MAC 2313 -- Analytic Geometry and Calculus III -- Section 3122
Spring 2010

Professor Paul Ehrlich

414 Little Hall
(352) 392-0281 ext 280
for messages -- 392-0281 ext 221
ehrlich@ufl.edu, ehrlich@bellsouth.net

Also by appointment

For routine questions about homework assignments, exam material, etc., it is efficient to contact me by e-mail with your questions. You may also request a meeting outside of regularly scheduled office hours by e-mail. I often read e-mail at home (and the Uf Exchange crashes often), so please always send a copy of your e-mail to me at ehrlich@bellsouth.net

Prerequisites and a Brief Discussion of the Course

Prerequisites -- MAC 3473, MAC 2312, or MAC 2512

Textbook

James Stewart, Calculus, Early Transcendentals, Sixth Edition
Brooks Cole Publishing Company, 2008.

There is also a Student Solutions Manual, Volume 2, which you may find helpful. (There have been occasional errors in past editions of the solutions manual.)

Based on student feedback, I am going to continue the practice of giving 6 quizzes during the semester on the Thursday meeting as part of helping you to keep up with the course and preparing for the hour exams.

We will cover Chapters 12 - 15 and most of 16.

Reading and homework will be assigned on the course web site and discussed in class.

Homework and Class Attendance Policy

August 24, 2009 -- We recognize that there are many uncertainties as to how the semester will unfold with potential H1N1 swine flu epidemics and will be guided by the University policy concerning absences from class and exams, quiz and test makeup from illness, etc. If you are not able to make it to a test or quiz as a result of illness, please contact me by e-mail before the test of quiz is given.

I am a firm believer in the traditional German academic concept of "Lehrnfreiheit" [= Learning Freedom], and so do not require class attendance nor take attendance in my sections. However, I DO, in accordance with the University of Florida policy, expect you to keep up with assigned homework, especially assignments posted on the course web site to be handed in. I will firmly adhere to any penalties for late written assignments or projects to be turned in, posted with such assignments. A statement that "I didn't know an assignment was posted on the web site" will not be treated as a valid argument. [This semester with the current organizational structure for the Problem Session, there will be no such assignments given.]

For a listing of day to day homework assignments, click on

For a tentative day by day syllabus in tabular form, click on

Tests -- all held in the regular classrooms including the Final Exam.

Test 1: Monday, February 1
Test 2: Monday, March 1
Test 3: Friday, April 2
Final: Thursday, April 29th from 5:30 pm - 7:30 pm in the usual classroom Little 221.

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Written medical documentation is required for makeup tests --modified by UF policy with regard to H1N1 swine flu problems.

Students requesting classroom accomodation must first register with the Dean of Students Office. The Dean of Students Office will provide documentation to the student who must then provide this documentation to the Instructor when requesting accomodation.

The University assume that you are familiar with the University's honesty policy regarding cheating and the use of copyright materials. These matters may be found at

Student Code of Conduct Including Academic Honesty

and the use of copyright materials is discussed at

Student Copyright Rules

The new UF policies concerning minus grades are discussed at

Grade Point Equivalencies and Minus Grades

We will not be assigning minus grades in this class.

460 Total Points

Four 100-point tests:	400 points
Six 10 - point quizzes:	60 points

Grading Scale

• A range is 414 - 460
• B+ range is 405 - 413
• B range is 368 - 404
• C+ range is 360 - 367
• C range is 322 - 359
• D+ range is 312 - 321
• D range is 276 - 311
• E range is below 275

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Comment of the Week

The week of January 4th -- Since I cannot draw arrows easily using this software, I will boldface letters to indicate that they are vectors.

Unlike the text, in writing on the board and in typing these notes, I like to distinguish between the absolute value |c| of a number c and the length ||v|| of the vector v because of the interplay between these two quantities, as in the identity ||cv|| = |c| ||v||

In my mind a key thing to note in this beginning material is that if v is a NONZERO vector, the u= v/||v|| is a unit vector in v's direction. Thus a vector of length c pointing in the same direction as v is given by cu and a vector of length c pointing in the opposite direction from v is given by -cu

Oliver Heaviside was a scientist with strong views -- especially, he believed that rather than doing proofs in high school by Euclidean geometry as we learn even today in 10th grade, that problems like No. 45 on page 778 should be done by vector methods as we are doing now. Another excellent example is provided by a vector proof that if a quadrilateral has one pair of opposite sides which are parallel and of equal length, then the other pair of opposite sides is parallel and of equal length, also, and hence the quadrilateral is a parallelogram. Note a significant advantage of vector methods over high school geometry. If we show AB = CD, then this vector equation contains both the Euclidean geometry assertions that the line segments AB and CD are parallel and also the assertion that the line segments have the same length. This is an example of the efficiency of vector methods over traditional Euclidean geometry that appealed to Heaviside.


