Multivariable Calculus and Vector Analysis

Cartesian, cylindrical and spherical coordinates in space. Cylindrical surfaces and second degree surfaces. Functions of many variables, limit, continuity, partial derivative of first or higher order of functions and geometrical meaning. Derivation rules, Schwartz’s theorem and directional derivative. Total differential and the conception of differentiation. Composite functions and homogeneous equations, complex forms and basic existence theorems. Jacobian determinant and functional dependence. Taylor’s and Maclaurin’s mean value theorems. Extremities of functions and bounded extremities, Lagrange’s multipliers. Vector analysis and vectors in space. Limit, continuity and derivative of vector functions of one and many variables. Elements of the differential geometry of curves in space. Position vector of particle, vector velocity and acceleration. Unit tangential and unit perpendicular vector of curve. Orthogonal coordinate system or trihedral Frenet-Serret, curvature and turning of curve. Gradient or grade of functions, tensor product, divergence and rotation or swirling of vector functions, their physical meaning and basic vector identities. Laplace’s differential operator, harmonic functions and partial differential equations of Helmholtz, wave and diffusion. Irrotational and solenoidal fields, Helmholtz’s decomposition theorem. Curvilinear coordinate systems, vector meaning of Jacobian determinant, special orthogonal and curvilinear coordinates, transformations and change of coordinates. Applications of partial derivatives to geometry, tangential plane and perpendicular straight line to surface, tangential straight line and perpendicular plane to curve. Multiple integration of functions, double and triple integrals, change of coordinate system and applications to the calculation of plane surface areas, of volumes of three-dimensional domains, of mass, of moments of inertia and of gravity center. Curve integrals of the first and of the second kind, application to the calculation of the force work and Green’s theorem for the plane. The meaning of the circulation of vector functions, curve integrals independent of the root of integration and applications. Surface integrals and surface parameterization, application to the calculation of the area of arbitrary surface in space. Gauss’ and Stokes’ or Green’s for the space integral theorems and their physical meaning.