DEPARTMENT OF MATHEMATICS FACULTY OF ENGINERING AND TECHNOLOGY SRM UNIVERSITY MA1001- CALCULUS AND SOLID GEOMETRY SEMESTER I ACADEMIC YEAR: 2014-2015 LECTURE SCHEME / PLAN The objective is to equip the students of Engineering and Technology with the knowledge of Mathematics and its applications so as to enable them apply in solving real world problems. The list of instructions (provided below) may be followed by a faculty relating to his/her own schedule includes warm-up period, controlled/free practice, and the respective feedback of the classes who handle. The lesson plan has been formulated based on high quality learning outcomes and the expected outcomes. Each subject must have a minimum of 56 hours, which includes 45 hours for lecture and rest of the hours for tutorials. The faculty has to pay more attention in insisting the students to have 95 % class attendance. UNIT I: MATRICES Lect. No L 1.1 Lesson schedule Learning outcomes Cumulative hours To refresh and enhance the 1 prerequisite mathematical knowledge for the course L1.2 Introduction to Matrix and its operations. Finding the characteristic equation. L.1.3 L.1.4 L.1.5 L.1.6 L.1.7 L.1.8 To find Eigen Values and Eigen vectors for Symmetric matrices (repeated and non repeated Eigen values) To find Eigen Values and Eigen vectors for Non-symmetric matrices (repeated and non repeated Eigen values). Properties of Eigen values and Eigen vectors. Problems based on the properties. Cayley Hamilton theorem and its applications (basically finding an inverse and higher powers of matrices) Orthogonal Matrices. Orthogonal transformation of symmetric matrices Quadratic form; Reduction of a quadratic form to canonical form using orthogonal transformation To motivate the students for the course. Eigen values and vectors are used to solve homogeneous linear differential equations with constant co-efficients and in optimization problems. To obtain the higher powers and the inverses of the given matrix. 2 3 4 5 6 7,8 9,10 Page 1 of 5
L.1.9 L.1.10 To find Rank, index, signature and nature of a quadratic form Revision of the topics covered in the first unit and solving problems UNIT II: FUNCTIONS OF SEVERAL VARIABLES L.2.1 L.2.2 L.2.3 L.2.4 Introduction to functions of several variables. Partial derivatives Definition and examples Chain rule for functions of several variables Total derivatives. Differentiation of Implicit functions Homogeneous functions. Euler s theorem To get familiar with functions of several variables which helps to find maxima and minima, calculus of variations, functional analysis which is applied in real world problems such as solving PDEs, pattern recognition, etc. 11 12 13 14 15 16 CYCLE TEST I DATE: 18.08.2014 L.2.5 Taylor s expansion for function 17 of two variables. L.2.6 Finding extreme values of the 18,19 function of two variables L.2.7 Method of Lagrangian 20,21 multiplier L.2.8 Jacobian 22 L.2.9 Properties of Jacobian 23 L.2.10 More problems to be solved in functions of several variables 24 UNIT III: ORDINARY DIFFERENTIAL EQUATIONS L.3.1 Ordinary Differential equation introduction Degree and Order of a differential equation L.3.2 To find the complementary function for Homogeneous differential equations depending on the nature of the auxiliary equation roots. L.3.3 L.3.4 L.3.5 equationtype-1, Type-2 equation Type-3,Type-4 equation Type-5,Type-6 To acquire a wide knowledge in solving differential equations. This plays a prominent role in theory of dynamical systems, heat and mass transfer operations.. 25 26 27 28 29 Page 2 of 5
