ma 242 section 11 Homepage

Welcome, please note that the homework and syllabus are linked just below.

• Course Syllabus for ma 242-011 ( grading scheme office hours etc...)
• Course Schedule ( homeworks assigned and test dates )
• ma 242-011 message board ( for homework discussion and test question speculation )
• Calculus III Online: Lectures by Dr. Norris, in real player format.
• Free tutoring in the MLC(Ha 244): Note that B. Suanmali, R. Petersen, J. Brown, R. Siskind, B. Benedict either have taught or are teaching ma 242.
• Constrained Partial Derivative Handout a few select hwks from Colley's 1st ed.

Test reviews:
The tests will follow closely the material emphasized in the reviews below. The best defense is to understand the homework and lecture examples, meaning you can do them cold without looking at a solution. Most of you will find the test is long so it is important that you learn how to calculate the problems both correctly and quickly. I have also posted the formula sheet that I will provide for you on the test day, you can keep the one I gave you in class and use it to study. You should pay specially close attention to the test format comments.
• Test One Overview and Formula Sheet:
• Test Two Overview and formula Sheet:
• Test Three Overview: and formula Sheet:
• Final Exam Overview: Please note that I am deleting problem two and decreasing the weight of the grad(f) in polars problem to 10pts, this way the closed-book points actually add to 20pts like I advertised in the Table.
• Take Home part of Final Exam:

Test solutions:
• Test One Solution:
• Test Two Solution:
• Test Three Solution:

• Required Homework Solutions:
These solutions are due to your peers and me, mostly me it turns out. We have selected the solution that we believe is most helpful to all and posted it for everyone's convenience,
• Problem Set One(Jan 18): hwks from 9.1-9.5
• Problem Set Two(Jan 25): hwks from 9.6,11.1,10.1-10.2
• Problem Set Three(Jan 30): hwks from 10.3-10.4
• Required homework solutions past Test I can be found in the "Recommended Homework Solutions", since you only turn in one problem the recommended hwk has become required. I expect you to be able to work those problems on the tests, with an exception of a difficult problem here or there. You may ask around test time if a given problem is unfair for the test, but you should probably be able to tell what's reasonable from the test reviews.

Recommended ( and a few Required) Homework Solutions:
These are my solutions to some of your homework. I have tried to select at least one of each type of problem you will encounter. These serve as additional examples to those given in lecture. You are of course free to ask me for further clarification if you find my solution to terse. Some of these problems are more advanced than the typical level of this course, I include those problems for your edification and my amusement. I have tried to include little remarks to alert you to the fact my solution is optional (meaning I don't expect you to do it the way I do it, for example anywhere I use the repeated index notation or "Einstein" notation you may ignore it if you like, but you should think about how to do it in your own brute-force way). Generally speaking you may choose the notation that you find most natural, sometimes I will use a notation that all of you find obtuse and obscure. I have my reasons, perhaps some of you will appreciate them. Those things which are "optional" are likely to show up as bonus questions on test ( just a point or two)

• [9.1-9.5]: Cartesian coordinates and vectors
• [9.5, 9.6, 11.1]: More vectors and functions of x,y and z
• [10.1-10.5]: Vector-valued functions and topics related.
• [11.2, 11.3, 11.5]: Partial derivatives.
• [11.4, 11.6]: Tangent planes and linearizations.
• [11.7, 11.8]: Min/Max and Lagrange multipliers.
• [12.2, 12.3, 12.7]: Double and Triple Cartesian Integrals.
• [10.5, 9.7, 12.9, 12.4, 12.8]: Jacobeans and Integration in other coordinates.
• [13.1, 13.5]: Curl, divergence, and vector fields.
• [13.2, 13.3]: Line integrals, FTC, conservative vector fields.
• [12.6, 13.6]: Surface area and surface integrals.
• [13.4, 13.7, 13.8]: Green's, Stokes', and Gauss' Theorems.

