Multivariable Calculus

Lecturer:

Tutor:

Supervisor:

Department of Natural Sciences

Synopsis:

The course is focused at elements of calculus in two and more variables and at real, complex and functional series.

Calculus in two variables: notion of a limit and continuity, partial derivative, differential and its applications. Derivative of a composed function, derivative of an implicit function. Higher order derivatives, local extremes. Constrained extremes, least squares method. Double and triple integrals, geometrical interpretation, Fubini theorem. Integration by substitution in double and triple integral.

Complex sequences, series of numbers. Convergence of complex series. Functional series and their convergence, power series. Taylor series. .

Requirements:

Credit condition - 70% presence, successful written test on 3. and 6. exercise. It is necessary to gain at least one half of maximum number of points.

Theme of 1. test: Domain of definition of two variable function, tangent plane, local extrema.

Theme of 2. test: Double and triple integrals, convergence of series.

Syllabus of lectures:

1. Multiple variable function, limit, continuity, partial derivative, higher order partial derivative, direction derivative, gradient.

2. Differential and its applications, tangent plane. Derivative of composed function, derivative of implicit function.

3. Local and constrained extrema, Lagrange multiplicators.

4. Double integrals, Dirichlet theorem, Fubini theorem, substitution in double integral, jacobian.

5. Triple integrals, substitution, spherical, cylindrical coordinates.

6. Curve and surface integrals, Green, Stokes and Gauss theorem.

7. Number and function series, convergence.

Syllabus of tutorials:

1. Domains of definition, limit of two variable function, partial derivative, direction derivative.

2. Total differential, tangent plane. Derivative of composed function, derivative of implicit function.

3. Extrema of function, 1. test.

4. Double integral, substitution, application.

5. Triple integral, substitution, application.

6. Curve and surface integral, 2. test.

7. Convergence of number and function series.

Study Objective:

To learn the elements of multivariable function calculus and of number and function series.

Study materials:

[1] http://mathworld.wolfram.com/topics/CalculusandAnalysis.html

Note:

Further information:

No time-table has been prepared for this course

The course is a part of the following study plans:

• Bakalářský studijní obor Biomedicínský technik - prezenční v angličtině (compulsory elective course)