• define and explain basic notions from topology and concepts as function, limit, continuity, partial derivative, extremal point, and multiple integral
• cite, explain and use central theorems such as the max-min existence theorem, differentiability implies existence of partial derivatives, the chain rule, Taylor's formula, the characterization of stationary points, the theorem on local maxima and minima with constraints, the implicit function theorem, and the theorem on change of variables in multiple integrals
• investigate limits, continuity, differentiability, and use the chain rule for transforming and solving partial differential equations
• explain the geometric significance of directional derivatives and gradients, and determine equations for tangent lines and tangent planes
• investigate local and global maxima and minima, with or without constraints
• explain the behavior of an implicitly given function, for example by Taylor expansion and implicit differentiation
• calculate multiple integrals by means of iterated integration and using various changes of variables, notably linear, plane polar and spherical
• investigate convergence of improper multiple integrals and calculate their values
• verify that results and partial results are correct or reasonable

The course will give proficiency in use of notions and relationships, e.g. ability in finding limits, differentiation of functions with applications to change of variables in derivatives, geometric problems, local and global maxima and minima problems and differentiation of implicitly defined functions, evaluation of double and triple integrals with applications to area, volume and center of mass problems. Applications will be given of mathematical models from various fields.