College of Engineering, Pune-5.
Department of Mathematics
(MA- ) VECTOR CALCULUS AND PARTIAL DIFFERENTIAL EQUATIONS (VCPDE)
S. Y. B. Tech. (All Branches), 2017-18
Teaching Scheme : Lectures: 2 hrs/week , Tutorial : 1 hr/week
Examination Scheme : Continuous Evaluation : 40 marks, End Semester Examination : 60 marks
Text book : Advanced Engineering Mathematics (10th Edition) by Erwin Kreyszig
Unit 1: Vector Calculus
Lesson Topic Section no.
no. of text
book
1 Vector and Scalar valued functions, scalar fields, vector fields (2D and 3D) 9.4
2 Plotting vector fields, isotherms / isobars (level curves of scalar field), 9.4, 9.5
parametric representation, tangent to the curve, arc length
parametrization
3 Gradient of scalar fields, Theorem 1, Laplace operator 9.7
4 Theroem2, directional derivative 9.7
Divergence of a vector field, curl of a vector field, Theorem 2(sec 9.9) 9.8, 9.9
Physical significance of divergence and curl (self-study) 9.8, 9.9
5&6 Line integrals 10.1
7 Path independence of line integrals, Theorem 1, Theorem 2, 10.2
Theorem 3
8 Green’s theorem in plane 10.4
9 Surface integral 10.5, 10.6
10, 11 Divergence theorem of Gauss , Stokes’s theorem 10.7, 10.9
& 12
Unit 2: Partial Differential Equations
Lesson Topic Section no.
no. of text
book
1 Introduction and classification of second order PDE, method of separation 12.1
of variables.
2 Physical derivation of the one dimensional wave equation 12.2
3&4 Solution of one-dimensional wave equation by method of separation of 12.3
variables
Physical derivation of the one dimensional heat equation (Self-Study) 12.5, 10.8
5&6 One dimensional heat equation 12.6
7&8 Potential equation for incompressible flow (Laplace equation) 12.6
Vibrations of circular membrane (two dimensional wave 12.10
equation)(Self-Study)
Unit 3: Laplace Transforms
Lesson Topic Section no.
no. of text
book
1&2 Laplace transform, inverse Laplace transform, linearity, first shifting 6.1
theorem, existence and uniqueness theorem
3 Transforms of derivatives and integrals, differential equations 6.2
4 Differentiation and integration of transforms 6.6
5 Unit step function, second shifting theorem 6.3
6 Dirac delta function, convolution theorem 6.4, 6.5
Solution of differential equation by partial fractions – Self Study 6.4
Reference Books:
1. Thomas’s Calculus (TC) Twelfth edition.
2. Advanced Engineering Mathematics by C. R. Wylie, McGraw Hill Publications, New Delhi
3. Functions of several variables by Wendell Fleming, Springer-Verlag, New York
4. Partial Differential Equations (4th edition) by Fritz John, Springer
5. Advanced Engineering Mathematics (7th edition) by Peter V. O’ Neil, Thomson, Brooks, Cole, Singapore.
6. Advanced Engineering Mathematics (2nd edition) by Michael D. Greenberg, Pearson Education.
Note:
1. 100% attendance is compulsory. Students having attendance below 75% will
be detained and will not be allowed to appear for the End Semester Exam.
2. Portion of syllabus and dates of T1, T2 exam will be announced one week in
advance as per the academic calendar and the allotted slot by the exam cell.
3. All students must register for the course on MOODLE and check for any notes,
tutorials, question banks, announcement of tests, syllabus etc on the same. You will
be provided useful links of video lectures and visual interpretations of concepts in
vector calculus only on MOODLE.
4. Tutorial problems will be unit wise and hence available from day 1 only on
MOODLE. Students must come prepared for the tutorial class.
5. COORDINATOR will not be responsible for any communication gap.
Outcomes : Students will be able to
1. know and recall core knowledge of the syllabus. ( To measure this outcome, questions
may be of the type- define, identify, state, match, list, name etc.)
2. understand basic concepts. ( To measure this outcome, questions may be of the type-
explain, describe, illustrate, evaluate, give examples, compute etc.)
3. analyze the problem and apply the appropriate concept. ( To measure this outcome,
questions will be based on applications of core concepts)
4. give reasoning. ( To measure this outcome, questions may be of the type- true/false with
justification, theoretical fill in the blanks, theoretical problems, prove implications or
corollaries of theorems, etc.)
5. apply core concepts to new situations. ( To measure this outcome, some questions will
be based on self-study topics and also comprehension of unseen passages.)
