MATH 231 section 1: Calculus III, calculus of several variables with analytic geometry
MATH 231 section 1 Homepage

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

• Guide for Test 3, in-class part mostly (yes there will be a take-home)
• Homework Project III Solution
• Homework from after Test 2 to end of course.
• Bail out package for Test 2
• corrections to Homework Project II solution
• Homework Project II
• Homework Project II Solution: contains some discussion of hyperbolic functions and a method for finding "nice" parametrizations via cross-sections.
• Revised (as of Oct 7)Required Homework up to Test 2
• Required Homework up to Test 2

• Course Syllabus
• Course Schedule ( due dates for assignments and test dates )
• Errata ( errors found so far )
• Constrained Partial Derivative Handout a few select hwks from Susan Colley's 1st ed. of Vector Calculus
• Corrections to 8/21/2008 lecture.
• A gift for Problem's One and Two of Homework Project I.
• .
• A note on how to compute determinants. We use these patterns to calculate cross-products for now, Jacobians a little later on.
• Homework Project 1 Solution
• Added notes on the general idea of differentiation these notes supplement the page 306... notes. I try to motivate the general derivative or Frechet Derivative by analogy to calculus I. The big idea is that the derivative is the best linear approximation. We covered these notes 305a,b,c,d,e,f on Tuesday Sept. 16, 2008. I may cover 305g on Sept. 18.
• Corrections to page 314

• Homework Quiz Solutions:

• Homework Quiz 1 Solution vectors.
• Homework Quiz 2 Solution lines and planes.
• Homework Quiz 3 Solution calculus of vector-valued functions of a real variable and motion in 3D.
• Homework Quiz 4 Solution partial derivatives.
• Homework Quiz 5 Solution fun with grad(f).
• Homework 6 Solution parametrizations and directional derivatives.
• Homework 7 Solution double and triple integrals in Cartesian coordinates.
• Homework 8 Solution Coordinate change and Jacobians.
• Homework 9 Solution double integrals in polar coordinates.
• Homework 10 Solution triple integrals using cylindrical coordinate change (use sphericals for #26 its easier).
• Homework 11 Solution integrations via spherical coordinates.
• Homework 12 Solution
• Homework 13 and 14 Solution
• Homework 15 Solution
• Homework 16 Solution
• Homework 17 Solution

• Test Solutions from this Semester:

• Solution for Test 1
• Solution for Test 2
• Solution for Bail Out Quiz
• Solution for Test 3 (in and out of class parts)

• Test Reviews:
• Review for Test 1
• Review for Test 2
• Review for in-class part of Test 3

• Select 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 (wait, maybe switch that). 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)

note: problem numbers probably do not match your text. These solutions were written originally for Calculus and Concepts which is a different, somehow, version of your text. You can look at the other version during office hours if you so desire. Additionally, I should mention that these solutions may in places use notation which I am avoiding in lecture. The main distinction is that I have made an effort to adorn every vector with the vector symbol. Strictly speaking this is just notation so leaving the vector off is no big deal so long as you say somewhere "HEY THIS IS A VECTOR". Of course, students are not always so careful and I will admit that my omission of the vector symbol has likely confused a few students. Some professors are not as open minded as I am on this point. For example, my Junior-level Mechanics professor started his whole class with an proclaimation that failure to write the vector over vectors would be graded as incorrect. Strict adherence to the vector notation can in principle help avoid making silly mistakes like dividing by a vector (almost never makes sense).

Homework Projects:

It would be foolish to not look over these well before the due date. These can be tricky in places. If you ask nicely I might help you when you are stuck...

Old Tests and Solutions:

The tests for our course will have a different focus, but perhaps these will give you some idea of the length of the tests.
• Old Test One Solution:
• Old Test Two Solution:
• Old Test Three Solution:

• Course Contact Information:

• Instructor: James S. Cook
• Office: Applied Science 105
• Office Hours: M-T-W-R-F from 7:00am-9:00am and M-W-F from 2:45pm-3:45pm
• email: jcook4@liberty.edu
• office phone: 434-582-2476
• all lectures are in DeMoss Learning Center room 1107
• lecture times: TR 12:25pm-1:40pm
• Course Notes for Calculus III:

• WARNING: these notes sometimes refer to the homework solution. Those numbers refer to problems from James Stewart's Calculus and Concepts. That text is nearly identical to your text, but the section numbers are different. If you are confused about which version is being referenced don't hestitate to stop by office hours or email me about it, I have both versions in my office.

