File archive. StudFiles
Finally, I got my hands on an extensive and long-awaited topic analytical geometry. First, a little about this section of higher mathematics…. Surely you now remembered the school geometry course with numerous theorems, their proofs, drawings, etc. What to hide, an unloved and often obscure subject for a significant proportion of students. Analytic geometry, oddly enough, may seem more interesting and accessible. What does the adjective "analytical" mean? Two stamped mathematical turns immediately come to mind: “graphic method of solution” and “analytical method of solution”. Graphic method, of course, is associated with the construction of graphs, drawings. Analytical same method involves problem solving predominantly through algebraic operations. In this regard, the algorithm for solving almost all problems of analytical geometry is simple and transparent, often it is enough to accurately apply the necessary formulas - and the answer is ready! No, of course, it will not do without drawings at all, besides, for a better understanding of the material, I will try to bring them in excess of the need.
The open course of lessons in geometry does not claim to be theoretical completeness, it is focused on solving practical problems. I will include in my lectures only what, from my point of view, is important in practical terms. If you need a more complete reference on any subsection, I recommend the following quite accessible literature:
1) A thing that, no joke, is familiar to several generations: School textbook on geometry, the authors - L.S. Atanasyan and Company. This school locker room hanger has already withstood 20 (!) reissues, which, of course, is not the limit.
2) Geometry in 2 volumes. The authors L.S. Atanasyan, Bazylev V.T.. This is literature for higher education, you will need first volume. Infrequently occurring tasks may fall out of my field of vision, and the tutorial will be of invaluable help.
Both books are free to download online. In addition, you can use my archive with ready-made solutions, which can be found on the page Download higher mathematics examples.
Of the tools, I again offer my own development - software package on analytical geometry, which will greatly simplify life and save a lot of time.
It is assumed that the reader is familiar with basic geometric concepts and figures: point, line, plane, triangle, parallelogram, parallelepiped, cube, etc. It is advisable to remember some theorems, at least the Pythagorean theorem, hello repeaters)
And now we will sequentially consider: the concept of a vector, actions with vectors, vector coordinates. Further I recommend reading the most important article Dot product of vectors, as well as Vector and mixed product of vectors. The local task will not be superfluous - Division of the segment in this regard. Based on the above information, you can equation of a straight line in a plane with the simplest examples of solutions, which will allow learn how to solve problems in geometry. The following articles are also helpful: Equation of a plane in space, Equations of a straight line in space, Basic problems on the line and plane , other sections of analytic geometry. Naturally, standard tasks will be considered along the way.
The concept of a vector. free vector
First, let's repeat the school definition of a vector. Vector called directed a segment for which its beginning and end are indicated:
In this case, the beginning of the segment is the point , the end of the segment is the point . The vector itself is denoted by . Direction is essential, if you rearrange the arrow to the other end of the segment, you get a vector, and this is already completely different vector. It is convenient to identify the concept of a vector with the movement of a physical body: you must admit that entering the doors of an institute or leaving the doors of an institute are completely different things.
It is convenient to consider individual points of a plane, space as the so-called zero vector. Such a vector has the same end and beginning.
!!! Note: Here and below, you can assume that the vectors lie in the same plane or you can assume that they are located in space - the essence of the material presented is valid for both the plane and space.
Designations: Many immediately drew attention to a stick without an arrow in the designation and said that they also put an arrow at the top! That's right, you can write with an arrow: , but admissible and record that I will use later. Why? Apparently, such a habit has developed from practical considerations, my shooters at school and university turned out to be too diverse and shaggy. In educational literature, sometimes they don’t bother with cuneiform at all, but highlight the letters in bold: , thereby implying that this is a vector.
That was the style, and now about the ways of writing vectors:
1) Vectors can be written in two capital Latin letters: 
etc. While the first letter necessarily denotes the start point of the vector, and the second letter denotes the end point of the vector.
2) Vectors are also written in small Latin letters:
In particular, our vector can be redesignated for brevity by a small Latin letter .
Length or module non-zero vector is called the length of the segment. The length of the null vector is zero. Logically.
The length of a vector is denoted by the modulo sign: ,
How to find the length of a vector, we will learn (or repeat, for whom how) a little later.
That was elementary information about the vector, familiar to all schoolchildren. In analytic geometry, the so-called free vector.
If it's quite simple - vector can be drawn from any point:
We used to call such vectors equal (the definition of equal vectors will be given below), but from a purely mathematical point of view, this is the SAME VECTOR or free vector. Why free? Because in the course of solving problems, you can “attach” one or another vector to ANY point of the plane or space you need. This is a very cool property! Imagine a vector of arbitrary length and direction - it can be "cloned" an infinite number of times and at any point in space, in fact, it exists EVERYWHERE. There is such a student's proverb: Each lecturer in f ** u in the vector. After all, not just a witty rhyme, everything is mathematically correct - a vector can be attached there too. But do not rush to rejoice, students themselves suffer more often =)
So, free vector- This a bunch of identical directional segments. The school definition of a vector, given at the beginning of the paragraph: “A directed segment is called a vector ...”, implies specific a directed segment taken from a given set, which is attached to a certain point in the plane or space.
It should be noted that from the point of view of physics, the concept of a free vector is generally incorrect, and the point of application of the vector matters. Indeed, a direct blow of the same force on the nose or on the forehead is enough to develop my stupid example entails different consequences. However, not free vectors are also found in the course of vyshmat (do not go there :)).