• Course Notes for Calculus III:
Posted below are links to my lecture notes from ma 242. I have tried to eliminate all the errors from the ma 242 material, but it's likely there is something wrong somewhere. If you find an error in the notes somewhere in pages 236-425 and you are the first to email me about it then I will give you a bonus point. Generally, the material in these notes matched up with the text, however, in several places, the text is not as detailed as I would like, so I add examples and discussion. I have intentionally tried to avoid duplicating examples that are given in your text. I do try to follow these notes fairly closely in lecture, modulo questions about homework.

• Cartesian coordinates and vectors: (9.1-9.4)
• [236-250] We define three dimensional Cartesian coordinates,Vectors, addition of vectors, scalar multiplication of vectors, dot and cross products of vectors. Also we discuss the meaning of the unit vector and the vector and scalar projections.
• Analytic geometry and graphing: (9.5-9.6 and 11.1)
• [251-262] Parametric equation of a line explained. Also equation of plane, graphs z=f(x,y), level surfaces, and contour plots introduced. We used Maple in lecture to view the standard examples of ellipsoids, paraboloids, hyperboloids and the cone.
• Curves in 3-d:
• [263-279] We develop the differential geometry of curves. Arclength, curvature and torsion characterize the shape of a given curve and they are calculated from the TNB (Frenet)-Frame which we identify as moving coordinate system. The osculating plane and circle as well as the TNB frame can be animated by Maple and we saw an example in class which is posted below.
• Motion in 3-d and Kepler's Laws:
• [280-289] Once we identify the parameter of the curve as time we may define velocity, speed, and acceleration. Then we use the TNB frame to break up the acceleration into its tangential and normal components, the familar equations of constant speed circular motion are derived rigoursly via calculus. Finally, we derive Kepler's Laws from scratch using Newton's Universal Law of Gravitation plus vector algebra ( skipped it, sadly).
• Partial Differentiation: (11.2,11.3,11.5 and more not in Stewart)
• [290-310] Limits generalized to two or more variables. Definition of partial differentiation plus geometric interpretation discussed, chain rule for several independent and/or several intermediate variables detailed. Problem of constrained partial differentiation introduced and remedied by clarifying which variables are dependent verses independent ( not quite in Stewart ). General idea of differentiation and the relation of the Jacobean matrix to the chain rule.
• Tangent Planes and Differentials (11.4,11.6)
• [311-319] Linearization introduced as a tool to approximate functions of several variables. The total differential introduced to aid in the estimation of error. The tangent plane discussed again, we find how to find its equation for the case z=f(x,y). Parametric surfaces introduced, then we see how to find the tangent plane in that alternate formulation for a surface.
• Multivariate Extrema: (11.7,11.8)
• [320-329] Theory for finding extrema for a function of two variables stated without proof, several examples given. Closed interval method generalized to a closed bounded region. Finally a novel new geometrically motivated approach known as the Lagrange Multiplier Method is used to solve constrained maximation problems.
• Multivariate Cartesian Integrals:(12.1,12.2,12.3,12,7)
• [330-342] Double and triple integrals over boxes defined. In short, the double and triple integrals require us to iterate the integrals, just as in partial differentiation the variables besides the integration variable are regarded as constants. Then we do more general double integrals over TYPE I and TYPE II regions. Graphing becomes very important. The order of iteration is dictated by the nature of the region, some are easier to characterize as TYPE I or TYPE II, if choose unwisely might have to split it up into several pieces. Finally, triple integrals over general regions discussed, again the order of integration should be based on how the region's graph can be understood in terms of bounding x,y,z in terms of each other.
• Multivariate Integrals in General Coordinates: (12.4,12.8,12.9)
• [343-359] Change of variables theorem for multivariate integrals given. A proof is sketched which is based on our general theory of differentiation from earlier. Standard curve-linear coordinate integrations are discussed, we derive the modified integration rules through calculating the appropriate Jacobean determinants. Finally a geometric motivation for the curvy area and volume elements is given. Differential forms are mentioned as an alternate method for calculating determinants.
• Differential Vector Calculus (13.1,13.5, and more not in Stewart)
• [360-384] We discuss the gradient, divergence and curl. Their geometric significance is pondered, and their algebraic structure is unfurled. Being discontent with Stewart's Cartesian-centric universe we derive formulas for cylindrical and spherical coordinates. This requires some effort, but in the process we learn more about vector algebra and coordinates in general. Your criticism of my confusing (,) with <,> gains merit as we consider cylindrical and spherical vector basis which have a coordinate depedence. Finally, we mention how the superior physics conventions compare to our distasteful mathy conventions. Throughout I sprinkle examples from electromagnetism to illustrate the use of the vector calculus. We also begin our discussion of Conservative vector fields and potential functions.
• Line Integrals: (13.2,13.3)
• [385-401] We define scalar line integrals along arclength, dx, dy and dz. Then we find how to integrate a vector field along a curve, this is called the line integral. The Fundamental Theorem of Calculus (FTC) for line integrals is given and applied to various problems. We find why the terminology "conservative vector field" is good. Objects under the sole influence of a conservative vector field have their net energy conserved. We derive the conservation of energy for such forces as a consequence of the FTC for line integrals.
• Parametric Surfaces and Surface Integrals: (10.5,12.6,13.6)
• [402-411] We revisit the parametric surface. We find how to calculate the surface area in view of the parametric characterization of the surface, graphs z=f(x,y) can also be treated as a special case where the parameters are simply "x" and "y".The definition of the surface integral of a vector field is given. I attempt to illustrate the "geometric" or intuitive approach in constrast to the calculations indicated from a straightforward application of the defintion. One shouldn't rely on either in all cases.
• Green's, Stoke's and Gauss Theorems: (13.4,13.7,13.8)
• [412-425] We begin with Stoke's Theorem and remark that Green's Theorem is simply a special case. However, it is an important case so we study a number of examples. Stoke's Theorem involves the curious trade of a surface integral for a line integral, this seems like magic. If you object to magic you can work through the proof in Stewart. Then the Divergence Theorem which is also known as Gauss's Theorem is discussed. This time a volume integral can be exchanged for a surface integral. Finally we conclude with a brief overview of the calculus of differential forms. I show how the exterior derivative unifies the distinct derivatives of vector calculus into a single operation. Then the Generalized Stoke's Theorem is seen to reproduce the FTC, Stoke's and Gauss's Theorems. You can see my ma 430 notes for more on differential forms.

