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Syllabus for Math 2057 "Calculus III"

Text : Calculus, Early Transcendentals 7th Edition by James Stewart

Math 2057 is a three-hour multi-dimensional calculus course designed primarily for engineering majors and certain other technical majors. The text is Calculus, Early Transcendentals by Stewart, 7th edition. The student is assumed to be capable and versed in the standard Calculus I and II topics, taking limits, continuity, taking derivatives of complicated functions, using derivatives, calculating the definite integral, integration techniques and applications of integration. Since this is a three hour class, the student should expect 3 hour exams and a final exam. The testing and evaluation policies for each class are entirely at the discretion of the instructor.  The calculator policy is also at the discretion of the instructor. Some sections may require the use of calculators and some sections may prohibit them.  Access to the WebAssign e-textbook can be provided for students whose instructor is not using WebAssign.  Contact George Cochran with any questions,

The student should finish this class with:

  1. an understanding of limits and continuity of functions of several variables;
  2. the ability to compute partial derivatives and directional derivatives;
  3. an understanding of linear approximation for multi-variable functions;
  4. an introduction to optimization of multi-variable functions using the second derivative test and Lagrange Multipliers;
  5. the ability to evaluate iterated integrals;
  6. the ability to use multiple integrals to calculate areas, volumes, masses and centers of mass for standard plane regions and solids;
  7. an introduction to line integrals, path-independence, potential functions and surface integrals; and
  8. an understanding of Green's Theorem, the Divergence Theorem and Stoke's Theorem.


One general comment: Many experienced instructors try to motivate 2057 topics by referring to the corresponding Math 1550 topic. You can present the multi-dimensional topic as an obvious extension of the associated single-variable topic. The students seem to respond well to this approach.