Actions with vectors. Collinearity of vectors
In the school geometry course, a number of actions and rules with vectors are considered: addition according to the triangle rule, addition according to the parallelogram rule, the rule of the difference of vectors, multiplication of a vector by a number, the scalar product of vectors, etc. As a seed, we repeat two rules that are especially relevant for solving problems of analytical geometry.
Rule of addition of vectors according to the rule of triangles
Consider two arbitrary non-zero vectors and : 
It is required to find the sum of these vectors. Due to the fact that all vectors are considered free, we postpone the vector from end vector : 
The sum of vectors is the vector . For a better understanding of the rule, it is advisable to put a physical meaning into it: let some body make a path along the vector , and then along the vector . Then the sum of the vectors is the vector of the resulting path starting at the point of departure and ending at the point of arrival. A similar rule is formulated for the sum of any number of vectors. As they say, the body can go its way strongly zigzag, or maybe on autopilot - along the resulting sum vector.
By the way, if the vector is postponed from start vector , then we get the equivalent parallelogram rule addition of vectors.
First, about the collinearity of vectors. The two vectors are called collinear if they lie on the same line or on parallel lines. Roughly speaking, we are talking about parallel vectors. But in relation to them, the adjective "collinear" is always used.
Imagine two collinear vectors. If the arrows of these vectors are directed in the same direction, then such vectors are called co-directional. If the arrows look in different directions, then the vectors will be oppositely directed.
Designations: collinearity of vectors is written with the usual parallelism icon: , while detailing is possible: (vectors are co-directed) or (vectors are directed oppositely).
work of a nonzero vector by a number is a vector whose length is equal to , and the vectors and are co-directed at and oppositely directed at .
The rule for multiplying a vector by a number is easier to understand with a picture: 
We understand in more detail:
1) Direction. If the multiplier is negative, then the vector changes direction to the opposite.
2) Length. If the factor is contained within or , then the length of the vector decreases. So, the length of the vector is twice less than the length of the vector . If the modulo multiplier is greater than one, then the length of the vector increases in time.
3) Please note that all vectors are collinear, while one vector is expressed through another, for example, . The reverse is also true: if one vector can be expressed in terms of another, then such vectors are necessarily collinear. Thus: if we multiply a vector by a number, we get collinear(relative to original) vector.
4) The vectors are codirectional. The vectors and are also codirectional. Any vector of the first group is opposite to any vector of the second group.
What vectors are equal?
Two vectors are equal if they are codirectional and have the same length. Note that co-direction implies that the vectors are collinear. The definition will be inaccurate (redundant) if you say: "Two vectors are equal if they are collinear, co-directed and have the same length."
From the point of view of the concept of a free vector, equal vectors are the same vector, which was already discussed in the previous paragraph.
Vector coordinates on the plane and in space
The first point is to consider vectors on a plane. Draw a Cartesian rectangular coordinate system and set aside from the origin single vectors and :
Vectors and orthogonal. Orthogonal = Perpendicular. I recommend slowly getting used to the terms: instead of parallelism and perpendicularity, we use the words respectively collinearity and orthogonality.
Designation: orthogonality of vectors is written with the usual perpendicular sign, for example: .
The considered vectors are called coordinate vectors or orts. These vectors form basis on surface. What is the basis, I think, is intuitively clear to many, more detailed information can be found in the article Linear (non) dependence of vectors. Vector basis.In simple words, the basis and the origin of coordinates define the entire system - this is a kind of foundation on which a full and rich geometric life boils.
Sometimes the constructed basis is called orthonormal basis of the plane: "ortho" - because the coordinate vectors are orthogonal, the adjective "normalized" means unit, i.e. the lengths of the basis vectors are equal to one.
Designation: the basis is usually written in parentheses, inside which in strict order basis vectors are listed, for example: . Coordinate vectors it is forbidden swap places.
Any plane vector the only way expressed as: 
, where - numbers, which are called vector coordinates in this basis. But the expression itself 
called vector decompositionbasis .
Dinner served:
Let's start with the first letter of the alphabet: . The drawing clearly shows that when decomposing the vector in terms of the basis, the ones just considered are used:
1) the rule of multiplication of a vector by a number: and ;
2) addition of vectors according to the triangle rule: .
Now mentally set aside the vector from any other point on the plane. It is quite obvious that his corruption will "relentlessly follow him." Here it is, the freedom of the vector - the vector "carries everything with you." This property, of course, is true for any vector. It's funny that the basis (free) vectors themselves do not have to be set aside from the origin, one can be drawn, for example, at the bottom left, and the other at the top right, and nothing will change from this! True, you don’t need to do this, because the teacher will also show originality and draw you a “pass” in an unexpected place.
Vectors , illustrate exactly the rule for multiplying a vector by a number, the vector is co-directed with the basis vector , the vector is directed opposite to the basis vector . For these vectors, one of the coordinates is equal to zero, it can be meticulously written as follows:
And the basis vectors, by the way, are like this: (in fact, they are expressed through themselves).
And finally: , . By the way, what is vector subtraction, and why didn't I tell you about the subtraction rule? Somewhere in linear algebra, I don't remember where, I noted that subtraction is a special case of addition. So, the expansions of the vectors "de" and "e" are calmly written as a sum: 
. Rearrange the terms in places and follow the drawing how clearly the good old addition of vectors according to the triangle rule works in these situations.
