This book endeavours to strike a balance between mathematical and numerical coverage of a wide range of mathematical methods and numerical techniques. It strives to provide an introduction, especially for undergraduates and graduates, to engineering mathematics and its applications. Topics include advanced calculus, ordinary differential equations, partial differential equations, vector and tensor analysis, calculus of variations, integral equations, the finit difference method, reaction-diffusion system, and probability and statistics. The book also emphasizes the application of important mathematical methods with dozens of worked examples. The applied topics include elasticity, harmonic motion, chaos, kinematics, pattern formation and hypothesis testing. The book can serve as a textbook in engineering mathematics, mathematical modelling and scientific computing.