Section 14.1
Functions of Several Variables: The students should understand domains and ranges without too much trouble. The graphs of functions with two independent variables will be more difficult for them. The typical student is not good at drawing or visualizing three-dimensional graphs. The instructor should make every effort to give lots of graphical examples to help the students master graphing of simple figures. The ideas of level curves and contour graphs are also very important.
Section 14.2
Limits and Continuity: The decision to use delta-epsilon is left to the instructor. The book stresses the sequential idea of limits. Students have trouble with the notion that a point can be approached along an infinite number of paths. They must be made aware that just because the limiting value is the same along a finite number of paths, does not mean the limit exists.
Section 14.3
Partial Derivatives: Most students will be able to calculate partial derivatives with a reasonable degree of skill. They will benefit from a geometric interpretation of these partial derivatives. The instructor should point out the various notations for partial derivatives and should discuss Clairaut's Theorem.
Section 14.4
Tangent Planes and Linear Approximations: Students should be made aware of tangent planes and how they are multi-dimensional analogues of tangent lines. dz can be interpreted geometrically to give the students a better feel for linear approximation. A contour map analysis of local linearity of a differentiable function can be very useful.
Section 14.5
The Chain Rule: The book employs a standard presentation. Students should be able to use the chain rule and write the formulae for the various cases. The material on implicit differentiation may be viewed as optional.
Section 14.6
Directional Derivatives and the Gradient: The students should be made aware of the geometric interpretation of the directional derivative. They should be shown how partial derivatives are specific types of directional derivatives. The gradient should be discussed and the usual formula for calculating directional derivatives should be presented. The interpretation of the gradient as the direction of maximum rate of growth and the perpendicularity of the gradient and level curves should be mentioned.
Section 14.7
Maximum and Minimum Values: Present the definitions concerning relative and absolute extrema and the notion of critical values. These ideas should be familiar to the students from 1550. The second-derivative test is vital in this section. The Extreme Value Theorem and finding extrema on compact regions will give students trouble. Optimizations on the boundary is difficult for most of the students. Try to keep the boundaries simple to minimize their headaches.
Section 14.8
Lagrange Multipliers: This is one of the most demanding sections for the students. Again, a geometric interpretation of why the method works may be useful. The main problem for the class will be solving the associated non-linear systems. They will beg the instructor for an algorithm for solving the non-linear systems and they are very lost when one is not produced. The instructor should exercise some care when using Lagrange Multipliers on a test and try to pick a reasonable problem.
Section 15.1
Double Integrals over Rectangles: This section introduces the student to double integrals. The notion of partitioning the domain and selecting a point inside each sub-domain should be familiar to the students. Treating this as a generalization of the 1550 Riemann sum works very well.
Section 15.2
Iterated Integrals: The students should be able to apply Fubini's Theorem and evaluate the resulting iterated integrals. I advise the instructor to not be surprised if the students do not remember how to compute many antiderivatives. Most students will have forgotten the 1552 chapter on techniques of integration.
Section 15.3
Double Integrals over General Regions: Aside from their difficulty with finding anti-derivatives, the students are also plagued with the inability to sketch or visualize surfaces. The instructor needs to be aware that the students are not as familiar with surfaces and some allowances for this should be made. 
Section 15.4
Double Integrals in Polar Coordinates: The students should be able to convert back and forth from rectangular and polar form for double integrals. The instructor should stress that polar form is typically useful when the domain involves circular parts or if the integrand is of a particular form. The famous Gaussian Integral can be shown to the students at this time. That presents a chance for the students to see a classic non-elementary integral evaluated through a technique they can now understand.
Section 15.5
Applications of Double Integrals:  These are the standard applications of mass, center of mass and moment of inertia.  The additional topics of probability and expected value of bivariate continuous random variables may be considered an optional topic. 
Section 15.6
Surface Area: The surface area of a function z = f(x,y) is developed and examined in this brief section. Some instructors skip this section and treat the surface area problem as a special case of surface integrals in section 16.7.
Section 15.7
Triple Integrals:  Most students have little trouble making the leap from double to triple integration. The section contains Fubini's Theorem and some of the applications of triple integrals. The instructor is advised (again) to use some discretion with the choice of the solids. The students will have some difficulty with the more exotic solids because their visualization skills are not very well developed.
Section 15.8
Triple Integrals in Cylindrical Coordinates: The students will not have seen cylindrical coordinate systems before this section. The instructor will have to present this coordinate system, as well as the triple integral involving cylindrical coordinates. The students should be able to determine when a cylindrical coordinate approach would be useful.
Section 15.9
Triple Integrals in Spherical Coordinates: The spherical coordinate system is more difficult for the students.  They cannot benefit from a similarity to polar coordinates to help them. They will need to gain familiarity with the spherical coordinate system, as well as learn how to use triple integration in spherical coordinates. Good graphic examples are very important in this section.  
Section 15.10
Change of Variables in Multiple Integrals: The students should be aware of the Change of Variables Theorem and how to use it for simple transformations. This section is relatively difficult for most students and they have a great deal of trouble trying to decide on a transformation on their own. If they are given the transformation, they are much more likely to be able to finish the problem.
Section 16.1
Vector Fields: Students should be able to draw two-dimensional vector fields and match vector fields and their plots. The students should be able to calculate a gradient field and should learn the terms conservative field and potential function.
Section 16.2
Line Integrals: The student should be able to compute line integrals along parametrically defined curves. The application of line integrals to mass and center of mass should be explored. Line integrals of vector fields should be covered and can be motivated through the concept of work.
Section 16.3
The Fundamental Theorem for Line Integrals: The student must be taught about independence of path. The section includes the discussion of how to compute a potential function from a gradient field. This technique, while troublesome for some students must be mastered before they can use the Fundamental Theorem. The section also includes a very nice discussion about the Law of Conservation of Energy.
Section 16.4
Green's Theorem: The proof for simple type I and type II regions is understandable by most students but the generalization to more complex regions should be avoided. The orientation issue gives some students problems while integrating. The section introduces the area formaulas which derive from Green's Theorem.  The planimeter is a nice mechanical exmple of Green's Theorem in use.  
Section 16.5
Curl and Divergence:  This standard section introduces the curl and divergence of a vector field.  The students usually have little trouble with this material. The section is finishes with the presentation of the vector forms of Green's Theorem.  
Section 16.6
Parametric Surfaces and Their Areas: The difficulty in dealing with parametric surfaces without the use of a CAS is well known. If you are using Mathematica, then this material is very interesting and the students respond much more positively. The development of the surface area integral for parametric surfaces is presented.  The students should be shown how this relates to the formula presented in section 15.6.
Section 16.7
Surface Integrals: The instructor covering surface integrals may wish to restrict the course to only covering surfaces defined as graphs of functions of x and y. While the book does present surface integrals over more general surfaces, time constraints and level of difficulty may make this presentation less desirable. The idea of an oriented surface is not too difficult for most students. Topics also include the vector surface integral and fluid flux. The physical interpretation of fluid flux is useful as a motivating factor in this section. Electric fields are also presented. Many students will be familiar with these topics from their physics courses.  
Section 16.8
Stokes' Theorem: Stokes' Theorem is introduced and verified for the special case of a simple plane region. The student's should be made aware of the relationship bewteen Stoke's and Green's Theorems. The notion of circulation is discussed and the relaltionship between the curl and circulation is explored. 
Section 16.9
The Divergence Theorem:  The Divergence Theorem is given and proved for simple solid regions. The fluid flow application of the Divergence Theorem is a very good tool to motivate this topic. The discussion at the end of this section is quite good and should be stressed to the student.  The terms source and sink are introduced.