Considered decomposition of the form 
sometimes called a vector decomposition in the system ort(i.e. in the system of unit vectors). But this is not the only way to write a vector, the following option is common:
Or with an equals sign:
The basis vectors themselves are written as follows: and
That is, the coordinates of the vector are indicated in parentheses. In practical tasks, all three recording options are used.
I doubted whether to speak, but still I will say: vector coordinates cannot be rearranged. Strictly in first place write down the coordinate that corresponds to the unit vector , strictly in second place write down the coordinate that corresponds to the unit vector . Indeed, and are two different vectors.
We figured out the coordinates on the plane. Now consider vectors in three-dimensional space, everything is almost the same here! Only one more coordinate will be added. It is difficult to perform three-dimensional drawings, so I will limit myself to one vector, which, for simplicity, I will postpone from the origin of coordinates: 
Any 3d space vector the only way expand in an orthonormal basis: 
, where are the coordinates of the vector (number) in the given basis.
Example from the picture: 
. Let's see how the vector action rules work here. First, multiplying a vector by a number: (red arrow), (green arrow) and (magenta arrow). Secondly, here is an example of adding several, in this case three, vectors: . The sum vector starts at the starting point of departure (the beginning of the vector ) and ends up at the final point of arrival (the end of the vector ).
All vectors of three-dimensional space, of course, are also free, try to mentally postpone the vector from any other point, and you will understand that its expansion "remains with it."
Similarly to the plane case, in addition to writing 
versions with brackets are widely used: either .
If one (or two) coordinate vectors are missing in the expansion, then zeros are put instead. Examples:
vector (meticulously 
) – write down ;
vector (meticulously 
) – write down ;
vector (meticulously 
) – write down .
Basis vectors are written as follows:
Here, perhaps, is all the minimum theoretical knowledge necessary for solving problems of analytical geometry. Perhaps there are too many terms and definitions, so I recommend dummies to re-read and comprehend this information again. And it will be useful for any reader to refer to the basic lesson from time to time for better assimilation of the material. Collinearity, orthogonality, orthonormal basis, vector decomposition - these and other concepts will be often used in what follows. I note that the materials of the site are not enough to pass a theoretical test, a colloquium on geometry, since I carefully encrypt all theorems (besides without proofs) - to the detriment of the scientific style of presentation, but a plus for your understanding of the subject. For detailed theoretical information, I ask you to bow to Professor Atanasyan.
Now let's move on to the practical part:
The simplest problems of analytic geometry.
Actions with vectors in coordinates
The tasks that will be considered, it is highly desirable to learn how to solve them fully automatically, and the formulas memorize, don't even remember it on purpose, they will remember it themselves =) This is very important, since other problems of analytical geometry are based on the simplest elementary examples, and it will be annoying to spend extra time eating pawns. You do not need to fasten the top buttons on your shirt, many things are familiar to you from school.
The presentation of the material will follow a parallel course - both for the plane and for space. For the reason that all the formulas ... you will see for yourself.
How to find a vector given two points?
If two points of the plane and are given, then the vector has the following coordinates: 
If two points in space and are given, then the vector has the following coordinates:
I.e, from the coordinates of the end of the vector you need to subtract the corresponding coordinates vector start.
Exercise: For the same points, write down the formulas for finding the coordinates of the vector. Formulas at the end of the lesson.
Example 1
Given two points in the plane and . Find vector coordinates
Decision: according to the corresponding formula:
Alternatively, the following notation could be used:
Aesthetes will decide like this:
Personally, I'm used to the first version of the record.
Answer:
According to the condition, it was not required to build a drawing (which is typical for problems of analytical geometry), but in order to explain some points to dummies, I will not be too lazy: 
Must be understood difference between point coordinates and vector coordinates:
Point coordinates are the usual coordinates in a rectangular coordinate system. I think everyone knows how to plot points on the coordinate plane since grade 5-6. Each point has a strict place on the plane, and they cannot be moved anywhere.
The coordinates of the same vector is its expansion with respect to the basis , in this case . Any vector is free, therefore, if necessary, we can easily postpone it from some other point in the plane. Interestingly, for vectors, you can not build axes at all, a rectangular coordinate system, you only need a basis, in this case, an orthonormal basis of the plane.
The records of point coordinates and vector coordinates seem to be similar: , and sense of coordinates absolutely different, and you should be well aware of this difference. This difference, of course, is also true for space.
Ladies and gentlemen, we fill our hands:
Example 2
a) Given points and . Find vectors and .
b) Points are given 
and . Find vectors and .
c) Given points and . Find vectors and .
d) Points are given. Find Vectors 
.
Perhaps enough. These are examples for an independent decision, try not to neglect them, it will pay off ;-). Drawings are not required. Solutions and answers at the end of the lesson.
What is important in solving problems of analytical geometry? It is important to be EXTREMELY CAREFUL in order to avoid the masterful “two plus two equals zero” error. I apologize in advance if I made a mistake =)
How to find the length of a segment?
The length, as already noted, is indicated by the modulus sign.
If two points of the plane and are given, then the length of the segment can be calculated by the formula
If two points in space and are given, then the length of the segment can be calculated by the formula
Note: The formulas will remain correct if the corresponding coordinates are swapped: and , but the first option is more standard
Example 3
Decision: according to the corresponding formula:
Answer: 
For clarity, I will make a drawing 
Line segment - it's not a vector, and you can't move it anywhere, of course. In addition, if you complete the drawing to scale: 1 unit. \u003d 1 cm (two tetrad cells), then the answer can be checked with a regular ruler by directly measuring the length of the segment.
