Calculus IV

Math 2454: Advanced Vector Calculus (Calculus IV)

(formerly Calculus IV MTH 2254)

Spring 2015

Synonym: 26086, Section: 001, Northridge 2244
Monday / Wednesday 3:40 pm - 5:25 pm

Mathematiea computer labs can be found here.

Course Content:

Course Description:  ADVANCED VECTOR CALCULUS/CALCULUS IV (4-4-0). This course develops the calculus of real- and vector-valued functions of one and several variables. Topics include matrix algebra and linear maps; vector-valued functions and their analysis; the geometry of Euclidean n-space; functions of several variables and their differentiation; gradients and directional derivatives; partial derivative; arc length; vector fields, divergence, and curl; Taylor's theorem for several variables; extreme of real-valued functions in n-space; LaGrange multipliers; multiple integrals and the chain rule; improper integrals; line integrals; area of surface; surface integrals; Green's Theorem; Gauss' Theorem; Stokes' Theorem; conservative fields

Instructional Methodology: This course is taught in the classroom primarily as a lecture/discussion course. There is also a computer lab component.

Course Rationale: This course serves as an extension of the traditional calculus sequence and contains additional topics relevant to students majoring in engineering, physics, and applied mathematics, including: 

   differentiability generalized to multivariable functions     an introduction to vector fields and flow lines     a description of four differential operators: gradient, divergence, curl, and Laplacian     the change-or-variables theorem for double and triple integrals     path integrals and line integrals;  parametrization of surfaces;   surface integrals     Green's theorem, Stokes' theorem, the divergence theorem     conservative vector fields; Maxwell's equations 


Please make sure you have the necessary prerequisites for this course. That means you need a C or better in MATH 2415 (Calculus III) or its equivalent.  If I feel you are not prepared for this course, I may choose to withdraw you. If you have any questions about your preparation for the course, please come and talk to me about it.

Course Materials:

Textbook: Vector Calculus, 6th edition (ISBN 1429215089), Marsden & Tromba, W.H. Freeman and Co., 2012.

Technology Required:  You must have access to technology that enables you to (1) Graph a function. (2) Find the zeroes of a function. (3) Do numerical integration.

Optional:  Mathematica version 9 (student edition) by Wolfram Research.  (This is available for your use at several locations on campus as well as most ACC campuses.)


There will be 4 exams during the semester.  In addition, you will be assigned several computer labs which will count for a substantial grade.  Grades will be distributed as:


Tests 1, 2, 3, 4




Computer labs and assignments







If you take any test late for any reason, there will be a penalty of 10 points off your test grade. However, no late tests will be allowed after I hand the graded tests back in class.  If you miss a test, you must try to take it during this lateperiod.  If you do not take the test during that period, you will receive a 0 for that grade.  Under

Grades will be assigned as follows:


90% or better


60% - 69%


80% - 89%


below 60%


70% - 79%





Withdrawn by student or instructor prior to last withdrawal date on school calendar


Incomplete grades (I) will be given only in very rare circumstances. Generally, to receive a grade of "I", a student must have taken all tests, be passing, and after the last date to withdraw, have a personal tragedy occur which prevents course completion. An incomplete grade cannot be carried beyond the established date in the following semester. The completion date is determined by the instructor but may not be later than the final deadline for withdrawal in the subsequent semester.


You should bring your homework to class every day.  It will be collected regularly.  There may also be in-class assignments or quizzes collected for a grade (as part of your homework grade). There will be a penalty on late homework. Homework that is more than a week late might not receive any credit.  If you do not follow the instructions that will be announced in class about how to organize and submit your homework, you may not receive full (or any) credit for it.


Attendance is required in this course.  It is extremely important for you to attend class regularly.  I MAY drop you from the course for excessive absences, although I make no commitment to do so.


It is the student's responsibility to initiate all withdrawals in this course.  The instructor may withdraw students for excessive absences (4) but makes no commitment to do this for the student. After the last day to withdraw, neither the student nor the instructor may initiate a withdrawal. It is the responsibility of each student to ensure that his or her name is removed from the roll should he or she decide to withdraw from the class.  The instructor does, however, reserve the right to drop a student should he or she feel it is necessary. The student is also strongly encouraged to retain a copy of the withdrawal form for their records.

Students who enroll for the third or subsequent time in a course taken since Fall, 2002, may be charged a higher tuition rate, for that course. State law permits students to withdraw from no more than six courses during their entire undergraduate career at Texas public colleges or universities.  With certain exceptions, all course withdrawals automatically count towards this limit.  Details regarding this policy can be found in the ACC college catalog.

The withdrawal deadline for Spring 2015 is April 27, 2015.