The week of January 11th -- One of the interesting things about lines and planes is the difference between two and three dimensions in studying the problem of whether or not two lines intersect. I have written a separate essay on this topic which you may access at

Comments on Lines in 3-Dimensions.

Another curious fact is that while neither the scalar product nor the cross product by themselves determine a vector, together they do -- see homework Problem 49 on page 793. Thus these two vector operations (all that we study) are indeed the fundamental concepts here.

A comparison of the properties of the scalar (or dot product) and the cross product may be found at

The Scalar and Cross Product

I have given an entirely algebraic derivation of the formula for the scalar projection of a vector b onto a nonzero vector a at

Algebraic Derivation of the Formula for comp_ab

based on more general techniques used in computational linear algebra, which avoids appealing to trigonometry.

In the Thursday Problem Session, we had a problem on finding the distance from a point to the specially tractable plane z = 3. In the next unit, you will see for yourself how the problem of minimizing the distance from a point to a more general plane works out, using the technique of Lagrange multipliers. The problem of finding the distance from a point to a general line is much easier to calculate. You may read an example of this problem worked by the methods of Calculus I at

The distance from a point to a line by calculus methods


In Chapter 16, the symbol n is reserved for a vector of length one which is perpendicular to a given surface. In working with planes in Chapter 12, it is usually unnecessary to work with a normal vector to the plane of length one. Hence, unlike the text I shall use N to denote any normal to a plane (not necessarily of length one) and reserve the symbol n for normal vectors of unit length.


The week of January 18th -- Note one subtle difficulty in determining spherical coordinates from cartesian coordinates, as in determining the spherical coordinates of the point P which has cartesian coordinates (1,-2,-(2)^1/2). One can calculate the rho of spherical coordinates unambiguously, and the angle phi unambiguously using a calculator, since phi is taken to be between 0 and pi. But when it comes to calculating theta, the arccos or arcsin function on the calculator can give the WRONG answer. In this problem we have to solve the equation cos(theta) = 1/(2)^1/2. The calculator would yield theta = pi/4. But then x = 2 sin(pi/4) sin(3 pi/4) = -1, but x is given rather as x = 1. Hence the key here is that ALSO theta = 7pi/4 satisfies cos(7 pi/4) = 1/(2)^1/2, but the arccos function on your calculator WON'T give you that choice. With THAT choice, x = 2 sin(7 pi/4) sin(3 pi/4) = +1 as given.

The week of January 25th -- Just a minor comment here as we now see the interplay between vector analysis and calculus, especially in a scientific calculation like the proof of Theorem 10 on page 833.

Nowdays the product rule for differentiation of real valued functions in Calculus I is often presented as in our current text on page 184 --

(fg) ' = fg ' + g f'
and this is a fine rule for commutative situations where xy = yx for any real numbers x,y makes this work correctly.

Now look at Theorem 3 on page 826, part (5) : there we find

d/dt(u(t) x v(t)) = u'(t) x v(t) + u(t) x v'(t).

Since a x b = - b x a, the order of the terms in the two factors very much matters. Thus when I teach basic calculus myself, I try to teach the product differentiation formula

(fg)' = f 'g + fg'
so that the cross product differentiation formula will then seem just like the basic formula from Calculus I. Note also that this more traditional ordering is also used in statement (4) for differentiation of the commutative scalar product.

I pointed out in class that for the straight line r(t) = r_o + t v that T'(t) = 0 , so that one cannot form the unit normal vector N(t) = T'(t)/ ||T'(t)||. A similar thing happens for the unit tangent and normal vector for the graph of a curve y = f(x) in the plane. Here the curvature vanishes at a point of inflection with f''(x) = 0, and in this case, also, one cannot form the vector N(x), cf. equation (11) on page 833.

The week of February 8th -- In limits and continuity for functions of several variables, things are a bit more complicated than for the calculus of one variable, because there are more possibilities of approach to the point (a,b) than just the left hand or right hand limit. A function f(x,y) has a limit (philosophically) as (x,y) ---> (a,b) only if no matter along which path we travel toward (a,b), we always get the same answer as we reach (a,b).