Yes, the solution is short, but there are a couple of important points in it that I would like to clarify:
First, in the answer we set the dimension: “units”. The condition does not say WHAT it is, millimeters, centimeters, meters or kilometers. Therefore, the general formulation will be a mathematically competent solution: “units” - abbreviated as “units”.
Secondly, let's repeat the school material, which is useful not only for the considered problem:
pay attention to important technical trick – taking the multiplier out from under the root. As a result of the calculations, we got the result and good mathematical style involves taking the multiplier out from under the root (if possible). The process looks like this in more detail: 
. Of course, leaving the answer in the form will not be a mistake - but it is definitely a flaw and a weighty argument for nitpicking on the part of the teacher.
Here are other common cases: 
Often a sufficiently large number is obtained under the root, for example. How to be in such cases? On the calculator, we check if the number is divisible by 4:. Yes, split completely, thus: 
. Or maybe the number can be divided by 4 again? . Thus: 
. The last digit of the number is odd, so dividing by 4 for the third time is clearly not possible. Trying to divide by nine: . As a result:
Ready.
Conclusion: if under the root we get a completely non-extractable number, then we try to take out the factor from under the root - on the calculator we check whether the number is divisible by: 4, 9, 16, 25, 36, 49, etc.
In the course of solving various problems, roots are often found, always try to extract factors from under the root in order to avoid a lower score and unnecessary troubles with finalizing your solutions according to the teacher's remark.
Let's repeat the squaring of the roots and other powers at the same time: 
The rules for actions with degrees in a general form can be found in a school textbook on algebra, but I think that everything or almost everything is already clear from the examples given.
Task for an independent solution with a segment in space:
Example 4
Given points and . Find the length of the segment.
Solution and answer at the end of the lesson.
How to find the length of a vector?
If a plane vector is given, then its length is calculated by the formula.
If a space vector is given, then its length is calculated by the formula 
.
All books can be downloaded for free and without registration.
NEW. PC. Rashevsky. Riemannian geometry and tensor analysis. 3rd ed. 1967 664 pp. djvu. 5.7 MB.
In this monograph, in a detailed presentation and with a comprehensive coverage of the subject, the author presents material that includes the most basic and important in the field of tensor analysis and Riemannian geometry.
A distinctive feature of the book is the way out of the field of pure tensor analysis and Riemannian geometry into mechanics and physics (special attention in this regard is paid to the theory of relativity). Pseudo-Euclidean and pseudo-Riemannian spaces, spaces of affine connection are considered. A number of examples give the basic ideas of the theory of geometric objects, including the theory of spinors in four-dimensional space. The exposition is also supplemented by a number of particular questions of fundamental importance (the theory of curves and hypersurfaces in a Riemannian space, etc.).
The book is intended for specialists in the field of tensor analysis and Riemannian geometry, engineers, and can also serve as a textbook for university students.
By its nature, this book is much closer to a textbook than to a monograph intended for specialists. The material is quite accessible to a third-year student of the university.
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NEW. IN AND. Filippenko. ELEMENTS OF FIELD THEORY. year 2009. 27 pp. PDF. 333 Kb.
The manual discusses the basic concepts of field theory: gradient, divergence, curl, circulation. Applications of the Gauss–Ostrogradsky and Stokes theorems are given. The conditions of potentiality and solenoidality of vector fields are indicated. Detailed solutions of typical examples for calculating the numerical characteristics of a vector field are given. A sufficient number of examples have been selected for independent solution by students.
The manual is intended for part-time students YURGUES.
I recommend reading in the study of electrical and magnetism in general physics.
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Akivis M. A., Goldberg V. V. Tensor calculus: Proc. allowance. 3rd ed., revised. 2003 304 pp. djvu. 2.0 Mb.
The foundations of tensor calculus and some of its applications to geometry, mechanics, and physics are outlined. As applications, a general theory of second-order surfaces is constructed, the tensors of inertia, stresses, and deformations are studied, and some questions of crystal physics are considered. The last chapter introduces the elements of tensor analysis.
For students of higher technical educational institutions.
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Yu.A. Aminov. Vector field geometry. 1990 215 pp. djvu. 5.1 MB.
Results on the geometry of vector fields in three-dimensional Euclidean space are presented, beginning with the work of Foss, Sintsov, Lilienthal, and others. Vector fields in r-dimensional space, systems of Pfaff equations, and external forms are considered. Some topological concepts are briefly outlined and de Rham's theorem is formulated. The Godbillon-Wey invariant of the foliation is introduced, and Whitehead's formula is proved. .
For students, graduate students and researchers in the specialty "geometry and topology". a).
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Anchikov AM Fundamentals of vector and tensor analysis. 1988 140 pages djv. 1.5 MB.
For students of physical and radiophysical specialties of universities and technical colleges who want to learn the course on their own.
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M.A. Akivis, V.V. Goldberg. Tensor calculus. 1969 352 pp. tdjvu. 3.4 MB.
The foundations of tensor calculus and some of its applications to geometry, mechanics and physics are outlined. As applications, a general theory of surfaces of the second order is constructed, the tensors of inertia, stresses, and deflection are studied. The last chapter introduces the elements of tensor analysis.
