The training manual presents practical exercises on vector analysis, matrix calculus and their application to the solution of systems of linear algebraic equations and reduction of the quadratic form to the sum of squares, as well as exercises on iterative methods for solving algebraic and transcendental equations, solving linear first-order partial differential equations .
The goal is to help you learn how to solve the tasks on these sections of the course of higher mathematics with the least amount of time.
All training material is divided into separate practical exercises
Practical assignments are divided into the following topics:
The first practical lesson. Numerical solution of algebraic equations.
The second practical lesson. Numerical solution of algebraic equations (continuation).
The third practical lesson. Solution of transcendental equations.
The fourth practical lesson. Basic definitions of the theory of matrices.
The fifth practical lesson. Matrix multiplication. Formulas for testing the multiplication of matrices. The inverse matrix and methods for obtaining it.
The sixth practical lesson. The inversion of a triangular matrix. The decomposition of a square matrix into a product of two triangular matrices. Calculation of the inverse matrix by representing it in the form of two triangular matrices.
Seventh practical lesson. Matrix representation of a system of linear algebraic equations. Numerical solution of linear algebraic
equations by elimination.
The eighth practical lesson. Characteristic equation of the matrix.
The trace of the matrix. Characteristic numbers and eigenvectors of the matrix. Rationing of the vector. Scalar product of two vectors. Orthogonal matrices. Transformation of the characteristic equation by the Leverrier method.
Ninth practical lesson. Transformation of the characteristic equation by the method of Academician AN Krylov. The Cayley-Hamilton theorem.
The tenth practical lesson. The application of matrices to the reduction of the quadratic form of two variables to the sum of squares (to the canonical form). Simplification of the equations of second-order curves.
The eleventh practical lesson. Level surfaces. Derivative direction. Gradient function.
The twelfth practical lesson. Vector field. Potential vectors. The potential of a vector field. Circulation of the vector. Linear integral. The vortex of the vector.
Thirteenth practical lesson. Flow of a vector field. Divergence of the vector. Ostrogradsky's formula.
Fourteenth practical lesson. Divergence properties. Exercises related to Ostrogradsky and Stokes formulas.
Fifteenth practical lesson. Harmonic functions. Green's formula.
Sixteenth practical lesson. The Hamilton operator.
The seventeenth practical lesson. Curvilinear coordinates. Orthogonal curvilinear coordinates. Recording in the orthogonal curvilinear coordinates of the basic differential field theory operations: gradient, divergence, rotor and Laplace operator. Expressions of the gradient, divergence, rotor and Laplace operator in cylindrical and spherical coordinate systems.
Eighteenth practical lesson. Integration of first-order linear differential equations with partial derivatives.
The goal is to help you learn how to solve the tasks on these sections of the course of higher mathematics with the least amount of time.
All training material is divided into separate practical exercises
Practical assignments are divided into the following topics:
The first practical lesson. Numerical solution of algebraic equations.
The second practical lesson. Numerical solution of algebraic equations (continuation).
The third practical lesson. Solution of transcendental equations.
The fourth practical lesson. Basic definitions of the theory of matrices.
The fifth practical lesson. Matrix multiplication. Formulas for testing the multiplication of matrices. The inverse matrix and methods for obtaining it.
The sixth practical lesson. The inversion of a triangular matrix. The decomposition of a square matrix into a product of two triangular matrices. Calculation of the inverse matrix by representing it in the form of two triangular matrices.
Seventh practical lesson. Matrix representation of a system of linear algebraic equations. Numerical solution of linear algebraic
equations by elimination.
The eighth practical lesson. Characteristic equation of the matrix.
The trace of the matrix. Characteristic numbers and eigenvectors of the matrix. Rationing of the vector. Scalar product of two vectors. Orthogonal matrices. Transformation of the characteristic equation by the Leverrier method.
Ninth practical lesson. Transformation of the characteristic equation by the method of Academician AN Krylov. The Cayley-Hamilton theorem.
The tenth practical lesson. The application of matrices to the reduction of the quadratic form of two variables to the sum of squares (to the canonical form). Simplification of the equations of second-order curves.
The eleventh practical lesson. Level surfaces. Derivative direction. Gradient function.
The twelfth practical lesson. Vector field. Potential vectors. The potential of a vector field. Circulation of the vector. Linear integral. The vortex of the vector.
Thirteenth practical lesson. Flow of a vector field. Divergence of the vector. Ostrogradsky's formula.
Fourteenth practical lesson. Divergence properties. Exercises related to Ostrogradsky and Stokes formulas.
Fifteenth practical lesson. Harmonic functions. Green's formula.
Sixteenth practical lesson. The Hamilton operator.
The seventeenth practical lesson. Curvilinear coordinates. Orthogonal curvilinear coordinates. Recording in the orthogonal curvilinear coordinates of the basic differential field theory operations: gradient, divergence, rotor and Laplace operator. Expressions of the gradient, divergence, rotor and Laplace operator in cylindrical and spherical coordinate systems.
Eighteenth practical lesson. Integration of first-order linear differential equations with partial derivatives.
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