NCC Multivariable Calculus Course Syllabus

NASSAU COMMUNITY COLLEGE
DEPARTMENT OF MATHEMATICS/COMPUTER SCIENCE/INFORMATION TECHNOLOGY
Course Syllabus for
MAT 225 Multivariable Calculus
Course Information
 Title Multivariable Calculus
 Credit Hours 4 Credits
 Number MAT 225
 Section EA1 CRN: 40951
 Semester/Term SPRING 2021 (1/19/2021 – 5/13/2021)
 Meeting time TR, 6:20PM – 8:20PM
 Location REMOTE
Instructor/Contact Information
 Name PROFESSOR A. Jorge Garcia
 Math Office telephone and fax numbers
516-572-7383/ 516-572-9715
 Email address alvar.garcia-fernandez@ncc.edu
 Blackboard link You can access the BlackBoard for this course through
the NCC Portal, following the link to NCC Online.
** Activate your NCC Email account!
** Math Center B130 – Under renovations and has moved to B109!
** Math Center B109 – Please use this resource for extra help. Bring your ID!
** Online Math Learning Center – https://ncc-zoom.zoom.us/j/91429552528
Course Description
 MAT 225 Multivariable Calculus
 Prerequisites: Students must have passes MAT 123 Calculus 2 with at least a C.
 Description: Curves and surfaces in three dimensional space, partial derivatives,
gradient, constrained and unconstrained optimization, vector fields, parametric
curves and surfaces. Integration topics include multiple integrals, volume, area,
line and surface integrals, flux, divergence
 *Calculator Requirement: TI84, TI86, TI89 or TI nSpire CX CAS (recommended)
 *Computing Requirement: Laptop or Tablet (http://sagecell.sagemath.org)
*Bring Your Own Device For Lectures & Exams
1

Learning Outcomes and Objectives
• OBJECTIVES: General
This course is designed to give the student skill for solving multivariable calculus problems . The
course is generally oriented toward problem solving techniques in engineering and the natural
sciences.
• OBJECTIVES: Specific
To enable the student to:
1. analyze graphs in a 3-dimensional Euclidean space.
2. operate algebraically with vectors.
3. analyze properties of functions with the vectors, directional derivatives and second order
derivatives.
4. establish local and global extrema of functions.
5. evaluate multiple integrals of multivariable functions.
6. solve line and flux integrals defined on vector fields.
7. solve line and flux integrals with Stoke’s Theorem and the Divergence Theorem in cases of special
geometry.
 SUNY General Education Goals & Outcomes ---- MAT 225 Multivariable Calculus
1. Functions of Several Variables
Students must algebraically analyze functions of 2 and 3 independent variables.
Outcome
1.1 Graphs
Students should be able to examine functions whose graphs are functions with 2-dimensional or 3-
dimensional domains.
2. Vectors
Students must to apply vector algebra to analyze multivariable functions.
Outcome
2.1 Algebraic Properties
Students should be able to algebraically operate with vectors.
2.2 Dot and Cross Products
Students should be able to construct and analyze curves and surfaces with vector algebra.
3. Optimization
Students must determine optimal properties of functions.
Outcome
3.1 Directional Derivatives/Gradient
Students should be able to use the gradient and directional derivative to establish rate of change
properties of functions.
3.2 Optimization
Students should be able to establish local and global extrema of functions with unconstrained
and constrained domains.
4. Multiple Integration of Multivariable Functions
Students must be able to integrate multivariable functions on their domains.
Outcome
4.1 Integration
Students should be able to evaluate multiple integrals by means of iterated integration.
2