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Averu J. et al. FOUR-DIMENSIONAL PIMAHOVA GEOMETRY. 175 pp. djvu. 3.9 MB.
Collective monograph written by a group of French mathematicians, edited by Arthur Besse. The book systematically presents results from the field of geometry and analysis, reflects their connection with modern problems of physics. For mathematicians of various specialties, theoretical physicists, graduate students and university students.
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3.T. BAZYLEV, K.I. DUNICHEV. Geometry 2. in 2 volumes. Uch. allowance 1975 368 pp. djvu. 5.4 MB.
Contents: PROJECTIVE SPACE AND IMAGE METHODS. FOUNDATIONS OF GEOMETRY. TOPOLOGY ELEMENTS. LINES AND SURFACES IN EUCLIDEAN SPACE.
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3. T. BAZYLEV, K. I. DUNICHEV, and V. P. IVANITSKA. Geometry. in 2 volumes. Uch. allowance for the 1st course. 1974 353 pp. djvu. 5.1 MB.
This geometry course, published in two books, was compiled on the basis of lectures given by the authors at the Mathematics Department of the Moscow Regional Pedagogical Institute. N. K. Krupskaya. It corresponds to the new program adopted in pedagogical institutes in 1970. The presentation of this course is fully consistent with the new program in algebra and number theory. The course is structured in such a way that such important concepts of modern mathematics as the concepts of set, vector space, mapping, transformation, mathematical structure, constitute a working tool in the study of geometry. The axiomatic method begins to be applied only in the chapter on n-dimensional affine and Euclidean spaces. Prior to this, the material is presented on the basis of those geometric ideas that students have developed while studying the school geometry course. The axiomatics of the school course of geometry and its connections with other axiomatics of geometry are considered in the section on the foundations of geometry (in the second part of the proposed course).
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Borisenko, Tarapov. Vector analysis and the beginnings of tensor calculus. Probably the best book on the subject. The presented material is quite enough to understand the sections of physics (especially useful for electricity and magnetism). There are many useful examples at the end of the book. Size 2.1 Mb.
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Wolf J. Spaces of constant curvature. 1982 480 pages djvu. 6.5 MB.
The book is devoted to classification problems of the theory of spaces of constant curvature-curvature and symmetric spaces. A prominent place in kei is occupied by the author's complete solution of the classical problem of spherical spatial forms. But a much wider range of problems is covered, including a partial classification of pseudo-Riemannian spaces of constant curvature. The first two chapters are an introduction to modern Riemannian geometry.
For scientists and graduate students specializing in geometry, topology, the theory of Lie groups, as well as theoretical physicists and specialists in mathematical crystallography. May be useful for undergraduate students.
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P.B. Gusyatnikov, S.V. Reznichenko. Vector algebra in examples and problems. Textbook. 1985 233 pages djvu. 4.1 MB.
The book is devoted to the age of mining calculus and its application to solving geometric problems. The necessary information from elementary geometry is given, vectors and linear operations on them, scalar, vector and mixed products of vectors are considered.
A fairly simple manual, but the material contained in it should be known to any engineering student.
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Dimitrienko Yu.I. Tensor calculus. Textbook. 2001 575 pages djvu. 5.1 MB.
The textbook covers the main sections of tensor calculus used in mechanics and electrodynamics of continuums, composite mechanics, crystal physics, quantum chemistry: tensor algebra, tensor analysis, tensor description of curves and surfaces, basics of tensor integral calculus. The theory of invariants, the theory of indifferent tensors defining the physical properties of media, the theory of anisotropic tensor functions, as well as the foundations of tensor calculus in Riemannian spaces and spaces of affine connection are presented.
For students and graduate students of higher educational institutions studying in physics, mathematics and engineering specialties.
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V.A. Zhelnorovich. Spinor theory and its applications. year 2001. 401 p.djvu. 3.1 MB
The book contains a systematic exposition of the theory of spinors in finite-dimensional Euclidean and Riemannian spaces; the application of spinors in field theory and relativistic continuum mechanics is considered. The main mathematical part is connected with the study of invariant algebraic and geometric relations between spinors and tensors. The theory of spinors and methods of tensor representation of spinors and spinor equations in four-dimensional and three-dimensional spaces are presented in a special and detailed way. As an application, we consider an invariant tensor formulation of some classes of differential spinor equations containing, in particular, the most important spinor equations of field theory and quantum mechanics; exact solutions of equations for relativistic spin liquids, Einstein-Dirac equations and some non-linear spinor field theory equations are given. The book contains a lot of factual material and can be used as a reference. The book is intended for specialists in the field of field theory, as well as for students and post-graduate students of physical and mathematical specialties.
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P.A. Zhilin. Vectors and tensors of the second rank. >1996. 275 pp. djvu. 1.5 MB.
The book is the first part of a slightly revised lecture notes on the course of theoretical mechanics, which is read by the author to students of the Faculty of Physics and Mechanics. The author had to take into account conflicting requirements. On the one hand, it is a modern advanced course taught to future mechanical engineers-researchers during the third, third and fourth semesters. On the other hand, when reading the course, the author could only count on the fact that students had a command of mathematics in the scope of the school curriculum.
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O.E. Zubelevich. Lectures on tensor analysis. 51 pp. PDF. 281 Kb.
