James Cook's Multivariable Calculus Archive
James Cook's Multivariable Calculus Archive

I: Miscelaneous:

• III. Test reviews and solutions from other semesters:
Our course will not be identical, but these give you some idea of the basic sort of questions I have asked on tests before. I'd like to think this semester will be more interesting. Here are a the tests from Spring 2009's 3hr calculus. Here are some tests from a 4-hr calculus III I taught in 2006. We do not cover certain topics that appear on these tests.

IV. Worked Problems:
You have a variety of resources below. I would recommend browsing through them from time to time. You should be able to find an example of most any standard question. To start, the following are the "Homework Projects" from Fall 2008 math 231.
The solutions posted below were originally given as homework problems for the Fall 2008.

What follows 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).

V. My 2006 Version of 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.

• [236-239] Cartesian coordinates and vectors: (13.1) We define three dimensional Cartesian coordinates and discuss distance between points. Also the point-vector correspondance is seen.

• [240-250] Vectors: (13.2-13.4) 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.

• [251-256] Lines and planes: (13.5) Parametric equation of a line examined. Also equation of plane motivated from vector viewpoint.

• [257-262] Analytic geometry and graphing: (13.6) Graphs z=f(x,y), level surfaces, and contour plots introduced.

• [263-268] Calculus for vector-valued functions of "t": (14.1-14.2) Concept of a vector-valued function of a real variable is introduced. Integrals and derivatives are done one component at a time.

• [269-279] Curves in 3-d: (14.3) 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.

• [280-283] Motion in 3-d: (14.4) 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.

• [283-289] Kepler's Laws: (14.4) 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.

• [290-291] Limits and continuity: (15.1-15.2) Limits generalized to two or more variables.

• [292-295] Partial Differentiation: (15.3) Definition of partial differentiation plus geometric interpretation discussed. Examples illustrating basic idea of partial differention are also given here.

• [296-299] Chain Rules for Partial Differentiation: (11.5) Chain rule for several independent and/or several intermediate variables detailed.

• [300-305] Constrained Partial Differentiation: (11.5 and more not in Stewart) 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".

• [305a-305f] 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.

• [306-310] Theory of derivatives:(not quite in Stewart) 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.

• [311-313] Tangent Planes and Differentials (15.4,15.6) Linearization introduced as a tool to approximate functions of several variables. The total differential introduced to aid in the estimation of error.

• [314-316] Directional Derivative (15.6) 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.

• [317-319] Tangent Planes and Parametrized Surfaces (15.6,17.6) 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.

• [320-324] Multivariate Extrema: (15.7) 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.

• [325-329] Lagrange Multipliers: (15.8) 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.

• [330-333] Multivariate Cartesian Integrals:(16.1,16.2,16.6) 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.

• [334-338] Double Integrals over nontrivial regions(16.3) 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.

• [339-342] Triple Integrals over nontrivial regions:(16.6) 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.)

• [343-359] Multivariate Integrals in General Coordinates: (16.4,16.7,16.8,16.9) 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.

• [360-365] Vector Fields: (17.1, and more not in Stewart) 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)

• [366-368] Gradient: (17.5, and more not in Stewart) 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.

• [369-372] Curl: (17.5, and more not in Stewart) 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.

• [360-384] Divergence: (17.5, and more not in Stewart) 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.

• [375-384] Curl and Divergence in curved coordinates: (not in Stewart) 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.

• [385-394] 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.

• [395-399] FTC for Line Integrals and Path Independence: (17.3) 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)

• {400-401] Work Energy Theorem: (17.3) 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.

• [402-406] Parametric Surfaces and their areas: (17.6) 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".

• [407-411] Surface Integrals: (17.7) 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.

• [412-419] Green's, Stoke's Theorems: (17.4,17.8) 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.

• [420] 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.

• [421-423] Gauss' or Divergence Theorem: (13.9) 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.

• [424-425] Differential Forms: (not in Stewart) 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.

VII. Further Practice Homework List:

These homeworks are not collected in the current course. You have been assigned "Problem Sets" which better reflect my vision for multivariable calculus. Stewart is a good starting point and this is why I leave these here to supplement my notes/lecture since additional examples usually help interested students.

Notice I have indicated which portion of my lecture notes as well as which part of the textbook is most relevant to the assigment. Beware, sometimes the homework is not exactly matched up with the lecture notes link, sometimes you need to look at the next few pages. The pdf's of my lecture notes are chopped up section by section, usually you can find what you need somewhere in that chapter. If you are lost send me an email, I'll try to point you in the right direction.