Corrections:
Running tab of errors big and small we find as I lecture:
1.) E4 on pg. 245 is incorrect, should have used A=(0,0,alpha)
2.) on pg. 253 I forgot to insist that skew line could not be parallel. A pair of lines are skew if they are nonintersecting and nonparallel.
3.) In the Proposition on pg. 273 it should be absolute value of dT/dt at time t_o. The theorem I was thinking about said positive curvature to rule out the case of zero curvature.
4.) In the formula for K(t) it should be a |r'(t)| in the denominator, NOT simply |T(t)|.
5.) In E45 on pg. 283 I forgot to put a T and an N on the LHS of those equalities. You can spot something is wrong because w/o modification we would be equating a scalar and a vector, not good.
6.) In E73 on pg. 313 the math is correct for R = 1/R1 + 1/R2, but physically its wrong, we should have 1/R. 7.) In E75 on pg. 315 the last paragraph incorrectly claims <1/sqrt(2),-1/sqrt(2)> is direction for max change in f, we should instead say the unit vector in the <1,1/2> direction gives max change, that unit vector is (1/sqrt(5))<1,1/2>. Likewise the unit vector for max negative change would be (-1/sqrt(5))<1,1/2> which is the unit vector opposite in direction to the gradient at (1,-1).
8.) In E82 in the notes I incorrectly suppose f_xy = 0 at one point on pg. 322. It doesn't change the outcome of the problem, but f_xy = 4 in fact so D=10-4=6 which leads to the same results. 9.) In E173 I forgot to put "rho"^2 in the dV, correct answer is (pi*R^5)/5

Bonus Projects:
These are not required. It is entirely possible to make an A+ w/o ever doing any of these. You may achieve a maximum of 15pts bonus. These are weighed the same as tests. It is much easier to earn these points on the test, but hey this is bonus so don't complain. These are due May 7 at the latest.
• Differential Forms Project (10pts)
• 90% of all the review exercises from a chapter we've covered.(5pts each chapter)