There are two chapters in the lectures: 1. Multilinear algebra, 2. Differential calculus of tensors.
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M.L. Krasnov, A.I. Kiselev, G.I. Makarenko. Vector analysis. Problems and examples with detailed solutions. Textbook. 2007 158 pages djvu. 944 Kb.
The proposed collection of problems can be considered as a short course in vector analysis, in which the main facts are reported without proof and illustrated with specific examples. Therefore, the proposed problem book can be used, on the one hand, to repeat the basics of vector analysis, and on the other hand, as a teaching aid for people who, without going into the proofs of certain proposals and theorems, want to master the technique of vector analysis operations. When compiling the problem book, the authors used the material contained in the available vector calculus courses and collections of problems. A significant part of the problems was compiled by the authors themselves. At the beginning of each section, a summary of the main theoretical provisions, definitions and formulas is given, and a detailed solution of 100 examples is also given. The book contains more than 300 tasks and examples for self-solving. All of them are provided with answers or instructions for the solution. There are a number of problems of an applied nature, which are chosen so that their analysis does not require additional information from the reader from special disciplines. The material of the sixth chapter, devoted to curvilinear coordinates and the basic operations of vector analysis in curvilinear coordinates, was included in the book in order to give the reader at least a minimum number of tasks to acquire the necessary skills.
The collection of problems is intended for students of daytime and evening departments of technical universities, engineers, as well as for part-time students familiar with vector algebra and mathematical analysis in the first two courses.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Download
V.F. Kagan. Fundamentals of the theory of surfaces in tensor presentation. In 2 parts. 1947-1948 years. djvu.
Part 1. Research apparatus. General foundations of the theory and internal geosetry of the surface. 514 pages 16.4 Mb.
Part 2. Surfaces in space. Mapping and bending surfaces. Special questions. 410 pages 14.8 Mb.
A book for those who want to thoroughly understand tensor analysis.
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1 of theoretical physics "I approve" Dean of the Faculty of Physics F. V. Titov 2012 Work program of the discipline Vector and tensor analysis for the specialty Physics, EN.F.3.4 Course: 1 Semester: 2 Lectures: 16 hours. Practical classes: 18 hours. Independent work: 36 hours. Total: 70 hours. Compiled by: Ph.D., Assoc. KTF KemSU Kravchenko N.G. Exam: 2nd semester Kemerovo 2013
3 1. Explanatory note The work program was compiled on the basis of the standard program of the course "Vector and tensor analysis" for the specialty "Physics", the direction "Physics", approved by the UMC in Physics of the UMO of classical universities (Moscow, 2001) and fully complies with the requirements of the State Educational Standard specialty "Physics" (direction "Physics"), approved in 2000. Relevance and significance of the course. Elements of vector and tensor analysis are widely used in all branches of physics. The course is aimed at developing ideas and skills in working with mathematical objects of a tensor nature, which form the basis of an invariant mathematical apparatus, widely used both in general (electricity and magnetism) and in theoretical physics (theoretical mechanics, electrodynamics, fundamentals of continuum mechanics, quantum mechanics). etc.). This course is also the basis for most of the special training courses. Purpose and objectives of the course. To systematize the previously acquired knowledge from mathematical analysis and analytical geometry (the concepts of a scalar, vector, transition from one coordinate system to another, integral theorems of Gauss-Ostrogradsky and Stokes, the concept of a vector flow and circulation of a vector field, etc.); gain new knowledge (the concept of a tensor, working with indices; the ability to work in curvilinear coordinates; differential operators rot, dv and grad; generalized integral theorems, etc.); be able to apply index forms of notation to solving applied problems (solving the simplest problems of electrodynamics, theoretical mechanics and continuum mechanics). The place of discipline in the professional training of specialists. The discipline is included in the professional cycle of general mathematical and natural sciences (EN.F.3.4). This discipline is logically and meaningfully connected with such disciplines and modules of the OOP as: "Analytical Geometry", "Linear Algebra", "Mathematical Analysis" and is necessary when studying the general physics course "Electricity and Magnetism", all courses of theoretical physics. The structure of the academic discipline. This course consists of two parts: Vector Analysis and Tensor Analysis. The issues that make up the main content of the course include: scalar and vector fields, Green's, Ostrogradsky-Gauss, Stokes' theorems, differential operators gradient, divergence, rotor, Laplace operator, basic operations of vector analysis in curvilinear coordinates, potential and solenoidal fields, multilinear functions vector
4 arguments, transformation of tensor coordinates when changing the basis of the linear space. Features of the study of discipline. This course is part of a large section of mathematics "Vector and tensor analysis", but is designed for physics students and a small number of hours are allocated for its study. Therefore, from this huge section of mathematics, the material that is necessary in the study of theoretical courses in physics has been selected. Based on the level of training of students studying at the Faculty of Physics of the KemSU, the traditions of teaching this course at the university there is no section "Elements of group theory". This is due to the separation of this section into independent courses "Group Theory" and "Theory of Symmetry". At the same time, an attempt was made to draw students' attention to the physical content of tensor calculus. Form of organization of lessons on the course. The organization of classes is traditional, for the course "Vector and Tensor Analysis" during one semester lectures are given and practical classes are conducted. However, classes are conducted every other week, which requires students to make certain efforts for the successful organization of practical classes and the assimilation of the material by students. The relationship between classroom and independent work of students. Classroom classes, lectures and practice involve independent work of students in this course. At the lectures, additional topics are offered for independent study, and some calculations are carried out independently. In practical classes, homework is given for independent problem solving and exercises. Requirements for the level of mastering the course content. Freely operate with such mathematical concepts as tensor, vector and scalar; vector field curl and divergence, scalar field gradient. Have the skills to work in different coordinate systems. Be able to apply knowledge of tensor and vector analysis to physical problems. Scope and timing of the course. The course "Vector and Tensor Analysis" is taught in the first year (2nd semester): lectures 1 hour per week (16 hours), practical classes 1 hour per week (18 hours), independent work of students (36 hours). Types of knowledge control and their reporting. The assimilation of the material presented at the lectures is controlled by holding five-minute "lecture dictations" on the basic concepts of previous lectures. The assimilation of each topic covered in the practical lesson is controlled by holding a five-seven minute test. During the semester, eight tests and seven lecture dictations are held. Topics submitted for independent study involve writing essays.