 Section # My Notes . Assignment Description / Hints Sec. 13.1 236-239 . 11, 13, 15, 20, 23-31(odds) 3d-Cartesian Coordinates Sec. 13.2 240-250 . 7, 13, 17, 21, 24, 31, 35, 37, 39, 40, 42 vectors Sec. 13.3 240-250 . 3, 8, 12, 13, 16, 20, 24, 23, 36, 40, 45, 50, 54, 56, 60 dot product Sec. 13.4 240-250 . 2, 6, 10, 14, 18, 20, 23, 25, 33, 39, 43, 45, 49 cross product Sec. 13.5 251-256 . 3, 6, 9 (no symmetric equation required for 6 or 9), 14, 16, 17, 18, 25, 26, 28, 32, 40, 55 lines and planes Sec. 13.6 257-262 . 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 34, 36 functions of several variables Sec. 14.1 263-268 . 8, 14(use Mathematica to graph curve, can pencil in direction if you wish), 25, 36, 38, 41, 42, 43 (use definition 1) vector-valued functions Sec. 14.2 263-268 . 3, 9, 11, 13, 14, 15, 16, 19, 27(use Mathematica), 28(use Mathematica), 33, 34, 43, 46, 49 calculus of vector-valued functions Sec. 14.3 269-279 . 1, 13, 43(you may use Mathematica to help calculate this if you wish) arclength and moving TNB-frame Sec. 14.4 280-283 . 9, 15, 19, 21, 35, motion in space Sec. 15.1 . . 21-26 graph with Mathematica, 36, 39, 43, 62, 64 graphing, functions of several variables Sec. 15.2 290-291 . 5, 7, 9 limits and continuity Sec. 15.3 292-295 . 16, 22, 32, 36, 37, 40, 43, 50, 51, 61, 70a-b, 76 basic partial derivatives Sec. 15.5 296-299 . 1, 2, 7, 14, 22, 25, 39, 40, 45, 53 chain rule for several variables Sec. 15.4 311-313, 317-319 . 1, 3, 11, 17, 29, 30, 42(we define the tangent plane at P to be the union of all tangent vectors at P) tangent plane and linearization Sec. 15.6 311-319 . 7, 12, 15, 20, 25, 28, 41a, 63 directional derivative Sec. 15.7 320-324 . 8, 11, 18, 29, 41, 55*** extrema in functions of several variables Sec. 16.2 330-342 . 6, 12, 13, 16, 22, 31 basic double integrals Sec. 16.3 330-342 . 8, 13, 15, 16, 22, 46 double integrals over general regions Sec. 16.6 339-343 . 3, 8, 12, 13, 14, 39 triple integrals Sec. 16.9 343-359 . 3, 4, 6, 7, 10, 13 the Jacobian Sec. 16.4 343-359 . 2, 9, 10, 16, 23, 25, 29, 36*** double integrals in polar coordinates Sec. 16.7 343-359 . 1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 20, 23a, 28 triple integrals in cylindrical coordinates Sec. 16.8 343-359 . 1, 4, 5, 6, 7, 8, 9, 10, 11, 13, 20, 21, 22, 27, 30, 40 triple integrals in spherical coordinates Sec. 17.1 360-365 . 2, 4, 6, 8, 10 (use Mathematica to plot vector fields in 2,4,6,8,10), 21, 23, 35** vector fields Sec. 17.5 366-368, 369-372, 373-374 . 1, 2, 5, 10, 12, 14, 16, 18, 21, 23, 26, 27, 29, 32(use 31 and 30's results if helpful), 37, 38, 33**, 34**, 39*** curl and divergence Sec. 17.2 385-394 . 1, 3, 4, 18, 20, 21, 32(use Mathematica), 45, 48 line integrals Sec. 17.3 395-401, 400-401 . 12, 15, 20, 21, 26, 30, 32, 33**, 34** FTC for line integrals, conservative forces Sec. 17.6 402-406 . 2, 5, 8, 19, 23, 24, 25, 26, 37, 43, 46, 55, 56, 57 parametrized surfaces and surface area Sec. 17.7 402-406 . 7, 21, 22, 24, 25, 29, 44, 47 surface integrals Sec. 17.4 412-419 . 1, 5, 8, 29** Greene's Theorem Sec. 17.8 412-419 . 5, 6, 9, 10, 11, 16, 19*,20*** Stoke's Theorem Sec. 17.9 421-423 . 6, 9, 12, 13, 14, 17, 25*, 26*, 27*, 28*, 29*, 30** Divergence Theorem

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