L.3.6 Solving the Linear differential equation with variable coefficients Euler s type L.3.7 Solving the Linear differential equation with variable coefficients Legendre s type L.3.8 Method of Variation of parameters L.3.9 Solving for Simultaneous linear differential equations L.3.10 Comparing different methods to solve linear differential equations which have been taught in the previous classes. 30,31 32 33,34 CYCLE TEST II : DATE: 19.09.2014 UNIT IV: GEOMETRICAL APPLICATIONS OF DIFFERENTIAL CALCULUS L.4.1 Brief overview of Differential calculus and the geometry behind it. L.4.2 Radius of curvature: Cartesian form L.4.3 Radius of curvature: Parametric form L.4.4 Radius of curvature: Polar form L.4.5 Centre of curvature Circle of curvature L.4.6 Evolute of a curve Involute of a curve L.4.7 Envelope of the family of curves To improve the ability in solving geometrical applications of differential calculus. The curvature is useful in Mathematical modeling, bending of beams and solving PDEs. 35 36 37 38 39 40 41,42 43,44 L.4.8 Properties of the envelopes 46 L.4.9 Relation between envelopes 47 and the evolutes. L.4.10 Review of the topics covered in the geometrical applications of differential calculus 48 SURPRISE TEST UNIT V: THREE DIMENSIONAL ANALYTICAL GEOMETRY L.5.1 Outline to Three dimensional analytical geometry, Direction cosines and direction ratios of a line segment. To grasp the knowledge of three dimensional space and basic geometrical objects, which will help us have deeper knowledge about higher dimensional spaces and have a strong basic intuition to study linear L.5.2 Equation of a sphere 50 L.5.3 Plane section of a sphere 51 L.5.4 Tangent plane 52 L.5.5 Orthogonal spheres 53 L.5.6 Equation of a cone 54 45 49 Page 3 of 5
L.5.7 Right circular cone algebra and algebraic curves 55 L.5.8 Equation of a cyclinder arising from engineering 56, 57 L.5.9 Right circular cylinder experiments. 58 L.5.10 Problems related to sphere, cone and cylinder 59-60 MODEL EXAM Date: 05.11.2014 (Duration: 3 Hours) LAST WORKING DAY : 21.11.2014 TEXT BOOKS: 1. Dr. K. Ganesan, Dr. Sundarammal Kesavan, Prof. K. S. Ganapathy Subramanian, Dr. V. Srinivasan, Engineering Mathematics I, Gamma Publications, 6 th Edition, 2014. 2. Kreyszig. E, Advanced Engineering Mathematics, 8th edition, John Wiley & Sons, Singapore, 2001 REFERENCES 1. Veerarajan T., Engineering Mathematics, Tata McGraw Hill Publishing Co., New Delhi, 2000 2. Venkataraman M. K., Engineering Mathematics - First Year (2nd edition), National Publishing Co., Chennai, 2000 3. Narayanan S., Manicavachagom Pillay T. K., Ramanaiah G., Advanced Mathematics for Engineering students, Volume I (2nd edition), S. Viswanathan Printers and Publishers, 1992 4. Kandasamy P etal. Engineering Mathematics, Vol. I (4th revised edition), S. Chand & Co., New Delhi, 2000 WEB RESOURCES: For unit 1: https://www.math.duke.edu/education/webfeatsii/lite_applets/eigenvalue/start.html http://www.cse.unr.edu/~bebis/mathmethods/linearalgebrareview/ http://academicearth.org/lectures/eigenvalues-and-eigenvectors http://web.mit.edu/18.06/www/ For unit 2: http://tutorial.math.lamar.edu/classes/calciii/multivrblefcns.aspx http://home.iitk.ac.in/~psraj/mth101/ http://synechism.org/drupal/cfsv/ http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/index.htm For unit 3: http://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/ http://tutorial.math.lamar.edu/classes/de/de.aspx http://archives.math.utk.edu/ctm/fifth/ricardo/paper.html For unit 4: http://www.calculus.org/ http://www.oid.ucla.edu/webcasts/courses/2009-2010/2010winter/math31a-1 http://ocw.mit.edu/courses/mathematics/18-013a-calculus-with-applications-spring-2005/ Page 4 of 5
For unit 5: http://www.geom.uiuc.edu/docs/reference/crcformulas/ node37.html#section02000000000000000000 http://mathforum.org/dr.math/faq/formulas/faq.analygeom.html Internal marks Total: 50 Internal marks split up: Cycle Test 1: 10 Marks Cycle Test 2: 10 Marks Attendance: 5 marks Model Exam: 20 Marks Surprise Test: 5 marks Dr. V. Srinivasan Dr. K. Ganesan Professor Professor & Head Course Co-ordinator Department of Mathematics Email: srinivasan.va@ktr.srmuniv.ac.in Email: hod.maths@ktr.srmuniv.ac.in Tel: +91-44-27417000 Ext: 2704 Tel: +91-44-27417000 Ext: 2701 Page 5 of 5