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# KEF/BMS

## Matematický seminář Course: Seminar in Mathematics

Department/Abbreviation: KEF/BMS

Year: 2018 2019

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