• Course Notes from Calculus I & II:
These are my course notes from Calculus I and II, somtimes I reference them in the calculus III notes, so there here if you need them.
• Basics of functions: (Chapter One)
• [1-9] Review of functions, their properties, and graphs. We will use these ideas thoughout the course.
• Motivations and limits: (2.2-2.5)
• [10-23] Need for derivative, finite limits, and limits involving infinity.
• Tangents and the derivative: (2.1 & 2.6-2.9)
• [24-34] Definition of tangent to curve, definition of derivative at a point, and the derivative as a function. Finally, the derivatives of the basic functions are calculated.
• Differentiation: (3.1-3.7)
• [35-50] Having established all the basic derivatives, we learn how to differentiate most any function you can think of. This is accomplished through the application of several rules and techniques: product rule, quotient rule, chain rule, implicit differentiation, logarithmic differentiation. These calculational tools form the heart of this course.
• Parametrized curves and more:(3.8-4.1)
• [51-58] Parametric curves discussed (please ignore 52, it is wrong), Approximation of a function by the "best linear approximation" explained, finally a number of related rates problems worked out.
• L'Hopital's Rule: (4.5)
• [59-64] We return to the study of limits again. With the help of differentiation, we are able to calculate many new limits through L'Hopital's Rule. We discuss a number of different indeterminant forms and see how to determine each.
• Graphing: (4.2-4.3)
• [65-75] We study local and global maxima and minima of functions. A number of geometric ideas are introduced, increasing, decreasing, concave up, concave down, critical points, inflection points, local maximum, local minimum, absolute(global) maximum, absolute(global) minimum. These ideas are studied through the revealing and powerful lense of calculus. The first and sencond derivative tests are presented and applied to some examples.
• Optimization: (4.6)
• [76-81] We apply differential calculus to a number of interesting problems. The first and second derivative tests are applied to some real-life problems.
• Basic Integration: (4.9-5.5)
• [82-103] We define the definite integral and see how it gives the "signed" area under a curve. While it is intuitively clear that the definite integral defined by the limit of the Riemann sum gives the area under a curve, it is almost impossible to directly calculate that limit for most examples. Fortunately, we find that the fundamental theorem of calculus (FTC) allows us to avoid the messy infinite limit. Instead of finding the limit of the Riemann sum, we merely must find the antiderivative of the integrand and use the FTC. Finding antiderivatives (aka indefinite integration) is a nontrivial task in general. We only begin the study listing the obvious examples from the basis of our study of differentiation. Then, we conclude our study by considering the technique of U-substitution. U-substitution is the most useful technique of integration since it is basically the analogue of the chain rule for integration.