• Cartesian coordinates and vectors: (13.1)
• [236-239] We define three dimensional Cartesian coordinates and discuss distance between points. Also the point-vector correspondance is seen.
• Vectors: (13.2-13.4)
• [240-250] We define 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.
• Lines and planes: (13.5)
• [251-256] Parametric equation of a line examined. Also equation of plane motivated from vector viewpoint.
• Analytic geometry and graphing: (13.6)
• [257-262] Graphs z=f(x,y), level surfaces, and contour plots introduced.
• Calculus for vector-valued functions of "t": (14.1-14.2)
• [263-268] Concept of a vector-valued function of a real variable is introduced. Integrals and derivatives are done one component at a time.
• Curves in 3-d: (14.3)
• [269-279] 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 help us visualize the curve.
• Motion in 3-d: (14.4)
• [280-283] 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.
• Kepler's Laws: (14.4)
• [283-289] We derive Kepler's Laws from scratch using Newton's Universal Law of Gravitation plus vector algebra. We will probably not have time for a lecture on this.
• Limits and continuity: (15.1-15.2)
• [290-291] Limits generalized to two or more variables.
• Partial Differentiation: (15.3)
• [292-295] Definition of partial differentiation plus geometric interpretation discussed. Examples illustrating basic idea of partial differention are also given here.
• Chain Rules for Partial Differentiation: (11.5)
• [296-299] Chain rule for several independent and/or several intermediate variables detailed.
• Constrained Partial Differentiation: (11.5 and more not in Stewart)
• [300-305] Sometimes which variables are dependent and which are independent is not clear. The notation introduced in the discussion of Constrained Partial differentiation helps reduce this confusion. Implicit differentiation is also discussed, in principle this is the same concept. Basically the overall question is how do you differentiate given that some algebraic relation also holds. In contrast, the earlier parts of the notes dealt with variables with no constraints. If there are no constraints then the variables are said to be "independent".
• Added notes on the general idea of differentiation these notes supplement the page 306... notes. I try to motivate the general derivative or Frechet Derivative by analogy to calculus I. The big idea is that the derivative is the best linear approximation. We covered these notes 305a,b,c,d,e,f on Tuesday Sept. 16, 2008. I may cover 305g on Sept. 18.
• Theory of derivatives:(not quite in Stewart)
• [306-310] I explain what the geometric idea is behind all these assorted partial derivatives we have seen. We simply insist that the derivative of a function is the best linear approximation. The concept of "best" implicits a "norm" and technically what I am describing here is an instance of the so-called "Frechet Derivative". Don't worry, we don't get too carried away, we simply explain how to differentiate other sorts of functions. The Jacobian matrix is introduced and calculated for a few explicit examples. We provide a very general proof for the chain rule on the basis of a little matrix/linear algebra argument. The results shown here are useful later in section 17.9.
• Tangent Planes and Differentials (15.4,15.6)
• [311-313] Linearization introduced as a tool to approximate functions of several variables. The total differential introduced to aid in the estimation of error.
• Directional Derivative (15.6)
• [314-316] Directional Derivatives are used to measure the rate of change of the function is some given direction. Because the directional derivative is linear it can be constructed from the partial derivatives if they are given. Geometrically, the graph will intersect a vertical plane to form some curve. The directional derivative measures the slope of that curve at some point. Pragmatically, the directional derivative tells us how quickly the function is changing. We need both a point and a direction to even ask the question. Pictures of this are fairly easy to see for f(x,y), but the concept extends to functions of many variables. The graphs of f(x,y,z) or f(w,x,y,z) lie in 4 or 5 dimensional space respectively, its hard to picture. Finally, it should be emphasized that the directional derivative is best remembered by its connection with the gradient of a function.
• Tangent Planes and Parametrized Surfaces (15.6,17.6)
• [317-319] The tangent plane discussed again, we find how to find its equation for the case z=f(x,y). Parametric surfaces are introduced, we see how to find the tangent plane in that alternate formulation for a surface. We compare and contrast the various viewpoints for describing surface as well as how the tangent plane is found in each setting.
• Multivariate Extrema: (15.7)
• [320-324] 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.
• Lagrange Multipliers: (15.8)
• [325-329] A novel geometrically motivated approach known as the Lagrange Multiplier Method is used to solve constrained maximation problems. I describe briefly how Lagrange Multipliers are used in the variational calculus of classical mechanics. I should emphasize that the technique of Lagrange multipliers is far larger than the particular application made in this section. That said we will not cover this topic this semester.
• Multivariate Cartesian Integrals:(16.1,16.2,16.6)
• [330-333] Double and triple integrals over boxes defined. In short, the double and triple integrals require us to iterate the integrals. The areas and volumes of integration considered here are simple rectangles or cubes. These are the easy ones.
• Double Integrals over nontrivial regions(16.3)
• [334-338] In short, the double integrals require us to iterate the integrals, just as in partial differentiation the variables besides the integration variable are regarded as constants. 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. Double integrals of a function give the signed volume of the functions graph over the region of integration.
• Triple Integrals over nontrivial regions:(16.6)
