The lectures are divided into three thematic blocks, each block is presented by mostly 4 lectures:

1. Definite integral of one variable, applications.
2. Real function of several variables. Basic concepts, composite function. Limit and continuity. Partial derivative.
3. Partial derivatives of a composite function, partial derivatives of higher orders. Directional derivatives, gradient. Total differentials.
4. Taylor polynomial. Space curve, tangent vector of a curve. Tangent plane and normal to a surface. Local extrema of a function of two variables.
5. Bound extrema, use of Lagrange multipliers. Global extrema of a function of two variables. Implicit functions of one and two variables.
6. Double integral, calculation, properties. Calculation according to Fubini's theorem and using transformations (polar coordinates).
7. Transformation and application of double integral. Example of the triple integral,.
8. Line integral in a scalar field. Vector field, divergence, rotation. Line integral in a vector field.
9. Work, circulation, Green's theorem. Independence of the curve integral on the integration path. Potential.
10. Ordinary differential equations (ODE), basic concepts. First-order equations, separated.
11. First-order equations, linear (and exact). Homogeneous DE of nth order.
12. Homogeneous linear DE with constant coefficients, Wronskian.
13. Inhomogeneous DE with special right-hand side and constant variation method. Application of DE in technical practice, boundary problems.

Exercises

The structure of the exercises corresponds to the lecture blocks. The last week is dedicated to completing the material and repeating some more demanding topics, such as differential equations. During the course, the student takes 2 tests. The recommended test duration is 45 minutes, the second teaching hour is dedicated to continuing the teaching.

1. Definite integral of one variable.
2. Domain of definition, partial derivative of a function of several variables.
3. Directional derivatives, gradient. Partial derivatives of a composite function of several variables. Total differential and its meaning.
4. Taylor polynomial, Normal and tangent plane. Local extrema...
5. Bound and global extrema. Implicit functions.
6. Recalling complex numbers. Test 1.
7. Calculation of double integral. Transformation of double integral and applications. Example of calculation of the triple integral.
8. Calculation of the curve integral in a scalar field. Calculation of the curve integral in a vector field.
9. Applications, work, circulation, Green's theorem and its applications. Independence of the curve integral on the integration path. Potential.
10. First-order DE, separated, linear.
11. Test 2. Homogeneous DE of nth order.
12. Inhomogeneous DE with special right side.
13. Method of variation of constants. Credit.

Learning outcomes

• Professional knowledge

After becoming familiar with the basic concepts of differential calculus of functions of two variables, or more variables, the student is able to navigate in specialized subjects of physical focus. Understanding partial derivatives, total differential or gradient is necessary to acquire the basics of higher mathematics for technical universities.

• Professional skills

The student will understand the basic concepts of integral calculus of functions of several variables and some applications using integrals such as applications for the length of a curve, work with a generally defined curve, moments, etc. Knowledge in the area of ​​analytical solution of differential equations is crucial.

• Competence

Familiarity with the presented teaching structure will enable students to orient themselves in the geometric and physical meaning of the mentioned issue. The concept of gradient or directional derivatives will expand the technical imagination of students.