The Euclidean space Rn. Functions between Euclidean spaces, limit and continuity of functions. Differentiation of vector-valued functions of a single variable, applications in mechanics and differential geometry, polar, cylindrical and spherical coordinates. Differentiable functions (partial derivatives, directional derivatives, differential). Vector fields. Gradient-divergence-curl. Fundamental theorems of differentiable functions (mean value theorem, Taylor). Inverse function theorem. Implicit function theorems. Functional dependence. Local and conditional extremes. Double and triple integrals: definitions, integrability criteria, properties. Change of variables, applications. Multiple integrals. Generalised multiple integrals. Contour integrals: Contour integral of the first and second kind, contour integrals independent of path, Green’s Theorem, simply and multiply connected domains of R2 and R3. Elements of surface theory. Surface integrals of the first and second kind. Fundamental theorems of vector analysis (Stokes and Gauss) applications.