5. Integration on Vector Fields
Students must be able to integrate vector functions.
Outcome
5.1 Vector Fields
Students should be able to determine properties of vector fields.
5.2 Conservative Fields
Students should be able to apply the Fundamental Theorem of the Calculus to evaluate line
integrals on conservative fields (establish potentials).
5.3 NonConservative Fields
Students should be able to use Green’s Theorem to evaluate closed curve line integrals on
nonconservative fields.
5.4 Flux Integrals on Closed Surfaces
Students should be able to evaluate a closed surface flux integral with the Divergence
Theorem.
5.5 Flux Integrals on Closed Curves
Students should be able to evaluate a flux integral defined on a closed curve in a plane with
Stoke’s Theorem.
• SUNY General Education Goals & Outcomes -------- Mathematics, A.S.
1. Draw Inferences from Mathematical Models
Students will demonstrate the ability to and draw inferences from mathematical models such as
formulas, graphs, tables, and schematics.
Outcome
1.1 Mathematical Interpretation
Students will interpret variables, parameters, and other specific information within a mathematical model.
1.2 Draw Inferences
Students will draw inferences about the situation being modeled mathematically.
1.3 Verbal Interpretation
Students will verbally interpret the results of their analysis of the mathematical model.
2. Represent Mathematical Information
Students will demonstrate the ability to represent mathematical information symbolically, visually,
numerically and verbally.
Outcome
2.1 Mathematical Information
Students will employ the appropriate representation to display the mathematical information.
2.2 Mathematical Terminology
Students will clearly define variables; draw, scale and label graphs; use correct mathematical terminology and/or
language.
3. Employ Quantitative Methods
Students will demonstrate the ability to employ quantitative methods such as arithmetic, geometry, or
statistics to solve problems.
Outcome
3.1 Identify Quantitative Methods
Students will be able to identify a specific numeric, algebraic, or statistical method(s) needed to solve a problem.
3.2 Applying Quantitative Methods
Students will apply the method identified, and correctly solve the problem.
4. Check Mathematical Results for Reasonableness
Students will demonstrate the ability to estimate and check mathematical results for reasonableness.
Outcome
4.1 Estimation
Students will estimate and justify a mathematical result to a problem.
4.2 Reasonableness
Students will articulate a justification for the estimate using a clearly defined logical plan.
3

5. Recognize Limits
Students will demonstrate the ability to recognize the limits of mathematical and statistical methods.
Outcome
5.1 Real Life Comparison
Students will describe how the results of the mathematical model may differ from the real-life situation it is
modeling.
5.2 Mathematical Assumptions
Students will articulate the assumptions made in developing a mathematical/statistical model.
Instructional Methods
This course is taught using a variety of instructional methods including lecture, class discussion and
examinations.
Please use the first four rows of the lecture hall. (Row one is on the floor at the desks.). Material
will be projected and written on the board.
For quizzes/exams, we will leave one seat between each student. This will require some people to
“temporarily” move their seat. You may return to your normal lecture seat after the test/quiz.
************************************************************************************
Textbook and Materials
 Calculus Multivariable, 3rd Edition, Briggs, W., Cochran, L., Gillett, B., Schulz, E.,
New York, 2019.
 References:
1. Elements of Calculus and Analytic Geometry by Thomas, G. B., Finney, R. L., Menlo Park, CA,
Addison-Wesley, 1981.
2. Multivariable Calculus by Barr, T. H., Edwards, C. H., Penney, D. E., Needham Heights, MA,
Pearson Custom, 2000.
3. Calculus: Multivariable by Smith, R. T., Minton, R. B., Boston, McGraw-Hill, 2002.
4. Multivariable Calculus, 6th Ed., McCallum et al., Wiley, New Jersey, 2013.
You need the text ASAP. You may obtain a hard copy or digital copy. Homework will be assigned after
the first lecture. The Math Center in B130 has copies you may use while in the Math Center. You need
your NCC ID to use the Math Center.
4

Student Responsibilities /Course Policies
 Participation: You are expected to arrive for class ON TIME and stay the whole
period. There are 28 sessions over 14 weeks (not including breaks). You should
actively participate by taking notes and asking questions. You should use the
Math Center in B109 (REMOTE) after class to go over any concepts you need
clarified before you go home for the day.
 Homework: Homework is assigned after each lecture. All homework should be
prepared for the next lecture. You are encouraged to check your work with the
answers at the end of the text or by using free help from online sites such as
Cheggs or WolframAlpha as well as the Math Center in B109.
 Group Work: Learning math is a TEAM SPORT; make friends early and set aside
some time to start the homework after class before you leave for the day or
review homework before class each morning.
 Exams/quizzes: There will be 5 exams (100 points each). All material for
each 2 hour exam will come from the homework assignments and the class
examples. Exam questions will include multi-step problems and applications.
You should review the list of learning outcomes and objectives on pages 2-4 of
this syllabus to make sure you can successfully complete each item when
tested on that material.
 Attendance/lateness policy: It is expected that you arrive ON TIME and stay
for the entire class. Attendance is part of your grade! A 6th
exam grade will be
based on your attendance. Everyone starts with 100 points on this exam grade.
Any absence loses 4 points. If you are late or leave early or miss class time for
any reason, every 30 minutes loses 1 point.
 Missed exams policy: There are NO MAKE-UP exams. The dates of the exams
are on the syllabus so that you may plan for them. Material for each exam will
be announced.
 Extra Credit: There is sufficient opportunity to demonstrate your knowledge
of the material. For this reason, extra credit is NOT offered in this class. You
must prepare for each test fully and do your best on each item.
 Partial Credit: Each question on a test is used to assess knowledge of a
particular concept. Simply writing something does not guarantee partial credit.
If it has been determined that understanding of the concept is NOT
demonstrated, no partial credit will be offered. Partial credit may be awarded if
all work is shown and there is a minor error in arithmetic or algebra.
5

