This second edition is a major revision based on AICTE 2018 Model Engineering Mathematics Curriculum. Physical scientists and engineers need mathematical tools in their everyday work. This book presents mathematics in a form appropriate for undergraduate courses in a friendly and lucid way. Basic to modeling any physical system, either in science or engineering, a techno-scientist needs a strong foundation of differential equations based on Calculus. The book sets foot in its journey from the mid-17th century Calculus of a single and several variables, invented independently by Isaac Newton and Gottfried Wilhelm Leibniz. Ordinary differential equations and a special class of them: Bessel, Legendre, Laguerre, Chebyshev, and Jacobi; partial and non-linear differential equations with applications to physical problems are central topics in this book. The theory of a complex variable with applications to the evaluation of improper integrals and the solution of potential problems is developed from an elementary to a reasonably advanced level. Numerical methods for interpolation, roots of equations, integrations, ordinary and partial differential equations, evaluation of eigenvalues and eigenvectors are treated in sufficient depth with MATLAB-based algorithms. Improper integrals, special functions, integral equations and calculus of variations are also outlined as they are important topics in science and engineering.