ABSTRACT
This chapter explains the application on differential equations in engineering and in mechanics, and on the associated solution techniques used in solving these equations. It reviews binomial theorem, differentiation, integration, Jacobian, complex variables, Euler’s polar form for complex variables, elementary functions like circular functions and hyperbolic functions, differentiation of complex functions, integration using the Cauchy integral formula and residue theorem, Gauss and Kelvin-Stokes theorems, series expansions, and vector calculus. The chapter discusses some more advanced topics that are unlikely being covered in elementary engineering mathematics courses, and they are the Frullani-Cauchy integral Ramanujan’s master theorem, Ramanujan’s integral theorem, Darboux’s formula, Mittag-Leffler’s expansion, Borel’s theorem, and tensor analysis. It also discusses both differential and integral forms of gradient, divergence, curl, and Helmholtz’s representation theorem. The chapter summarizes some essential mathematical backgrounds needed for later analysis of differential equations, and explores the definitions of algebraic functions and transcendental functions.