 Academic Dishonesty & Plagiarism from NCC College Policy)
Academic dishonesty, which includes plagiarism and cheating, will result in
some form of disciplinary action that may lead to suspension or expulsion under
the rules of the Student Code of Conduct. Cheating can take many forms
including but not limited to copying from another
student on an examination, using improper forms of assistance, or receiving
unauthorized aid when preparing an independent item of work to be submitted
for a grade, be it in written, verbal or electronic form. Anyone who assists or
conspires to assist another in an act of plagiarism or any
other form of academic dishonesty may also be subject to disciplinary action.
Plagiarism is a particular type of academic dishonesty that involves taking the
words, phrases or ideas of another person and presenting them as one's own.
This can include using whole papers and paragraphs or even sentences or
phrases. Plagiarized work may also involve statistics, lab
assignments, art work, graphics, photographs, computer programs and other
materials. The sources of plagiarized materials include but are not limited to
books, magazines, encyclopedias or journals; electronic retrieval sources such
as materials on the Internet; other individuals; or paper writing services.
A student may be judged guilty of plagiarism if the student:
(a) Submits as one's own an assignment produced by another, in whole or in
part.
(b) Submits the exact words of another, paraphrases the words of another or
presents statistics, lab assignments, art work, graphics, photographs, computer
programs and other materials without attributing the work to the source,
suggesting that this work is the student's own.
Allegations of student plagiarism and academic dishonesty will be dealt with by
the appropriate academic department personnel. It is the policy of Nassau
Community College that, at the discretion of the faculty member, serious acts
will be reported in writing to the Office of the Dean of Students, where such
records will be kept for a period of five years beyond the student's last
semester of attendance at the College. These records will remain internal to the
College and will not be used in any evaluation made for an outside individual or
agency unless there is a disciplinary
action determined by a formal ruling under the Student Code of Conduct, in
which case only those records pertaining to the disciplinary action may apply. A
student whose alleged action is reported to the Office of the Dean of Students
will be notified by that office and will have the right
to submit a letter of denial or explanation. The Dean will use his/her discretion
in determining whether the alleged violation(s) could warrant disciplinary action
under the Student Code of Conduct. In that case the procedures governing the
Code of Conduct will be initiated.
6

