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Department of Computer Science and Information Technology

THE UNIVERSITY OF LAHORE

MA-2117 MULTIVARIATE CALCULUS FALL-2015 Course Title

Multivariate Calculus

SCU

3 Credit(s)

Co-requisite (s)

None

Pre-requisite(s)

Calculus & Analytic Geometry

Weekly tuition pattern 2 sessions (90 minutes each) Teaching Team

Humaira Muslim Zuriat Zahra

[email protected] [email protected]

Syllabus Designed By: IMRAN HASHIM [email protected]

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Course Description: This course focuses on two basic applications: Differential Calculus and Integral Calculus. Under these, we will study different techniques and some Fundamental theorems of calculus in multiple dimensions for example Stokes' theorem, Divergence theorem, Green's theorem, Other topics of discussion are Limits and Continuity, Extreme values, Fourier series and Laplace transformations. These mathematical methods are used extensively in the physical sciences, engineering, economics and computer graphics.

Objectives: This course covers vector and multi-variable calculus. As its name suggests, multivariable calculus is the extension of calculus to more than one variable. Single variable calculus is a highly geometric subject and multivariable calculus is the same, maybe even more so. In your calculus class you studied the graphs of functions y=f(x) and learned to relate derivatives and integrals to these graphs. In this course we will also study graphs and relate them to derivatives and integrals. One key difference is that more variables means more geometric dimensions. This makes visualization of graphs both harder and more rewarding and useful. By the end of the course you will know how to differentiate and integrate functions of several variables. In single variable calculus the Fundamental Theorem of Calculus relates derivatives to integrals. We will see something similar in multivariable calculus and the capstone to the course will be the three theorems (Green's, Stokes' and Gauss') that do this.

Student Learning Outcomes: Upon successful completion of this course, students will be able to: I. II. III. IV. V. VI. VII. VIII. IX. X. XI.

Determine Limits and Continuity of multi-variable function Evaluate Partial Differentiation and will know the related techniques. Apply the concept of Extreme-Values of multi-variable functions to real world problems. Solve Double Integral for Cartesian and Polar co-ordinates and can do their interconversions. Find Triple Integrals in rectangular, spherical and cylindrical co-ordinates. Apply Multiple Integrals for area and volume problems. Apply elementary operations on Vector-Valued function. Compute arc-length and solve problems regarding change of parameter. Evaluate Line, Surface and Volume integral. State Green’s Theorem, Divergence Theorem and Stoke’s Theorem and show how these theorems are applied. Find Fourier Series of given periodic function.

Course Structure: I. II. III.

Presentation by lecturer Class Activities and Home assignments Quiz Page 2|6

Course Duration: This course will be held twice a week of 1 hour 30 min class duration.

Course style: The course will be delivered in a classroom environment.

Additional Course Requirement: In addition to the objective of this course, students are expected to gain skills in Multivariate Calculus through a variety of learning activities that include text study, problem solving, whole class discussion, small group collaboration, and by using the technology, for this purpose they are suggested to use software Microsoft Math. The most critical component of each of these experiences, and key to their success and achievement, will be their determination to thoughtfully engage in these activities with energy, interest, perseverance and a collegial spirit.

Text and Other Resources: I. II. III. IV.

Calculus with Analytic Geometry by Howard Anton, 10th edition, John Wiley and sons, Inc. Calculus by Thomas Finney, 12th edition, Addison-Wesley. Multivariable Calculus by James Stewart, 7th edition, Cengage Learning. Advanced Engineering Mathematics by Erwin Kreyszig, 10th edition, John Wiley and sons, Inc.

Course Outline: The lecturers are supposed to complete the following topics before the mid/final term examination as prescribed in the course outline below: Weeks

Lectures

Topics/Sub-Topics

1

1

PARTIAL DERIVATIVES

2 3

2 3 4 5 6

4

7

5

8 9

6 7

10 11 12 13

Function of two or more variables Limit and continuity of function of two variables Partial derivatives Differentiability and chain rule of function of two variable Directional derivatives Gradient for function of two variables Differentiability Directional derivatives Gradient for function of three variables Tangent planes and normal vectors Total differential for function of two variables Maxima and minima of function of two variables

MULTIPLE INTEGRALS Double integral over rectangular region Double integral over non-rectangular region, Reversing the order in Double integrals, Area by Double integral Double integral in polar coordinates Conversion of Double integrals from rectangular to polar coordinates Page 3|6

8

14 15 16

Introduction to Triple integrals Triple integral over rectangular region Triple integral over non-rectangular region

