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Synopsis This book is our attempt to enrich and enliven the teaching of multivariable and mathematical methods courses for scientists and engineers. From one perspective, the subject of multivariable calculus only exists because it can be applied to important problems in science. It can equally well be used in the mathematical methods for scientists and engineers. The subject is traditionally called, Vector Analysis and its Application, or Multivariable Calculus. The usual content: Preliminary theory of vectors: addition and subtraction of vector, dot and cross product of vectors. Vector-valued functions: derivatives and integrals of vector-valued functions of one variable; space curves; tangent and normal. Calculus of vector fields: derivatives and integrals of vector-valued functions line, surface and volume integrals; fundamental theorem of line integrals; green's, stokes', and divergence theorems. Partial derivatives: directional derivatives; gradients; divergence, curl of vector valued functions. This book play a major role as basic tools in differential geometry, mechanics, fluid mathematics. The bulk of the book consists of five chapters on vector analysis and its applications. Each chapter is accompanied by a problem set. The problem sets constitute an integral part of the book. Solving the problems will expose you to the geometric, symbolic, and numerical features of multivariable calculus.