• [339-342] In short, the triple integrals require us to iterate the integrals, just as in partial differentiation the variables besides the integration variable are regarded as constants. Triple integrals over general regions are 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. If we integrate the constant function f=1 then that triple integral calculates the volume of the integration region. However, the integral of f(x,y,z) generally represents some 4-volume (or hypervolume) which represents a sum of the function's values over the region of integration. For example, integrals of charge or mass density yield the total charge or mass contained in the region of integration. (Here I am speaking of volume densities.)
• Multivariate Integrals in General Coordinates: (16.4,16.7,16.8,16.9)
• [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 Jacobian 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. Cartesian coordinates are often preferred because we are most familar with them. However, it is known that using other coordinates which reflect the symmetry of a problem can both reduce calculational difficulty and emphasize the conceptual beauty of the problem.
• Vector Fields: (17.1, and more not in Stewart)
• [360-365] 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. The convenient (Cartesian) correspondance of points and vectors is no longer valid with respect to the natural coordinate basis for non-Cartesian coordinates. We see that the cylindrical and spherical coordinate vector basis have a coordinate depedence. One cannot simply identify (,) with <,> in non-Cartesian coordinates. (the business about other coordinate systems is not to be found in your text, however the material is complete in these notes so no worries)
• Gradient: (17.5, and more not in Stewart)
• [366-368] We discuss the gradient. Its geometric significance is pondered, and its 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. This ongoing discussion of non-Cartesian coordinates is not in your text, but again the material is complete within these notes.
• Curl: (17.5, and more not in Stewart)
• [369-372] We discuss the curl. Its geometric significance is pondered, and its algebraic structure is unfurled. We also begin our discussion of Conservative vector fields and potential functions.
• Divergence: (17.5, and more not in Stewart)
• [360-384] We discuss the divergence. Its geometric significance is pondered, and its algebraic structure is unfurled. I include two important examples from electromagnetism to illustrate the use of the vector calculus.
• Curl and Divergence in curved coordinates: (not in Stewart)
• [375-384] 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. The convenient (Cartesian) correspondance of points and vectors is no longer valid with respect to the natural coordinate basis for non-Cartesian coordinates. We see that the cylindrical and spherical coordinate vector basis have a coordinate dependence. Derivatives act on the coordinate vectors as well as the component functions of a vector field, this complicates the formulas considerably. The fact remains that these formulas are essential building blocks for real-world problems in physics where using Cartesian coordinates is cumbersome. Instead, sphericals or cylindrical coordinates are better. And now since we have done all the heavy lifting here you can be sure that those weird looking formulas for grad, curl and div are actually quite reasonable. It is just a matter of persistence to derive them. Perhaps you will have the opportunity to derive the one that is missing in these notes... Finally, at the end of this section we summarize the formulas in both the math conventions and the superior more tasteful physics conventions.
• Line Integrals: (17.2)
• [385-394] 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.
• FTC for Line Integrals and Path Independence: (17.3)
• [395-399] 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. (see next section)
• Work Energy Theorem: (17.3)
• [400-401] 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 their areas: (17.6)
• [402-406] 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".
• Surface Integrals: (17.7)
• [407-411] 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 definition. One shouldn't rely on either in all cases. I should mention these are often called "flux integrals" because the surface integral calculates the flux of a field which cuts across the surface.
• Green's, Stoke's Theorems: (17.4,17.8)
• [412-419] 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.
• Conservative Vector Fields: (many sections)
•  We collect our thoughts about conservative vector fields. As it turns out there are a number of distinct methods to characterize such fields. Each has its place, interestingly topology becomes important in this discussion.
• Gauss' or Divergence Theorem: (13.9)
• [421-423] 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. This allows the density of some quantity to be related the flux of a vector field.
• Differential Forms: (not in Stewart)
• [424-425] 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. The concept of a differential form is important to both modern theoretical physics and math.

Bonus Point Policy:

What is the name of the scientist pictured below ? It's worth a point on any test if you can tell me. It is also possible to earn bonus points by asking particularly good questions or suggesting corrections to errors in notes and materials on the course website. This does not include spelling or grammatical errors, those are provided for your amusement. Once I notify the class of the error you may no longer ask for that point. To start with check our page of known errors (errata page)

Bonus Projects:
These are not required. It is entirely possible to earn an A without completing these. You may achieve a maximum of 10pts bonus (put towards a test I don't drop).

• Differential Forms Project (10pts)
• If you are interested in some topic involving calculus III then suggest and alternate project

• 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.

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