5 Criteria for assessing students' knowledge of the course. To obtain admission to the exam for the course "Vector and Tensor Analysis", you need to attend classroom classes and complete control tasks for a practical and theoretical course. The system for evaluating the work of students is scoring, students who score at least 25% of the maximum possible score are allowed to take the exam. The mark "good" is given when solving two problems of the examination paper. The problem is considered solved if its complete, correct, step-by-step solution with an oral explanation is given. To get an “excellent” grade, in addition to solving the problem, it is necessary to fully and understandably answer two theoretical questions of the ticket. The exam is conducted orally. 2. Thematic plan. Volume of hours Title and content of sections, topics, modules General Classroom work Lectures Practical Laboratory Self-study Forms of control Elements of vector algebra Lecture dictation, verification work 2 Tensor algebra Lecture dictation, verification work 3 Vector analysis - basic definitions 4 Integral theorems of vector analysis, differential characteristics of vector fields Lecture th dictation, verification work Lecture th dictation, verification
6th work 5 Basic operations of vector differentiation 6 Green's formulas and the main theorem of vector analysis Lecture s dictation, test work Lecture s dictation 7 Curvilinear coordinate systems Abstract 8 Elements of group theory Abstract Total: The content of the discipline. Theoretical course. Elements of vector algebra. Scalars. Vectors - definition, addition rule. Opposite vector. Zero vector. Projection of a vector onto an axis. Linear dependence of vectors. Condition of linear independence of three vectors. Decomposition of vectors. Vector basis. Cartesian basis. Scalar, vector, mixed, double cross product of vectors - definition, calculation in Cartesian coordinate system. Transformation of orts of two orthogonal bases. Orthogonal transformations. Orthogonal matrices. Tensor Algebra. General definition of a tensor. Transformation law for orthogonal transformations of coordinate systems. Covariance of tensor equations. Examples. Algebra of tensors: addition, multiplication, convolution of tensors. Symmetric and antisymmetric tensors. is the Kronecker symbol. A sign of the tensority of a quantity. Own and improper orthogonal transformations. Pseudotensors. Levi-Civita pseudotensor. Vector analysis - basic definitions. Vector function of scalar argument. Derivative of a vector function of a scalar argument. tensor field. Differentiation of the tensor field with respect to the coordinate. scalar field. Directional derivative. Gradient. Vector field. Vector lines. Vector line equation. Integral theorems of vector analysis, differential characteristics of vectors. Vector field flow. Ostrogradsky's theorem
7 Gauss for vector fields. Divergence of a vector field. Circulation of a vector field. Stokes' theorem for vector fields. Vector field rotor. Basic operations of vector differentiation. Hamilton operator (). Recording the basic operations of vector differentiation in vector form with an operator and in a Cartesian coordinate system. Recording the basic operations of vector differentiation in tensor form. Vector differential operations of the second order. Laplace operator. Green's formulas and the main theorem of vector analysis. Consequences from integral theorems: 1st and 2nd Green's formulas. The main theorem of vector analysis is the construction of potential and solenoidal vector fields. Curvilinear coordinate systems. Definition. Lame coefficients. local basis. Cylindrical, spherical coordinate systems. Gradient, divergence, rotor, Laplace operator in curvilinear coordinate systems. Elements of group theory. abstract groups. Group theory axioms. Subgroup, conjugate sets. Classes. Isomorphism and homomorphism of groups. Direct product of groups. Group multiplication tables. Practical exercises 1. Vector algebra (vectors, scalars, basic operations with vectors: scalar, vector, mixed product of vectors). 2. Tensor algebra. -Kronecker symbol, Einstein summation rule, differentiation of functions of several variables using index notation (j, x,) x 3. Tensor algebra. Tensors: definition, transformation law (problems on the transformation law, invariant tensors on the example of a -symbol). Optional: differentiation (lesson 2). 4. Tensor algebra. Levi-Civita pseudotensor, even and odd permutations, notation of vector expressions in tensor form. 5. Vector analysis. Gradient: definition (cartesian coordinate system). 1 Consideration of basic examples: r, (a, r), (, a) r in the Cartesian system r Coordinate-free differentiation ((r) r,) r 6. Vector analysis. Divergence of a vector field: definition (cartesian coordinate system), physical meaning with examples. Main tasks dv r 3,