9 10

MID-TERM EXAMINATION 18

11 12

13 14 15

VECTOR CALCULUS

17

19 20 21 22 23 24 25 26 27

Triple integral in cylindrical coordinates Triple integral in spherical coordinates Vector fields, Conservative fields and Potential functions Curl and divergence Line integrals Work as a Line integral, Fundamental theorem for line integrals Conservative field test, finding Potential functions Green’s theorem, Work and Area using Green’s Theorem Parametric surfaces and their areas Surface integrals Application of surface integrals: Flux Divergence theorem, Finding flux using the divergence Theorem Stokes’ theorem, Calculating work using Stokes’ Theorem

FOURIER ANALYSIS Euler formulas Fourier series of the functions having period

Functions of any Period P=2L Transition from period to any period 16

28

Fourier series of even and odd functions Half range expansion

29 30

Fourier cosine and sine transforms Transforms of derivatives

17

FINAL-TERM EXAMINATION

Assessment Criteria: No. 1. 2 3 4

Assessment Assignments Quizzes Midterm Final Total

Percentage 10% 10% 30% 50% 100%

Detail of Assignments: The detail of all assignments and quizzes are in the course FOLDER.

Attendance Requirements: You are expected to attend all lectures because that’s the only way to ensure your full marks in Multivariate Calculus. Similarly the quizzes may be unannounced and no retaking of a missed quiz will be considered except the exceptional case. You are responsible for your attendance, so Page 4|6

be attentive when the attendance is called.

Submission and Collection of Assignment: All assignments should be submitted according to the due date and time i.e. beginning of the class. Late submission will be accepted with 50% deduction during a definite period of time. All the assignments will be returned after marking.

General Information: Students are required to be familiar with THE UNVERSITY OF LAHORE Code Conduct, and to abide by its terms and conditions. Copying of Copyright Material by Student: A condition of acceptance as a student is the obligation to abide by the University’s policy on the copying of copyright material. This obligation covers photocopying of any material using the University’s photocopying machines, and the recording off air, and making subsequent copies, of radio or television broadcasts, and photocopying textbooks. Students who flagrantly disregard University policy and copyright requirements will be liable to disciplinary action under the Code of Conduct. Academic Misconduct: Please refer to the Code of Conduct for definitions and penalties for Academic Misconduct, plagiarism, collusion, and other specific acts of academic dishonesty. Academic honesty is crucial to a student's credibility and self-esteem, and ultimately reflects the values and morals of the University as a whole. A student may work together with one or a group of students discussing assignment content, identifying relevant references, and debating issues relevant to the subject. Academic investigation is not limited to the views and opinions of one individual, but is built by forming opinion based on past and present work in the field. It is legitimate and appropriate to synthesize the work of others, provided that such work is clearly and accurately referenced. Plagiarism occurs when the work (including such things as text, figures, ideas, or conceptual structure, whether verbatim or not) created by another person or persons is used and presented as one’s own creation, unless the source of each quotation or piece of borrowed material is acknowledged with an appropriate citation. Encouraging or assisting another person to commit plagiarism is a form of improper collusion and may attract the same penalties. To prevent Academic Misconduct occurring, students are expected to familiarize themselves with the University policy, the Subject Outline statements, and specific assignment guidelines. Students should also seek advice from Subject Leaders on acceptable academic conduct. Guidelines to Avoid Plagiarism Whenever you copy more than a few words from any source, you must acknowledge that source by putting the quote in quotation marks and providing the name of the author. Full details must be provided in your bibliography. If you copy a diagram, statistical table, map, etc., you must acknowledge the source. The recommended way is to show this under the diagram. If you quote any statistics in your text, the source should be acknowledged. Again full details must be provided in your bibliography. Whenever you use the ideas of any other author you Page 5|6

should acknowledge those, using the APA (American Psychological Association) style of referencing. Students are encouraged to co-operate, but collusion is a form of cheating. Students may use any sources (acknowledged of course) other than the assignments of fellow students. Unless your Subject Leader informs you otherwise, the following guideline should be used: Students may work together in obtaining references, discussing the content of the references and discussing the assignment, but when they write, they must write alone. Referencing For Written Work Referencing is necessary to acknowledge others' ideas, avoid plagiarism, and allow readers to access those others’ ideas. Referencing should: I. Acknowledge others' ideas. II. Allow readers to find the source. III. Be consistent in format and IV. Acknowledge the source of the referencing format To attain these qualities, the school recommends use of either the Harvard or American Psychological Association style of referencing, both of which use the author/date. Referencing Standards APA style referencing Approval Checked by,

Approved by,

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