Calculus was first invented to meet the mathematical needs of scientists of the sixteenth and seventeenth centuries, needs that mainly mechanical in nature. Nowadays it is a tool used almost everywhere in the modern world to describe change and motion. Its use is widespread in science, engineering, medicine, business, industry, and many other fields. Calculus also provides important tools in understanding functions and has led to the development of new areas of mathematics including real and complex analysis, topology, and non-euclidean geometry.
The objective of this course is to introduce the fundamental ideas of the differential and integral calculus of functions of several variables.
Power series, Taylor Polynomials, Taylor Series, Maclaurin series, Binomial series, Lines and planes, Functions of several variables, Limits and Continuity, Partial Differentiation, Chain Rule, Tangent plane, Critical points, Global and Local Extrema, Directional Derivatives, Gradient, Divergence and Curl, Multiple integrals with applications, Triple integrals with applications, Triple integrals in Cylindrical and Spherical coordinates, Line-, Surface- and Volume Integrals, Independence of path, Green’s Theorem, Conservative Vector Fields, Divergence Theorem, Stoke’s Theorem.
RELATIONSHIP WITH THE OTHER COURSES
This course provides the mathematical background for engineering students and is very important, for instance, for advanced courses on partial differential equations, probability and statistical analysis or numerical analysis.
On succesful completion of the course, the students should be able to:
- find the radius and the interval of convergence of a power series, indicating at which points the series converges absolutely/conditionally;
- construct Taylor and Maclaurin series for a given function;
- use Taylor and Maclaurin series for approximation of functions and estimate the error;
- use power series to calculate limits;
- understand and apply two and three dimensional Cartesian coordinate system;
- recognize and classify the equations and shapes of quadratic surfaces;
- use the properties of vectors and operations with vectors;
- recognize and construct the equations of lines and planes;
- operate with vector functions, find their derivatives and integrals, find the arc length;
- understand and use the concept of a function of several variables, find it’s domain;
- calculate the limits of multivariable functions and prove the nonexistence of a limit;
- find partial derivatives using the properties of differentiable multivariable functions and basic rules;
- apply partial derivatives for finding equations of tangent planes, normal lines, and for extreme values;
- evaluate double integrals in Cartesian and polar coordinates and triple integrals in Cartesian and cylindrical coordinates;
- apply multiple integrals for computing areas and volumes;
- understand and use integration in vector fields;
- find line integrals and flux using Green’s Theorem;
- find circulation of a vector field using Stoke’s theorem;
- use Divergence Theorem to find the flux of a vector field.