Classroom behavior:

Classroom behavior should support and enhance learning. Behavior that disrupts the learning process will be dealt with appropriately, which may include having the student leave class for the rest of that day. In serious cases, disruptive behavior may lead to a student being withdrawn from the class. ACC's policy on student discipline can be found in the Student Handbook on the web at:

Class participation:


All students are expected to actively participate in this class. This can include asking relevant questions in class, participating in class discussions and other in-class activities, helping other students, coming to office hours with questions, and doing other things that contribute to the class.

Keeping up:

Please, try to keep up with the homework and with the lecture in class. There just isn't much time to catch up. This means you have to be sure to allow yourself plenty of time to do the homework and to study.

Ask questions:

Please, please, please, if you don't understand something, or you aren't clear about something, or if you think I (or the book) have made a mistake (it has been known to happen), or if you have any other questions, please ask. Don't let confusion accumulate. If you don't want to ask in class, come to our office hours (or call) and ask. It is much easier to ask a question now than to miss it on the test.  I expect all students to participate in class discussions and other activities. Trust me, you will get much more out of the class if you become actively involved in it.

Always show your work:

It is much more important that you understand the processes involved in solving problems than that you just give me the right answer. If I see from your work that you understand what you are doing, I will usually give partial credit for a problem, even if you made a mistake somewhere along the line. If you don't show your work (unless I believe you could reasonably do it in your head), I may not give you full credit, even if the answer is right. If you can really do something in your head, that's great, but when in doubt, write it down.  It is also very important that you write what you mean. I will correct your notation the first few times, but I will start counting it wrong if you continue to write things incorrectly. In addition, please write clearly and legibly. If I can't read it, I won't grade it.

Time required and outside help:

To do homework and study requires two or three times as much time outside of class as the time you spend in class in order to succeed in this course. If you need more out-of-class help than you can obtain in your instructor's office hours, free tutoring is available in any of ACC's Learning Labs.

ACC main campuses have Learning Labs which offer free first-come, first-serve tutoring in mathematics courses. The locations, contact information and hours of availability of the Learning Labs are posted at:





A student who has taken this course should be able to:


1.     Compute limits, derivatives, gradients, directional derivatives, divergence, curl, and double and triple integrals of appropriate scalar and vector-valued functions.

2.     Use the Jacobian and the change of variables theorem to convert regions and integrals into alternate coordinate systems for solution.

3.     Demonstrate an understanding of and be able to derive expressions for parametric curves and parametric surfaces, as well as determining appropriate orientations for each.

4.     Compute line integrals and surface integrals of scalar and vector-valued functions over parametric curves and parametric surfaces.

5.     Demonstrate an understanding of and be able to use the theorems of Green, Stokes, and Gauss to simplify and solve appropriate integrals.

6.     Use the methods developed in the course to solve problems in the sciences.


Additional information about ACC's mathematics curriculum and faculty is available on the Internet at


ACC College Policies

Statement on Scholastic Dishonesty

A student attending ACC assumes responsibility for conduct compatible with the mission of the college as an educational institution.  Students have the responsibility to submit coursework that is the result of their own thought, research, or self-expression.  Students must follow all instructions given by faculty or designated college representatives when taking examinations, placement assessments, tests, quizzes, and evaluations.  Actions constituting scholastic dishonesty include, but are not limited to, plagiarism, cheating, fabrication, collusion, and falsifying documents.    Penalties for scholastic dishonesty will depend upon the nature of the violation and may range from lowering a grade on one assignment to an Fin the course and/or expulsion from the college.  See the Student Standards of Conduct and Disciplinary Process and other policies at


Student Rights and Responsibilities

Students at the college have the rights accorded by the U.S. Constitution to freedom of speech, peaceful assembly, petition, and association. These rights carry with them the responsibility to accord the same rights to others in the college community and not to interfere with or disrupt the educational process. Opportunity for students to examine and question pertinent data and assumptions of a given discipline, guided by the evidence of scholarly research, is appropriate in a learning environment. This concept is accompanied by an equally demanding concept of responsibility on the part of the student. As willing partners in learning, students must comply with college rules and procedures.


Statement on Students with Disabilities

Each ACC campus offers support services for students with documented disabilities.  Students with disabilities who need classroom, academic or other accommodations must request them through the Office for Students with Disabilities (OSD).   Students are encouraged to request accommodations when they register for courses or at least three weeks before the start of the semester, otherwise the provision of accommodations may be delayed.  


Students who have received approval for accommodations from OSD for this course must provide the instructor with the Notice of Approved Accommodationsfrom OSD before accommodations will be provided.   Arrangements for academic accommodations can only be made after the instructor receives the Notice of Approved Accommodationsfrom the student.  


Students with approved accommodations are encouraged to submit the Notice of Approved Accommodationsto the instructor at the beginning of the semester because a reasonable amount of time may be needed to prepare and arrange for the accommodations.   