 Copyright statement: The Higher Education Opportunity Act of 2008 (HEOA)
requires the College to address unauthorized distribution of copyrighted
materials, including unauthorized peer-to-peer file sharing.
Thus, the College strictly prohibits the users of its networks from engaging in
unauthorized distribution of copyrighted materials, including unauthorized peer-
to-peer file sharing. Anyone who engages in such illegal file sharing is violating
the United States Copyright law, and may be subject to criminal and civil
penalties. Under federal law, a person found to have infringed upon a
copyrighted work may be liable for actual damages and lost profits attributable
to the infringement, and statutory damages of up to $150,000. The copyright
owner also has the right to permanently enjoin an infringer from further
infringing activities, and the infringing copies and equipment used in the
infringement can be impounded and destroyed. If a copyright owner elected to
bring a civil lawsuit against the copyright infringer and ultimately prevailed in
the claim, the infringer may also become liable to the copyright owner for their
attorney's fees and court costs. Finally, criminal penalties may be assessed
against the infringer and could include jail time, depending upon the severity of
the violation. Students should be aware that unauthorized or illegal use of
College computers (such as engaging in illegal file sharing and distribution of
copyrighted materials), is an infraction of the Student Code of Conduct and may
subject them to disciplinary measures. To explore legal alternatives to
unauthorized downloading, please consult the following website:
http://www.educause.edu/legalcontent.
 Course Resources
Labs and learning centers: MATH CENTER REQUIREMENT
If needed, students are encouraged to avail themselves of further study and/or
educational assistance available in the Math Center located in B109
(REMOTE). These activities and use of the resources provided are designed to
help the student master necessary knowledge and skills.
 Assessments and Grading Methods
Grades will be assigned as follows:
ATTENDANCE: 20%
TEST AVERAGE: 60% (4 tests, 20% each, lowest grade dropped)
FINAL EXAM: 20%
 A (90-100) B+ (85 - 89) B(80 – 84) C+ (75 – 79) C (70 – 74) D+(65 – 69) D
(60 – 64) F (< 60)
 A W grade will only be awarded if the proper paperwork is filed in a timely
manner. File a W form anytime up to the CLASS BEFORE the final.
 If you stop attending class and fail to withdraw, you will receive a UW.
 Americans with Disabilities Statement & Non-Discrimination Statement
(NCC Required)
"If you have a physical, psychological, medical, or learning disability that may have an
impact on your ability to carry out the assigned coursework, I urge you to contact the
staff at the Center for Students with Disabilities (CSD), Building U, (516)572-7241, TTY
(516)572-7617. The counselors at CSD will review your concerns and determine to
what reasonable accommodations you are entitled as covered by the Americans with
Disabilities Act and section 504 of the Rehabilitation Act of 1973. All information and
documentation pertaining to personal disabilities will be kept confidential.”
7

Course Schedule and Important Dates: (Tentative Schedule (May be subject to change))
PART01: VECTORS+MATRICES
TUESDAY 2021.0119 (Screencast 101)
DAY01 REVIEWA– LIMITS I
DAY01 UNIT01 – VECTORS & DOT PRODUCTS
DAY01 UNIT02 – DETERMINANTS & CROSS PRODUCT
HWK01 (CHAP13) Read Sections, Complete End Of Section Exercises: mult5
THURSDAY 2021.0121 (Screencast 102)
DAY02 REVIEWB – LIMITS II
DAY02 UNIT03 – MATRICES & INVERSES
DAY02 UNIT04 – SQUARE SYSTEMS & EQUATIONS OF PLANES
DAY03 REVIEWC – PRODUCT RULE
DAY03 UNIT05 – PARAMETRICS: EQUATIONS FOR LINES & CURVES
DAY03 UNIT06 – MORE PARAMETRICS: POSITION, VELOCITY & ACCELERATION
DAY03 UNIT06 – MORE PARAMETRICS: KEPLER (OPTIONAL)
HWK03 (CHAP13/CHAP14) Complete End Of Chapter Review Exercises: mult5
THURSDAY 2021.0128
DAY04 HWK REVIEW
HWK04 (preTEST1A)
TUESDAY 2021.0202
DAY05 UNIT07 – REVIEW01
DAY05 preTEST1A (VECTORS+MATRICES)
HWK05 (Study preTEST1A)
THURSDAY 2021.0204
DAY06 TEST1B (VECTORS+MATRICES)
8

PART02: PARTIAL DERIVATIVES
DAY07 REVIEWD – QUOTIENT RULE
DAY07 UNIT08 – LEVEL CURVES, TANGENT PLANES & PARTIAL DERIVATIVES
DAY07 UNIT09 – MIN/MAX & LEAST SQUARES (OPTIONAL)
DAY08 REVIEWE – CHAIN RULE
DAY08 UNIT10 – THE SECOND DERIVATIVE TEST & BOUNDARIES
DAY08 UNIT11 – DIFFERENTIALS & CHAIN RULE
DAY09 REVIEWF – s(t), v(t), a(t)
DAY09 UNIT12 – GRADIENTS, DIRECTIONAL DERIVATIVE & TANGENT PLANES
DAY09 UNIT13 – LAGRANGE MULTIPLIERS
DAY09 UNIT13 – SNELL (OPTIONAL)
DAY09 UNIT14 – NON-INDEPENDENT VARIABLES (OPTIONAL)
HWK09 (CHAP15) Complete End Of Chapter Review Exercises: mult5
THURSDAY 2021.0218
DAY10 HWK REVIEW
HWK10 (preTEST2A)
*** BREAK: TUESDAY (2021.0223)
*** BREAK: THURSDAY (2021.0225)
TUESDAY 2021.0302
DAY11 UNIT15 – REVIEW02
DAY11 preTEST2B (PARTIAL DERIVATIVES)
THURSDAY 2021.0304
DAY12 TEST2B (PARTIAL DERIVATIVES)
9

