Course: Seminar in Mathematics

Department/Abbreviation: KEF/BMS

Year: 2021 2022

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