This chapter is a concise yet rigorous presentation of three-dimensional vector and tensor analysis. Subjects are specifically chosen for the need of continuum mechanics and geophysics. Besides the definitions, the algebraic operations, the operations of the vector differential operator nabla, and the basic field theory, also included are the fourth order isotropic tensor, the reduction of a symmetrical second order tensor to the diagonal form, Green’s identities, and Helmholtz’s theorem. The curvilinear coordinate system is treated to some length. The general curvilinear coordinate system is formulated first, and then the orthogonal curvilinear coordinate system is treated as a special case. The spherical and cylindrical coordinate systems are considered as instantiations. Formulae of the nabla operator in the curvilinear coordinate system are systematically derived. Throughout the chapter, the formulations are rigorous, i.e., there is no jump of logic or derivation. However, the learning curve is steep, as the text is concise. Less experienced learners of mathematics are suggested to read a book providing a less steep learning curve in parallel, such as those recommended by the end of the chapter.