учебники и пособия для вузов

математика

учебно-методические пособия

английский язык

физика и математика

This practical class focuses on applying the chain rule to composite functions of several variables, finding derivatives of implicitly defined functions, and computing gradients and directional derivatives.

The class includes theoretical review, five problem blocks with solutions, independent work, and homework. Block 1 covers the chain rule for functions with one and multiple variables. Block 2 addresses implicit differentiation, including singular points. Block 3 presents directional derivative calculations. Block 4 explores gradient properties: direction of steepest increase and maximum growth rate. Block 5 features applied problems from physics and geography.

The class concludes with independent work, homework assignments, problem-solving recommendations, analysis of typical mistakes, and self-assessment questions. This material benefits students in technical and natural sciences programs studying multivariable calculus.

электроника

Practical Session 3. Differentiation of Composite Functions

The lecture focuses on differential calculus for functions of several variables. It begins with the chain rule for differentiating composite functions, covering cases where arguments depend on one or multiple variables. Theory is supported by proofs and examples.

Special attention is given to differentiating implicitly defined functions. Students learn to find derivatives of functions defined by equations, including cases with one or two independent variables. Conditions for theorem applicability are discussed with examples.

The concept of directional derivative is introduced, characterizing the rate of function change in a specified direction. The formula using partial derivatives and direction cosines is derived, with examples in two and three dimensions.

The lecture concludes with a preview of the gradient — a vector pointing in the direction of steepest function increase. Its geometric interpretation and connection to directional derivatives are explained.

современная русская литература

Lecture 3. Differentiation of Composite and Implicit Functions

This lecture introduces functions of several variables, explaining the roles of independent and dependent variables, domain, and range. It covers methods for visualizing such functions, including graphs in three-dimensional space and level lines on a plane.

The core topic is the limit of a function of two variables. For a limit to exist, it must be the same regardless of the path taken to approach the point. Examples demonstrate how different curves can yield different limit values, proving that a limit does not exist. The relationship between double and iterated limits is also discussed.

Finally, the lecture addresses continuity, defining when a function is continuous at a point and classifying discontinuities. It concludes with the Weierstrass and Cauchy theorems, explaining why properties like closedness and connectedness are essential for continuous functions on sets.

книги для подростков

детская познавательная и развивающая литература

саморазвитие и советы

Lecture 1. Introduction to the Analysis of Functions of Several Variables

This lecture explores differentiability for functions of several variables. It begins by defining first-order partial derivatives, explaining that to compute them, all other variables are treated as constants. The geometric meaning is illustrated: a partial derivative represents the slope of the tangent line to a curve created by slicing the surface with a plane.

The core of the lecture introduces the total differential. A function is differentiable if its total increment can be broken down into a linear part plus a negligible error term. This linear part, called the differential, has coefficients that are precisely the partial derivatives.

The relationship between differentiability, continuity, and partial derivatives is examined. While differentiability guarantees both continuity and the existence of partial derivatives, the reverse is not true. The lecture concludes with a sufficient condition: continuous partial derivatives imply differentiability.

социология

Lecture 2. Differentiability of Functions of Several Variables and Partial Derivatives

This practical class reinforces theoretical concepts through hands-on problem-solving. The session focuses on three key skills: finding domains of definition for functions with square roots, logarithms, and fractions by identifying constraints like non-negative radicands and non-zero denominators; computing limits at points where direct substitution fails; and investigating continuity.

A major emphasis is placed on evaluating limits, where students learn why checking multiple paths of approach is essential—different paths yielding different values proves that a limit does not exist. Polar coordinates are introduced as a technique for certain limits.

The class also examines piecewise functions for continuity at specific points by comparing the function's value with the limit along various approaches. The material includes homework problems, solution recommendations, common mistakes to avoid, and control questions for self-assessment.

общественно-политические журналы

современное политическое положение

Practical Class 1. Introduction to FSV

This practical class develops skills in differentiating functions of several variables. Students learn to compute first-order partial derivatives by treating all variables except one as constants.

The session covers finding the total differential, which combines partial derivatives into a linear approximation of the function's change. This tool is applied to approximate calculations near known points using small increments.

A key theoretical point: while differentiability guarantees both continuity and the existence of partial derivatives, the converse is not true—a function can have partial derivatives at a point yet fail to be differentiable there.

The class also includes finding equations of tangent planes and concludes with homework problems, solution tips, and review questions.

сказки

детская фантастика

Practical class 2. Differentiation of Functions of Several Variables

This practical class consolidates key concepts from the first two lectures. Students practice finding domains of definition by solving inequalities involving square roots, logarithms, and inverse trigonometric functions, then sketch the resulting regions.

Limit evaluation is a major focus, with students learning to test different approach paths and use polar coordinates when appropriate. They investigate continuity by comparing function values with limits along various paths.

The session reinforces differentiation skills through computing first-order partial derivatives and total differentials. Students apply these tools to approximate calculations, estimating values near convenient reference points.

