the endpoints of $ C$. This is true in two and three dimensions, and, in two dimensions, in order to decide if $ \vec F=(P,Q)$ is a gradient field we need to verify that $ P_y = Q_x$ when the domain $ \Omega$, where $ \vec F$ is defined, is simply connected (i.e. connected and with no ``holes''). We saw several applications of that theorem as well as how to find $ f$ from $ \vec F$.

Do problems §17.2: 1-4, 12, 13, 16, 17, 21, 22, 24-28.

4.41 M, 11/22/04: Work and conservation of energy

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.

Do problems §17.3: 1-3, 6, 7.

4.42 W, 11/24/04: New way of writing line integrals; integrals with respect to arc-length

We saw an alternative way of writing the line integral $ \int_C \vec{h}\cdot d\vec r$ as $ \int_C Pdx + Qdy + Rdz$, where $ \vec h = (P,Q,R)$. We also saw the line integral w.r.t. arc-length, denoted by $ \int_C f ds$, where $ f$ is a scalar function.

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

4.43 M, 11/28/04: Green's theorem

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).

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

4.43.1 ANNOUNCEMENT: No office hours tomorrow

My office hours this week will be held on Thursday, 9-11, instaed of Tuesday.

4.43.2 ANNOUNCEMENT: Form of final exam

I just realized that the duration of the final exam (W, 12/8/04, at 11:30-2:20, in Skiles 271) is 2 hours and 50 minutes. So what I had told you about how many problems is not valid, as I thought the final was less than 2 hours long. You should expect at least 8 problems on the final. Probably half of these will cover Chapter 17.

4.44 W, 12/1/04: Applications of Green's formula to evaluation of some double integrals

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 volume of the solid that arises if we rotate a polygonal region (which is part of the right half plane $ x\ge 0$) about the $ y$-axis.

We'll have a review on Friday. Please come prepared to ask questions.

4.45 F, 12/3/04: Review

He had a very short discussion about the form of the final test and several procedural matters. We agreed that I'll try to grade all papers and set the grades by Thursday night and post them on the web. Then, we should settle all complaints by Friday.

There will most likely be 8 problems on the final exam four of which will cover Chapter 17 (through §17.5), and the remaining will cover Chapters 13-16.

4.45.1 Practice one hour exam for Chapter 17

Do the following problems in an hour with closed books (no calculators will be needed). In any test you write for this class you'll have to show all your work.

  1. Problem 16, p. 1031 of your book.
  2. Problem 32, p. 1040 of your book.
  3. Problem 19, p. 1049 of your book.
  4. Problem 7, p. 1048 of your book.

4.46 W, 12/8/04: Final exam

We had our final exam which lasted 2 hours and 20 minutes. Of the following 8 problems I ended up not taking problem 2 into account.

  1. Find the point of maximal curvature of the curve $ y = \ln x$, for $ x>0$.
  2. Assume that $ u = u(x,y)$ has continuous second order partial derivatives. Show that

    $\displaystyle u_{xx} + u_{yy} = u_{rr} + \frac{1}{r^2} u_{\theta\theta} + \frac{1}{r} u_r, $

    where $ r$ and $ theta$ are the usual polar coordinates.
  3. Find the points on the sphere $ x^2+y^2+z^2 = 1$ that are the closest to and farthest from $ (2,1,2)$. Use the method of Lagrange multipliers.
  4. Using polar coordinates find the volume of the solid bounded above by the plane $ z=y+b$, below by the $ xy$-plane, and on the sides by the circular cylinder $ x^2+y^2=b^2$.
  5. Let $ \vec h(x,y,z) = (2xz+\sin y, x\cos y, x^2)$. Show that $ \vec h$ is a gradient field and evaluate the line integral

    $\displaystyle \oint_C \vec h \cdot d\vec r $

    where $ C$ is the curve given by the parametrization $ \vec r(t) = (\cos t, \sin t, t)$, $ 0\le t \le 2\pi$.
  6. A homogenous wire of mass $ M$ winds around the $ z$-axis as

    $\displaystyle C:\ \ \vec r(t) = (a \cos t, a \sin t, bt),\ \ 0\le t \le 2\pi. $

    Find the length of the wire, the center of mass and the moment of inertia around the $ z$-axis, in terms of the quantities $ a, b, M$.
  7. Let $ C$ be a smooth curve that bounds a domain $ \Omega$ of area $ A$. Calculate

    $\displaystyle \oint_C (ay+b)\,dx + (cx+d)\,dy. $

  8. Suppose that $ f$ and $ g$ have continuous first order partial derivatives in a simply connected open domain $ \Omega$. Show that if $ C$ is any smooth simple closed curve in $ \Omega$, then

    $\displaystyle \oint_C \left(f(\vec r)\vec\nabla g(\vec r) + g(\vec r)\vec\nabla f(\vec r)\right)\cdot d\vec r = 0. $

You can find the results of the final as well as the final letter grades here. I'll be in my office on Friday, 12/10/04, from 2:30 on for a couple of hours for questions and complaints.

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Mihalis Kolountzakis 2004-12-09