Thus we have the important principal in the box on page 872 -- if f(x,y) tends toward two DIFFERENT limits along two different paths as we approach (a,b), then f(x,y) fails to have a limit at (a,b).

Unfortunately, one has to understand the negation of this correctly -- if we take two different trial paths of approach toward (a,b) and get the same answer, this does NOT mean that f has a limit at (a,b). One must prove that this answer is gotten along all POSSIBLE paths, not just along straight lines passing through (a,b), and/or parabolas, like in the text examples and homework. Typically a "Squeeze Theorem" approach is easier to follow in establishing that a limit exists, than a direct delta-epsilon proof, so that's how I am teaching it on your first brush with this theory.

In considering continuity at (a,b), we first REQUIRE that f(a,b) be defined. [In considering limits, it is NOT necessary that f(a,b) be defined.] Then it is essential to note that even though f(a,b) is defined, if lim f(x,y) as (x,y) ---> (a,b) does NOT exist, then f(x,y) FAILS to be continous at (a,b), even though f(a,b) is defined.

Those of you who are particularly brave may enjoy skimming the essay on my web site on the historical developments of continuity at

The Historical Development of Continuity

The week of February 15th -- This week I will write a bit on the theory of calculus of several variables and the importance of some of the concepts in Section 14.4. Recall from last week my favorite example --

f(x,y) = xy / (x² + y²) if (x,y) is not (0,0) and f(0,0) = 0.


[ Thus f(x,y) is a function with domain |R².]

We have seen that this function does not have a limit at (0,0), since if we approach (0,0) along C₁ : x-axis, then f(x,0) = 0 has limit 0, whereas if we approach along C₂ : line y = x, then f(x,x) = 1/2 has limit 1/2. Hence, from our studies in Section 14.2, we are definitely convinced that f(x,y) is NOT continuous at (0,0), since f(x,y) has no limit at (0,0).

I showed in class that working from the definition of partial derivative, one may calculate directly that for this function, we have f_x(0,0) = 0 and f_y(0,0) = 0. Hence, this function provides an example of a function of several variables which FAILS to be continuous at (0,0), even though both partial derivatives of f exist at (0,0). [Contrast this with the simpler situation in Calculus I where one shows that if f '(a) exists, then f(x) is continuous at x = a]. Therefore, a fancier notion of differentiability is needed in several variables than merely assuming that the partial derivatives exist at (a,b).

We see this definition on Page 895 of the text, Definition 7, which I won't retype. Then I showed in class, that with THIS definition of differentiability, it may be shown (Homework Problem 45 on Page 900) that if f(x,y) is differentiable at (a,b), then f is continuous at (a,b). Thus with this definition of "differentiability" as given, one recovers the desired result akin to Calculus I that "differentiability at (a,b)" implies "continuity at (a,b)." [For y = f(x), f(x) is differentiable at x = a if f '(a) exists.]

Now one wants to have an easily checkable criterion for f(x,y) to be differentiable at (a,b). This is provided by Theorem 8 on Page 895, which asserts that if f_x and f_y are continuous AT (a,b) and defined at all points "NEAR" (a,b), then f is differentiable at (a,b).

An important consequence of this Theorem 8 is the following practical result:

Helpful Fact: Suppose f_x and f_y are continuous at all points of the domain D of f(x,y). Then f is differentiable (hence continuous) at all points of its domain D.



The week of February 22nd -- This week we do one of the big sections of calculus of several variables, perhaps one of the most helpful topics apart from optimization in several variables.

Since I can't type upside down triangles too easily, I will use another common notation for the gradient vector field,to which it is good to introduce you -- grad(f).

There are two ways that defining the directional derivative may be approached. First, this may be defined for any non-zero vector v at (x_o,y_o,z_o) and then it may proved for differentiable functions using the chain rule as I did in class that

D_vf(x_o,y_o,z₀) = [grad(f)(x_o,y_o,z_o) o v]/ ||v||

Dividing by the factor ||v|| takes into account the fact that it is not required that v be a unit vector.

The calculus books tend to take a second approach to this problem. Given a nonzero vector like v = i + j + k, they advocate forming a unit vector u = v/||v|| out of v, and then using the formula for directional derivative for a differentiable function and unit vector u calculate:

D_uf(x_o,y_o,x_o) = grad(f)(x_o,y_o,z_o) o u.