• U-substitution (5.5)
• [98-104] U-substitution: remember when changing the variable of integration that you must convert the measure (dx) and the integrand to the new variable.
• Integrating powers of sine or cosine (5.7)
• [105-106] Integrating powers of sine and cosine. Odd-powers amount to an easy u-substitution while even powers require successive applications of the double-angle formulae.
• Trig-substitution (5.7)
• [107-111] Trig-substitution: same game as U-substitution except that the substitutions are implicit rather than explicit. Typically, trig substitutions are used to remove unwanted radicals from the integrand.
• Integration by parts (5.6)
• [112-115] Integration by parts: IBP is integration's analogue of the product rule. We note that the heuristic rule "LIATE" is useful to suggest our choice of U and dV.
• Partial Fractions (5.7 & Appendix G)
• [116-122] This special technique helps us to rip apart rational functions into easily digestable pieces. I explain how to break up a rational function into its basic building blocks. Additionally, I explain explicitly how to integrate any of the basic rational functions that can result from the method. The overall result is that we can integrate any rational function with the help of the partial-fractal decomposition. Besides being useful for integrating rational functions, the algebra introduced will be useful in later course-work ( for example Laplace Transforms...)
• Numerical integration (5.9)
• [123-125] Simpson's rule and trapezoid rule, brief discussion of errors.
• Improper integration (5.10)
• [127-131] Integrals to infinity and integrals of infinity. Both of these must be dealt with by limits. We examine how these integrals suggest that some shapes that have infinite length can have a finite area.
• Areas bounded by curves (6.1)
• [132-139c] Graph, draw a picture to find dA, then integrate. We calculate the area of the triangle, circle and ellipse and more.
• Finding volumes (6.2)
• [140-146g] Graph, draw a picture to find dV, then integrate. We calculate the volume of the cone, sphere, torus and more.
• Arclength (6.3)
• [147-150] How to find the length of an arbitrairy curve and how to take the average of a function.
• Average of a function (6.4)
• [151-153] How to find the length of an arbitrairy curve and how to take the average of a function.
• Applications of calculus to physics (6.5)
• [154-165] We see how notions from highschool physics are generalized with the help of calculus. We calculate work done by a non-constant force - the net force applied by a non-constant pressure. We conclude with a discussion of how to find the center of mass for a planar region of constant density.
• Probability (6.6)
• [165b-165c] The definition for a probability distribution of a continuous variable x is given. Then the probability for a particular range of values is defined. Finally, the mean and median are defined and discussed briefly.
• Introduction to differential equations (7.1)
•  Differential equations are described in general. The definition of the terms: ordinary, order, autonomous, linear, homogeneous, nonhomogeneous and solution, are given and discussed as they apply to differential equations.
• Direction Fields and Euler's Method (7.2)
• [167-170] Basic terminology and graphing DEqs. Euler's method is then discussed. We explicitly see how to construct an approximate solution by using Euler's method. In essence, this is nothing more than tracing out a solution curve in the direction field. Equilibrium solutions are also defined and discussed in several examples.
• Separation of variables (7.3)
• [171-177] Separate then integrate. A number of physically interesting examples given.
• Exponential growth (7.4)
•  Perhaps the most naive growth model, yet it is a good approximation of many real world processes. We study the differential equation that it arises from and derive the solutions by math we've learned in earlier sections.
• Logistic growth (7.5)
• [179-182] The logistic equation is slightly less naive than simple exponential growth. We analyze the logistic differential eqn. in two ways. First, we study implications of the DEqn directly and find some rather interesting general conclusions about any solution. Second, we solve the logistic DEqn directly and find the general form of the solution. An example of how you might try to apply it is then given (realistic modeling of population growth has not proven to be very reliable historically, so I'll abstain from anything but the math here... )
• Homogeneous 2nd order linear ordinary DEqns (7.7)
• [183-187] We begin by carefully analyzing the possible solutions to the homogeneous case. We find three possiblities corresponding to the three types of solutions to the quadratic characteristic equation. In each of the cases I, II, and III we find two linearly independent fundamental solutions. The general solution is then formed by taking a linear combination of the fundamental solutions.
• Nonhomogeneous 2nd order linear ordinary DEqns (7.8)
• [183-187] We begin by reviewing the possible solutions to the homogeneous case. Next, we explain the form of the general solution to a nonhomogenous DEqn is the sum of a complementary and particular solution. The complementary solution is found by the same technique as in section 7.7. Then, we see how to find the particular solution through the method of undetermined coefficients. We begin with several examples, next a general algorithm describing the method is given (hopefully exposing some of the subtleties avoided in the first few examples), and after that yet more examples are given. Finally, we conclude by explaining how the complementary and particular solutions combine to make the general solution (a proof long overdue at this point in the notes).
• Springs and RLC circuits (7.9)
• [192-195] We study the motion springs in a viscous media and three cases result (under/over/critical damping), just like in the last section. It is the same math. Then we study springs that are pushed by an outside force and we encounter the interesting phenomenon of "resonance". Finally, we note the analogy between the RLC circuit and a spring with friction.
• Sequences (8.1)
• [196-200] Definitions and examples to begin. Then we discuss how to take the limit of a sequence using what we learned about limits in calculus I. The squeeze theorem and absolute convergence theorem help us pin down some otherwise tricky limits.
• Series (8.2)
• [201-203] The series is a sum of a sequence. We give a careful definition of this - a series is the limit of the sequence of partial sums. When the sequence of partial sums converges, we say the series converges. We discuss telescoping and geometric series - these two are the easiest series to actually calculate. The n-th term test is given - this is likewise the easiest and quickest of all the convergence tests to apply. Finally, we give some general properties of convergent series.
• Convergence Tests (8.3-8.4)
• [204-210] The task of determining whether or not a given series converges or diverges is a delicate question and we try to develop some intuition by examining a number of examples. We go over a number of tests which can be used to prove that a series converges or diverges. Lastly, we summarize the tests at our disposal in this course.
• Estimating a series (8.3-8.4)
• [211-213] We explain how close a particular partial sum is to the series. This is important because it's not always possible to calculate the limit of the sequence of partial sums. The alternating series error theorem is especially nice.
• Power series (8.5)
• [214-216] A power series is a function which is defined pointwise by a series. We study a number of power series and discover what elementary functions they correspond to. The set of real numbers for which the power series converges is known as the "interval of convergence" (IOC). We learn that the IOC must be a finite subinterval of the real numbers, or the whole real line. This means the IOC can be described by its center point and the radius of convergence (R). We also discover that a power series allows us to differentiate and integrate term by term.
• Geometric series tricks for power series (8.5-8.6)
• [217-218] The term by term calculus theorem (from the last section of notes) along with the geometric series allows us to find power series expansions for a number of functions. Admitably, this method is a bit awkward, however, if you take a course in complex variables (I recommend Ma 513), you'll find that these calculations are quite important later on.
• Taylor series (8.7)
• [219-226] The Taylor series explicitly connects the power series expansion of a function to the derivatives of that function. The Taylor series method simply generates the power series by taking some derivatives and evaluating them at the desired center point for the power series expansion. We establish the standard Maclaurin series and discuss how to generate new series from those basic series. While this section allows us to generate the power series in a straightforward fashion, it is not always the case that this is the most efficient method. The last section, while awkward, is quicker for the examples it touches.
• Binomial series (8.8)
• [227-228] This beautiful theorem shows us how to raise binomials to irrational powers. It is very important to many applications in engineering and physics where it is enough to keep just the first few terms in the binomial series. Since the series expansion is unique (if it exists), we find some of the same results as we did with Taylor series and before.
• Applications of Taylor series (8.9)
• [229-235] I discuss error a bit more and we see why sin(x)=x up to about 20 degrees to an accuracy of 0.01 radians(this claim should be familar to you from the pendulum in freshman physics). Then we continue an example from calculus I and we see how to calculate the square root using the power series expansion of sqrt(x). Then we calculate power series solutions to some otherwise intractable integrals. Then we conclude the course by examining how some series we've covered are used in physics.