PART03: DOUBLE INTEGRALS & LINE INTEGRALS
*** EVENING ACTIVITY HOUR: TUESDAY (2021.0309)
DAY13 REVIEWG – a(t), v(t), s(t)
DAY13 UNIT16 – DOUBLE INTEGRALS – CARTESIAN
DAY13 UNIT17 – DOUBLE INTEGRALS – POLAR
DAY13 UNIT17 – CENTER OF MASS (OPTIONAL)
DAY13 UNIT18 – CHANGE OF VARIABLES
DAY13 UNIT18 – JACOBIAN DETERMINANTS (OPTIONAL)
DAY14 REVIEWH – DEFINITE INTEGRALS I
DAY14 UNIT19 – VECTOR FIELDS & LINE INTEGRALS IN THE PLANE
DAY14 UNIT20 – PATH INDEPENDENCE & CONSERVATIVE FIELDS
DAY14 UNIT21 – GRADIENT FIELDS & POTENTIAL FUNCTIONS
DAY15 REVIEWI – DEFINITE INTEGRALS II
DAY15 UNIT22 – GREEN’S THEOREM
DAY15 UNIT23 – FLUX & NORMAL FORM OF GREEN’S THEOREM
DAY15 UNIT24(start) – SIMPLY CONNECTED REGIONS
TUESDAY 2021.0323
DAY16 HWK REVIEW
HWK16 (preTEST3A)
THURSDAY 2021.0325 (DOES NOT INCLUDE GREEN'S THEOREM FOR FLUX)
DAY17 UNIT24(finish) – REVIEW03
DAY17 preTEST3A (DOUBLE & LINE INTEGRALS)
TUESDAY 2021.0330 (DOES NOT INCLUDE GREEN'S THEOREM FOR FLUX)
DAY18 TEST3B (DOUBLE & LINE INTEGRALS)
10

PART04: TRIPLE INTEGRALS & ALL THE 4 GREAT THEOREMS (NO TESTS)
DAY19 REVIEWJ – BY PARTS
DAY19 UNIT25 – TRIPLE INTEGRALS – CARTESIAN & CYLINDRICAL
DAY19 UNIT26 – SURFACE AREA
DAY19 UNIT26 – SPHERICAL COORDINATES (OPTIONAL)
*** BREAK: TUESDAY (2021.0406)
*** BREAK: THURSDAY (2021.0408)
DAY20 REVIEWK – PARTIAL FRACTIONS
DAY20 UNIT27 – VECTOR FIELDS IN 3D, SURFACE INTEGRALS & FLUX
DAY20 UNIT28 – DIVERGENCE THEOREM I
DAY21 REVIEWL – POWER SERIES I
DAY21 UNIT29 – DIVERGENCE THEOREM II
DAY21 UNIT29 – DIFFUSSION EQUATION (OPTIONAL)
DAY21 UNIT30 – LINE INTEGRALS IN SPACE, CURL, EXACTNESS & POTENTIAL
DAY22 REVIEWM – POWER SERIES II
DAY22 UNIT31 – STOKE’S THEOREM
DAY22 UNIT32a – STOKE’S THEOREM & PATH INDEPENDENCE
DAY22 UNIT33 – TOPOLOGY & MAXWELL’S EQUATIONS (OPTIONAL)
THURSDAY 2021.0422
DAY23 HWK REVIEW
HWK23 (preTEST4A)
11

PART05: REVIEW&FINAL (TEST4B on PART04 & TEST5B on PART01-PART04 = FINAL EXAM)
TUESDAY 2021.0427
DAY24 UNIT32b – REVIEW04 preTEST4A
THURSDAY 2021.0429
DAY25 TEST4B (TRIPLE INTEGRALS & THE GREAT THEOREMS)
HWK25 (preTEST5A)
TUESDAY 2021.0504
DAY26 UNIT34+UNIT35 – REVIEW05 preTEST5A (FINAL REVIEW)
THURSDAY 2021.0506
DAY27 TEST5B (FINAL EXAM)
TUESDAY 2021.0511
DAY28 POST TEST5B REVIEW
*** POSSIBLE SNOW MAKE UP DAY: THURSDAY (2021.0513)
12

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