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# Matematický seminář

**Course:** Seminar in Mathematics

**Department/Abbreviation:** KEF/BMS

**Year:** 2020

**Guarantee:** 'Mgr. Jan Říha, Ph.D.'

**Annotation:**

- Introduction to vector algebra
- Scalar and vector physical quantities, their properties
- Definition of a vector, vector space
- Arithmetical and geometrical definition of a vector
- Linear combinations of vectors, linearly-dependent and independent system vectors, bases and dimensions of vector space
- Operation with vectors - scalar, vector and mixed product of vectors
- Transformations of vector coordinates in curvilinear systems of coordinates used in physics
- Use of vector calculus in physics

- Introduction to tensor calculus
- Anisotropic media, tensor physical quantities, their properties
- Definition of a tensor
- Algebraic operations with tensors
- Transformations of tensor components
- Tensors in physics

- Differential calculus of a function with one variable
- Real function of one real variable, basic types of functions, their properties
- Limit of a function, basic rules for calculation of function limits
- Differentiation of a function, its physical and geometrical interpretation
- Differential of a function, its physical and geometrical interpretation
- Differentiations of higher orders, physical interpretation of the second differentiation

- Differential calculus of a function with two and more variables
- Real function of more real variables
- Partial differentiation of the first order and higher orders
- Total differential of the first order and higher orders

- Integral calculus of a function with one variable
- Primitive function, indefinite integral
- Basic methods and rules of integration
- Definite integral and its calculation
- Use of definite integral in geometry and physics

- Introduction to solving of differential equations
- Definition of differential equation
- Solving of basic types of the first-order differential equations - equations with separable variables, homogeneous equations, linear equations
- Solving of the second-order differential equations with constant coefficients

- Integral calculus of a function with two and more variables
- Double integral and its calculation
- Triple integral and its calculation

**Course review:**

- Introduction to vector algebra
- Scalar and vector physical quantities, their properties
- Definition of a vector, vector space
- Arithmetical and geometrical definition of a vector
- Linear combinations of vectors, linearly-dependent and independent system vectors, bases and dimensions of vector space
- Operation with vectors - scalar, vector and mixed product of vectors
- Transformations of vector coordinates in curvilinear systems of coordinates used in physics
- Use of vector calculus in physics

- Introduction to tensor calculus
- Anisotropic media, tensor physical quantities, their properties
- Definition of a tensor
- Algebraic operations with tensors
- Transformations of tensor components
- Tensors in physics

- Differential calculus of a function with one variable
- Real function of one real variable, basic types of functions, their properties
- Limit of a function, basic rules for calculation of function limits
- Differentiation of a function, its physical and geometrical interpretation
- Differential of a function, its physical and geometrical interpretation
- Differentiations of higher orders, physical interpretation of the second differentiation

- Differential calculus of a function with two and more variables
- Real function of more real variables
- Partial differentiation of the first order and higher orders
- Total differential of the first order and higher orders

- Integral calculus of a function with one variable
- Primitive function, indefinite integral
- Basic methods and rules of integration
- Definite integral and its calculation
- Use of definite integral in geometry and physics

- Introduction to solving of differential equations
- Definition of differential equation
- Solving of basic types of the first-order differential equations - equations with separable variables, homogeneous equations, linear equations
- Solving of the second-order differential equations with constant coefficients

- Integral calculus of a function with two and more variables
- Double integral and its calculation
- Triple integral and its calculation