Course information

Introduction: The Euclidean space Rn. The topology of Rm. Sequences. Functions between Euclidean spaces, their graphical representation, level sets. Limit and continuity of functions. Connected and path-connected sets. Derivatives of Vector Functions of One Variable: Derivatives of vector functions of one variable. Applications in Differential Geometry and Mechanics. (Frenet, curvature, torsion). Curvilinear coordinates and their corresponding unit vectors. Derivatives of Functions Rn→ Rm: Partial derivative. Higher-order partial derivatives. Schwarz s Theorem, Directional derivative. Differentiable functions, related theorems, 1st order differential and optimal linear approximation, tangent plane of a surface. Composite function derivative and applications. Gradient of a real function. Divergence and curl, Laplacian, Streamlines of a Vector Field, Material derivative. Mean value theorem. Higher order differentials Taylor formula. Inverse function theorem. Implicit functions. Functional dependence. Extrema: Extrema of functions. Constrained extrema. Lagrange multipliers.