Bonus Point: First person to email me the identity of the scientist/tree farmer pictured, Times Up, one of you guessed it. This is David Pittman of Mayland Community College. The Hannah says do your homework early and ask good questions. You can't argue with the Hannah, she's always right. Wild world of maple sheets:
I have gathered together some simple applications of Maple to aid you in completing your homework. Of course, you can find much more in the Maple help, but this ought to get you started. We can add to this list as the semester goes on with your help. I want to post more very simple Maple sheets that get straight to the point of how to do this or that with Maple. So if you know something else that would be good to add here, email me your idea ( and attach the sheet).

• Integration indefinite and definite integrals.

• posted below are Maple examples from class, without output.

• Quadric surfaces and more: 3d implicit plots of Quadric surfaces, contour mappings and the corresponding 3d-plots. And a token level surfaces plot. Presented with the 251-262 notes on sections 9.6 and 11.1 on thursday 1-18-07. Click her to see a static html version: 3d-plots and contours
• 3-d animation of TNB frame, osculating circle and plane.
• Animating parametric 2-d curves:

• TNB-frame around a donut:

• In this you can see the TNB-frame follow the path that wraps around the donut, the circle is the osculating circle. You can decipher the Maple code to see how my brother cooked this one up, we've covered the requisite theory, you just need to think about the Maple coding to make it all animated and such, you could change the code to do other curves as well.

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