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This latter formula has the following implications for what grad(f) accomplishes when it is nonzero:

Four important properties of grad(f)

(i) grad(f)(p), if nonzero, gives the direction of greatest increase of f at p, and the maximum rate of increase of f at p is ||grad(f)(p)||.

(ii). - grad(f)(p), if nonzero, gives the direction of greatest decrease at p, and the maximum rate of decrease of f at p is given by - ||grad(f)||.

Here is a more advanced consequence of the above formula for the directional derivative which is not mentioned in the calculus books, but which is quite important in working out mathematical optimization theory (or linear programming) --

(iii). Suppose grad(f)(p) is nonzero, and we travel forward along a line starting at p whose direction vector u makes an acute angle with grad(f)(p). Then f(x,y,z) is increasing along this line (near p).

(iv). grad(f) is perpendicular to the level surfaces

S = {(x,y,z) in |R³ ; f(x,y,z) = k }.
Hence, if grad(f)(p) is nonzero, it is a normal vector for the tangent plane to S at p.

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Now let us revisit something I did when we were covering Section 14.4 and I derived a formula for a normal vector to the tangent plane to a surface which is a graph z = f(x,y). I used an interpretation of the partial derivatives to claim that two vectors were tangent to the surface, then I calculated their cross product to derive a certain formula. Let us see how easily this result may be obtained by using (iv) above. Given the graph z = f(x,y), form the new function F(x,y,z) = f(x,y) - z. Then the graph z = f(x,y) is exactly the level surface F(x,y,z) = 0. Hence, by (iv), grad(F) forms a normal vector to this surface. But

grad(F) = f_x i + f_yj - (1)k.
Hence, by this machinery, we reobtain the result I derived for you that

N(x,y,z) = f_xi + f_y j - k

is a normal vector to the tangent plane to the surface z = f(x,y).


The week of March 1st -- Especially when using the method of Lagrange multipliers to do max-min problems, it is important to study the constraint equation when beginning to work on the problem to see what it has to say about the permissible values of the variables. This then is often helpful in being able to throw away some possible valuess of x,y,z or lambda which arise in solving the equation systems.

Consider for example in the first two problems I assigned in past semesters for you to turn in on ??, we are extremizing the distance from the origin to the surface x² y² z = 1. Hence the requirement that "the point lies on the surface" translates into z > 0 and x and y are both nonzero. Thus the expressions one gets for f_x and f_y make sense, because x and y are never 0.

Since we are studying Max-Min Problems, you may enjoy reading a short essay on how the lawyer Pierre de Fermat himself worked max-min problems in the pioneering days of the development of calculus, before the more systematic work of Newton and Leibniz.

How Fermat Did Max-min Problems Himself .


The week of March 15th -- As we begin to concentrate on integration in several variables, let me stress the importance of being able to read off the region of integration from the limits of integration. In Section 15.2 we have the specialized application of Fubini's Theorem which in general is only valid for RECTANGLES in the boxed formula on Page 961, where we can just interchange the limits of integration in changing from a dydx integral to a dx dy integral. Contrast this with Example 5 on page 970 of Section 15.3 or Problem 50 on Page 973 which I emphasized in class. For these NONrectangular regions, one must first sketch the region of integration and understand its bounding curves in order to change the limits of integration.

I like to think of the limits of integration in the following way myself. Look at Problem 45 on Page 973. I think of the limits here as meaing -- y varies between y = 0 and y = 1, and for each fixed y, we have x varying between x = 3y and x = 3.

Especially, when we change from Euclidean coordinates to polar coordinates to evaluate certain integrals, one must figure out the region of integration and then redescribe it in terms of polar coordinates, not just plug in x = r cos(theta), y = r sin(theta) in the xy- equations of the limits of integration.


The week of March 22nd -- First, note on Page 1019 of the book, an analytic calculation of the spherical coordinate volume element is obtained, by using the Jacobian transformation technology, rather than those pictures of Figure 7 on Page 1007. In that Jacobian calculation on Page 1019, if you do it in detail, you will see an essential use of the basic trigometric identity sin²(x) + cos²(x) = 1 for both of the spherical coordinate angles.

Second, I want to comment on mathematics notation versus physics notation. In the standard physics texts (Gaussian surfaces, and electro magnetism portion), one has the use of the symbol dA for what was classically called the vector surface area element, which shows up in expressions like E o dA.

In Section 15.6 of the prior edition of our book, we had a technology for calculating surface area in the case of a surface which is a graph z = f(x,y), which is summarized in the boxed formula (9) of page 1077, this topic now being put into Chapter 16.

