– Introduction to the theory of curves: parametric representation of a curve, arc length, tangent and normal plane, curvature, torsion, Frenet frame.
– Introduction to the theory of surfaces: parametric representation of a surface, first fundamental quadratic form, metric tensor, covariant and contravariant components, surface element.
– Curvilinear coordinates: coordinate surfaces and curves, area element, volume element, Cartesian, spherical and cylindrical coordinates, gradient, divergence and curl.
– Double integrals: definition and properties of the double integral, geometric interpretation, computation of the area of a plane region.
– Double integrals: change of variables in integration, applications.
– Triple integrals: definition and properties, change of variables in integration, applications.
– Introduction to line integrals of the first and second kind: definitions and properties of line integrals, relation between first and second kind, applications.
– Green’s Theorem – Potential function and irrotational field in the plane – Line integrals over multiply connected domains.
– Surface area – Surface integrals of the first and second kind.
– Gauss’s and Stokes’s Theorems.
– Applications of Gauss’s and Stokes’s Theorems – Potential function and irrotational field, applications to multiply connected domains.
– Applications of double and triple integrals – Computation of mass, moment of inertia, center of mass, gravitational potential and Coulomb potential.