A theoretical highlight is investigating differentiability, where students see that a function may have partial derivatives at a point yet fail to be differentiable there. The material includes homework problems, solution strategies, and common pitfalls.

историческая фантастика

социальная фантастика

история философии

книги по философии

Practical Class 3: Functions of Several Variables: Limits and Differentiation

Higher Mathematics – II semester

This lecture introduces line integrals, extending integration from intervals to curves. Two types are covered: scalar line integrals with respect to arc length and vector line integrals. Physical motivations include finding the mass of a wire with variable density and calculating work done by a force field.

Green's Theorem is presented as a powerful tool relating line integrals around closed curves to double integrals over enclosed regions. It is proved for simple regions and applied to compute areas.

The concept of path independence is explored through conservative vector fields. The Fundamental Theorem for Line Integrals shows that conservative fields yield integrals dependent only on endpoints. Conditions for conservativeness are established, including the component test.

Examples illustrate evaluating line integrals, applying Green's Theorem, finding potential functions, and working with both conservative and non-conservative fields.

магия / колдовство

православные традиции

всемирная история

Lecture 1: Line Integrals and Related Theorems

This lecture introduces surface integrals, extending integration from curves to two-dimensional surfaces. Two main types are covered: scalar surface integrals for functions like mass density, and vector surface integrals (flux) for fields like fluid flow.

The concept of parameterized surfaces is presented, along with tangent vectors, normal vectors, and the surface area element. Special attention is given to surfaces given as graphs.

Stokes' Theorem is introduced as a generalization of Green's Theorem, relating line integrals around space curves to surface integrals of curl over bounded surfaces.

The Divergence Theorem relates flux through closed surfaces to volume integrals of divergence over enclosed regions. It is applied to compute flux and derive Archimedes' principle.

The lecture concludes with a summary table of the three great theorems: Green's Theorem, Stokes' Theorem, and the Divergence Theorem.

личная эффективность

саморазвитие / личностный рост

мотивация

Lecture 2: Surface Integrals and Advanced Theorems

This practical session focuses on applying line integral concepts through handsoll problem-solving.

Students work through six problems covering scalar line integrals, work calculations, Green's Theorem applications, area computation using line integrals, and conservative vector fields with potential functions. Each problem includes complete step-by-step solutions.

Key skills developed include parameterizing curves, computing arc length elements, evaluating work done by force fields, applying Green's Theorem to simplify closed curve integrals, using the area formula derived from Green's Theorem, testing vector fields for conservatism, finding potential functions, and applying the Fundamental Theorem for Line Integrals.

The session concludes with homework problems and a summary of key formulas for reference.

общая психология

Practical Session 1: Line Integrals and Related Theorems

This practical session focuses on applying surface integral concepts through hands-on problem-solving.

Students work through seven problems covering scalar surface integrals, surface area calculation, flux integrals, Stokes' Theorem, and the Divergence Theorem. Each problem includes complete step-by-step solutions.

Key skills developed include parameterizing surfaces in cylindrical and spherical coordinates, computing normal vectors and surface area elements, evaluating scalar surface integrals for mass and area calculations, computing flux through cylinders, spheres, and planar surfaces, applying Stokes' Theorem to convert line integrals to surface integrals, using the Divergence Theorem to simplify flux calculations through closed surfaces, and verifying these theorems through explicit computation.

The session concludes with homework problems and a summary of key formulas for reference.

Practical Session 2: Surface Integrals and Advanced Theorems

Higher Mathematics – IV semester

This lecture opens a new section of the mathematical analysis course dedicated to the theory of series. Students will be introduced to the fundamental concepts needed to understand how infinite sums can lead to finite results.

The first part thoroughly examines numerical sequences and their limits—the foundation for the further study of series. Simple examples are used to explain how to determine whether a sequence converges or diverges.

Next, the concept of an infinite series is introduced through the sequence of partial sums. Special attention is given to geometric series—a crucial class of series for which a simple sum formula exists. Telescoping series, where most terms cancel each other out, are also discussed.

The lecture covers the divergence test and teaches students to avoid the common mistake of assuming that a term tending to zero guarantees the convergence of the series.

The material is illustrated with examples from physics and the theory of infinite decimals.

любовное фэнтези

боевое фэнтези

героическое фэнтези

Lecture 3: Introduction to Series

This practical session is designed to reinforce the theoretical material from the first lecture on series. Students will practice calculating sequence limits and learn to determine the convergence of series through concrete examples.

The main focus is on geometric series: students will learn to recognize them among other expressions and find their sum when possible. Problems involving the representation of infinite repeating decimals as common fractions are discussed—a classic application of geometric series.

A separate section is dedicated to telescoping series, where through clever transformations most terms cancel out, and the sum is found using elementary methods. Students will also practice applying the divergence test—the first and simplest tool for checking series.

The session concludes with problems for independent study, allowing students to consolidate the material covered.

S.V. Podkletnova