### Engineering Mathematics-I

 Shivaji University, Kolhapur First Year B. Tech ENGINEERING MATHEMATICS - I Course Objective To teach Mathematical methodologies and models. To develop mathematical skills and enhance logical thinking power of students. To provide students with skills in matrix, differential calculus , complex numbers & curve fitting which would enable them to devise engineering solutions for given situations they may encounter in their profession. To produce graduates with mathematical knowledge, computational skills and the ability to deploy these skills effectively in the solution of problems, principally in the area of engineering. Course Outcome Students in this course will apply the Procedure and methods to solve technical problems. Student can understand how to model real world scenario using Mathematics Students will be able to solve computational problems using Scilab/Matlab Unit 1 Matrix Theory                                                                                                                                 8 hrs  Rank of a matrix, Normal form of a matrix, Consistency of the system of linear equations, linear dependence and independence of vectors, Eigen values and eigen vectors, Cayley-Hamilton Theorem and its applications.                Unit 2 Differential Calculus                                                                                                                     8 hrs Successive differentiation, Leibnitz’s Theorem and its applications, Taylor's and Maclaurin's series, indeterminate forms.                                              Unit 3 Partial Differentiation                                                                                                                  8 hrs Partial derivatives of first and higher order, total differentials, differentiation of composite and implicit functions. Euler’s Theorem on Homogeneous functions with two and three independent variables. Deductions from Euler’s Theorem.   Unit 4 Applications of Partial Differentiation                                                                                        8 hrs                    Errors and Approximation, Jacobian, Properties of Jacobian, Jacobian of Implicit function, Maxima and Minima of functions of two variables   Unit 5 Complex Numbers                                                                                                                          8 hrs Complex number, representation of a complex number in Cartesian & Polar co-ordinate systems, Argand’s diagram, De’Moivre’s Theorem and its applications, circular and inverse circular functions, hyperbolic and inverse hyperbolic functions, logarithm of complex numbers.   Unit 6 Curve fitting                                                                                                                                     8 hrs Fitting of Curves by method of Least-squares for linear, parabolic, and exponential, Coefficient of correlation, Spearman’s rank correlation, coefficient and lines of regression of bivariate data.   Suggested list of Tutorials/Assignments- To find rank of the matrix Solution of system of linear equations Eigen values and Eigen Vectors Applications of Leibnitz theorem Indeterminate form Euler’s Theorem on Homogeneous functions Applications of partial differentiation Applications of De’Moivres theorem Fitting of curves Coefficient of Correlation Lines of Regression Introduction to scilab/Matlab   General Instructions: Batch wise tutorials are to be conducted. The number of students per batch should be as per the practical batches. Students must be encouraged to write Scilab/Matlab Programs in tutorial class only. Each Student has to write at least 4 Scilab tutorials (including print out) and at least 6 class tutorials on entire syllabus. SciLab/Matlab Tutorials will be based on To find rank of the matrix Solution of system of linear equations Eigen values and Eigen Vectors Fitting of curves Coefficient of Correlation Lines of Regression Reference Books- B. S. Grewal, “Higher Engineering Mathematics”, Khanna Publications, New Delhi. C.R.Wylie, “Advanced Engineering Mathematics”, McGraw Hill Publication, New Delhi. Erwin Kreyszig, “Advanced Engineering Mathematics (7th Edition)”, Wiley Eastern Ltd., Bombay. P. N. Wartikar and J. N. Wartikar, “A Text Book of Engineering Mathematics (Volume-I & II)”, Pune Vidyarthi Griha Prakashan, Pune. Shanti Narayan, “Differential Calculus” S. Chand and company, New Delhi. S. S. Sastry, “Engineering Mathematics (Volume-I)”, Prentice Hall Publication, New Delhi. B.V. Ramana, “Higher Engineering Mathematics”, Tata McGraw-Hill.