Additional information about the Office for Students with Disabilities is available at


Safety Statement

Austin Community College is committed to providing a safe and healthy environment for study and work. You are expected to learn and comply with ACC environmental, health and safety procedures and agree to follow ACC safety policies. Additional information on these can be found at Because some health and safety circumstances are beyond our control, we ask that you become familiar with the Emergency Procedures poster and Campus Safety Plan map in each classroom. Additional information about emergency procedures and how to sign up for ACC Emergency Alerts to be notified in the event of a serious emergency can be found at


Please note, you are expected to conduct yourself professionally with respect and courtesy to all. Anyone who thoughtlessly or intentionally jeopardizes the health or safety of another individual will be dismissed from the days activity, may be withdrawn from the class, and/or barred from attending future activities.


You are expected to conduct yourself professionally with respect and courtesy to all. Anyone who thoughtlessly or intentionally jeopardizes the health or safety of another individual will be immediately dismissed from the days activity, may be withdrawn from the class, and/or barred from attending future activities.


Use of ACC email

All College e-mail communication to students will be sent solely to the students ACCmail account, with the expectation that such communications will be read in a timely fashion. ACC will send important information and will notify you of any college related emergencies using this account.  Students should only expect to receive email communication from their instructor using this account.  Likewise, students should use their ACCmail account when communicating with instructors and staff.  Instructions for activating an ACCmail account can be found at


Testing Center Policy

Under certain circumstances, an instructor may have students take an examination in a testing center.  Students using the Academic Testing Center must govern themselves according to the Student Guide for Use of ACC Testing Centers and should read the entire guide before going to take the exam.  To request an exam, one must have:


    ACC Photo ID

    Course Abbreviation (e.g., ENGL)

    Course Number (e.g.,1301)

    Course Synonym (e.g., 10123)

    Course Section (e.g., 005)

    Instructor's Name


Do NOT bring cell phones to the Testing Center.  Having your cell phone in the testing room, regardless of whether it is on or off, will revoke your testing privileges for the remainder of the semester.  ACC Testing Center policies can be found at


Student And Instructional Services

ACC strives to provide exemplary support to its students and offers a broad variety of opportunities and services.  Information on these services and support systems is available at:


Links to many student services and other information can be found at:


For help setting up your ACCeID, ACC Gmail, or ACC Blackboard, see a Learning Lab Technician at any ACC Learning Lab.

Course Outline and Approximate Calendar:
Please note:  schedule changes will almost certainly occur during the semester.
Any changes will be announced in class.







MLK Holiday No class



1.1 2.1

Review of lines and planes in 3-D, dot product, cross product, determinants, n-dimensional vectors, graphing functions of 2 variables, contour plots, slicing



2.1 2.2

More on graphing, limits and continuity of functions of several variables




Differentiation a new approach for higher dimensions, partial derivatives, the derivative matrix, the tangent approximation



2.4, 2.5

Paths and curves, properties of the derivative, the chain rule



2.6, 3.1

The directional derivative, gradients, iterated partial derivatives, equality of mixed partials



3.2, 3.3

Taylors Theorem, extrema of real-valued functions, first derivative test, second derivative test, absolute extrema




Constrained extrema and Lagrange Multipliers

Test 1 – Dates to be announced in class



4.1, 4.2

Curves and acceleration, Newtons Second Law, arc-length and velocity



4.3, 4.4

Vector fields, divergence and curl



5.1 - 5.4

Double integrals over rectangular regions, Double integrals over general regions, describing arbitrary regions, reversing the order of integration




Triple integrals: describing arbitrary regions, reversing the order of integration




Changing coordinate systems describing maps from R2 -> R2, linear vs. non-linear maps, one-to-one and onto maps




The Jacobian determinant, the Change of Variables Theorem, polar/cylindrical/spherical coordinate transforms in integrals



6.2, 6.3

More on change of variables, Applications: average value, center of mass, moments of inertia




Heavy lifting with Mathematicamore in-depth look at how to use Mathematica beyond the basics we have seen so far

Test 2 – Dates to be announced in class

3/16/153/22/15SPRING BREAK (No class)



7.1, 7.2

The path integral integrals of scalar-valued functions over curves. The line integral integrals of vector-valued functions over curves




Parameterized surfaces




More parameterized surfaces, area of a surface




Integrals of scalar-valued functions over a surface




Integrals of vector-valued functions over a surface




Compare and contrast: summing up how to extend integration in different ways, the role of the differential of arc-length and the differential of surface area




Greens Theorem a surprising relationship between the outside and the inside, orientation




StokesTheorem taking the idea and extending it

Test 3 – Dates to be announced in class




Orientation and parametric surfaces/curves




Conservative fields How to save yourself a lot of work




GaussTheorem moving up a dimension, orienation




Differential forms and manifolds - an introduction




The wedge product and the derivative of forms




The Fundamental Theorem of Calculusrevisited, bringing it all together



Test 4, Part 1 – Given in class



Test 4, Part 2 – Given in class



This webpage was created by Marcus McGuff.
It was last updated on January 22, 2016 .