In class, I gave a shorthand notation for this formula as

dS = [(f_x)² +(f_y)² + 1]^1/2 dx dy

and I said that the symbol dS was classically called the surface area element, which is a shorthand for saying that it is exactly the thing which one integrates in order to calculate surface area, exactly as in the book's formula.

Now corresponding to physics, one can define a vector symbol dS, which I could call the vector surface area element. And this should be a vector with the property that its length ||dS|| = dS = that thing which calculates the surface area, and its direction is perpendicular to the surface in question. [This quantity which I am writing dS here in this note is exactly the dA of physics.] Thus if one introduces the standard notation of n for the unit outer normal to the surface (hopefully as in physics), one has the symbolic equation

dS = dS n,

but this is more conventionally written dS = n dS.


Hence,

F o dS = (F o n)dS =
(normal component of F)(that quantity which calculates surface area).

Because of this formula, ignoring the second term dS, one may learn in physics that F o dS is the normal component of the force F.

The week of April 5th -- People sometimes find that they can calculate line integrals, yet are puzzled about what a line integral means. A specific answer can be given in two particular cases: (i) if we have a function f(x,y,z) and we integrate f(x,y,z)ds over a curve C, then dividing this number by the length of the curve C gives the average value of the function f(x,y,z) over C. Or if we imagine f(x,y,z) as giving the mass density at the point (x,y,z) of C, then the line integral of f over C is simply calculating the total mass. (ii) if we integrate F o dR over a curve C, then this answer divided by the length of C yields the average value of the tangential component of F over C. Hence, it is important conceptually to realize that (for non conservative vector fields at least) the line integral depends BOTH on the vector field F or function f in question and on the particular curve C being traversed.

The expression F o dR can be interpreted in two ways. The first,

F o dR = (F o T) ds
is conceptually important in enabling us to see the line integral as representing an integration of the tangential component of F along the curve. A second formulation, for F = P(x,y,z)i + Q(x,y,z)j + R(x,y,z)k, as

F o dR = P(x,y,z)dx + Q(x,y,z)dy + R(x,y,z)dz
is a more helpful interpretation in computational terms.

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In Chapter 16, we study three basic operations on vector fields and functions; our friend the gradient and now two new operations basic in physics, the divergence and curl (defined in Section 16.5).

Here are the basic philosophical interpretations of these three objects:

curl(F) of a vector field F is a vector field which measures circulation (swirling) of F per unit volume.

div(F) of F is a function ("scalar field") which measures the net outflux of F per unit volume

grad(f) of a function is a vector field which measures the maximum rate of change of f.

Why do we concentrate on curl(F) and div(F) ?? One answer is that these two quantities together with boundary values of F determine the vector field F, hence they are all that one need study.

Here is a more exact statement of what I mean here, a result which can be found in vector analysis books, like we use for MAS 4156 --

Theorem -- Suppose F and G are two vector fields which are differentiable on a solid domain D in |R³ which satisfy the following three conditions:
(i) F o n = G o n at all points of the surface S bounding D.
(ii) curl(F) = curl(G) at all points of D.
(iii) div(F) = div(G) at all points of D.

Then F = G throughout D.


The week of April 12th -- Here is some philosophy for this week. So one starts this unit by learning how to calculate a line integral by parametrizing the curve, substituting in, and doing an integration (Section 16.2). No one really enjoys this, philosophicaly speaking, so as one advances in vector analysis, the idea is to AVOID directly calculating a line integral.

In what we are able to cover of this Chapter, two separate theories are presented. First, in Section 16.3, one learns the wonderful result that for a conservative vector field F, that if C is any curve from P to Q and f(x,y,z) is a potential for F, then the line integral of F o dR over C is simply given by the potential difference f(Q) - f(P). Hence, for conservative vector fields, we can calculate line integrals by finding a potential function. Now most vector fields are NOT conservative, but yet many that occur in basic physics ARE conservative, so that this is a useful concept.

The second technology may be used for arbitrary vector fields, but the curve MUST be a closed curve. Then for the case of a closed curve in the plane, and F(x,y) = P(x,y)i + Q(x,y)j, we have Green's Theorem relating the line integral of F o dR over C to a double integral. More generally, for a closed curve in |R ³, if we take any surface S which has C as a boundary, then the line integral may be calculated as a surface integral of curl(F) o n over the surface S, employing Stokes' Theorem.

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Paul Ehrlich
ehrlich@ufl.edu
last revised December 21, 2009