1. Vector Calculus:
This chapter covers topics such as the parametrization of curves, computing arc length of curves in space, line integrals, vector fields, and their applications in work, circulation, and flux. It also deals with path independence, potential functions, and concepts related to the smoothness and connectedness of domains. The fundamental theorem of line integrals and conservative fields are discussed, along with exact differential forms. Additionally, the chapter introduces the concepts of divergence (Div) and curl and includes Green’s theorem in the plane (without proof).
2. Laplace Transform:
In this chapter, students learn about the Laplace transform and its inverse. The concept of linearity in Laplace transforms is covered, as well as the first shifting theorem (s-shifting). Transforms of derivatives and integrals are explored, along with their application to ordinary differential equations (ODEs). The unit step function (Heaviside function) and the second shifting theorem (t-shifting) are explained. The Laplace transform of periodic functions, short impulses, and Dirac’s delta function are also included. The chapter concludes with discussions on convolution, integral equations, and solving ODEs with variable coefficients, as well as systems of ODEs.
3. Fourier integral:
This chapter introduces Fourier integrals, Fourier cosine integrals, and Fourier sine integrals. The focus is on understanding the principles and applications of Fourier analysis.
4. First-Order Ordinary Differential Equations (ODEs):
Students learn about solving first-order ODEs, including exact, linear, and Bernoulli’s equations. The chapter covers equations solvable for different variables (p, y, x) and explores Clairaut’s type of differential equations.
5. Ordinary Differential Equations of Higher Orders:
This chapter deals with higher-order ODEs, specifically homogeneous linear ODEs with constant coefficients, Euler–Cauchy equations, and the existence and uniqueness of solutions. Concepts like linear dependence and independence of solutions, Wronskian, and nonhomogeneous ODEs are discussed. The methods of undetermined coefficients and variation of parameters are presented for solving these equations.
6. Series Solutions of ODEs and Special Functions:
The final chapter focuses on series solutions of ODEs using the power series method. It introduces special functions such as Legendre’s equation and Legendre polynomials, Frobenius method, Bessel’s equation, and Bessel functions of the first kind, along with their properties.