8 dv[ a, r] 0, vector lines. Non-coordinate "vector" differentiation using the properties of divergence: (dv(A B) dva dvb, dv A dva (, A),) 7. Vector analysis. Vector field rotor: definition (Cartesian system), physical meaning with examples. Main tasks: rotr 0, rot[ a, r] 2a. Examples for non-coordinate "vector" differentiation using the properties of the rotor (rot(A B) rota rotb, rot A rota [, A],). 8. Solution of problems on vector differentiation 1 9. Curvilinear coordinate systems. Consideration of basic examples (r, r dvr r, dv, rotr) in cylindrical and spherical coordinate systems. n r 4. List of basic educational literature 1. Gordienko A.B., Zolotarev M.L., Kravchenko N.G. Fundamentals of vector and tensor analysis: a tutorial. Tomsk: from TSPU, p. 2. Zhuravlev Yu.N., Kravchenko N.G. Introduction to the theory of symmetry: teaching aid / GOU VPO "Kemerovo State University". Kemerovo: Kuzbassvuzizdat, p. 3. Keller I. E. Tensor calculus. / St. Petersburg: Lan, 2012, 176 p. (accessed from 4. Fikhtengolts G.M. Course of differential and integral calculus: Textbook. In 3 vols. Volume 3. 9th ed. / St. Petersburg: Lan, 2009, 656 p. (accessed from Additional literature. 1. A. B. Gordienko, M. L. Zolotarev, and Yu. I. Polygalov, Fundamentals of Vector and Tensor Analysis, Part I, Vector Algebra, Guidelines for Independent Work of Students, Kemerovo, KemSU, G. M. Fikhtengol’ts, Course of Differential and integral calculus, M.: Fizmatlit, 2003, vol.3, 723 pp. 3. Polygalov, Yu.I., Guidelines for the course Fundamentals of vector and tensor analysis, Kemerovo, Publishing House of the KemGU, 1988, 82 pp. 4. Gordienko A.B., Zolotarev M.L., Polygalov Y.I. Fundamentals of vector and tensor analysis. Part 2. Fundamentals of vector analysis. Guidelines for independent work of students. Kemerovo, KemGU, Forms of current, intermediate and boundary control. A ) Questions for the exam 1. Scalars.Vectors - definition, addition rule.Opposite vector.Zero vector.Projection onto the axis.
9 2. Linear dependence of vectors. Condition of linear independence of three vectors. Decomposition of vectors. Vector basis. Cartesian basis. 3. Scalar, vector, mixed, double vector product of vectors - definition, calculation in the Cartesian coordinate system. 4. Transformation of orts of two orthogonal bases. Orthogonal transformations. Orthogonal matrices. 5. General definition of a tensor. Transformation law for orthogonal transformations of coordinate systems. 6. Covariance of tensor equations. Examples. 7. Algebra of tensors: addition and multiplication of tensors. 8. Algebra of tensors: convolution of tensors. 9. Symmetric and antisymmetric tensors. -Kronecker symbol (definition, transformation law, rank). 10. A sign of the tensority of a quantity. 11. Own and improper orthogonal transformations. Pseudotensors. 12. Levi-Civita pseudotensor. Writing a vector product in tensor form. 13.Vector-function of scalar argument. Derivative. 14. Tensor field. Differentiation of the tensor field with respect to the coordinate. 15. Scalar field. Directional derivative. Gradient. 16.Vector field. Vector lines. Vector line equation. 17. Flow of a vector field. 18. Ostrogradsky-Gauss theorem for vector fields (statement). Divergence of a vector field. 19. Circulation of a vector field. Stokes' theorem for vector fields. Vector field rotor. 20. Recording the basic operations of vector differentiation in vector form with an operator and in a Cartesian coordinate system. 21. Recording the basic operations of vector algebra and vekton differentiation in tensor form: A, B, A B, A, B, C, dva, rota. 22.Vector differential operations of the second order. 23. Consequences from integral theorems: Green's first formula. 24. Consequences from integral theorems: Green's second formula. 25. Main theorem of vector analysis. Construction of a solenoidal vector field. 26. Main theorem of vector analysis. Construction of a potential vector field. 27. Curvilinear coordinates. 28. Scalar field gradient in orthogonal curvilinear coordinate systems. 29. Divergence of a vector field in orthogonal curvilinear coordinate systems.
10 30. Vector field rotor in orthogonal curvilinear coordinate systems. 31. Laplace operator in orthogonal curvilinear coordinate systems. 32.Axioms of group theory. 33. Subgroup, conjugate sets. Subgroup index. 34. Classes. 35. Direct product of groups. An approximate list of tasks submitted for the exam 1. Operations with vectors. 1.1 Calculate [ A, B ] and A, B) for vectors: A 5 6 j 3 and A 1 1j A 5 4 j 3 and A 3 j Calculate (C,[ A, B]) for vectors: (1) A 11 6 j 2, B 10 7 and C A j 2, B 10 7 and C 3 2 j 1.3 Show by direct calculation that [ A,[ B, C]] [[ A, B], C] : (1) A 11 6 j 2, B 10 7 and C A j 2, B 10 7 and C 3 2 j 1.4 Show by direct calculation that [ A,[ B, C]] B(A, C) C(A, B) : ( 1) A 11 6 j 2, B 10 7 and C 2 3 j A j, B 10 7 and C 2 j 1.5 Calculate the volume of the pyramid ABCD, whose vertices have coordinates: (2) A(1,-1,0), B(2,3,1), C(-1,1,1), D(4,3,-5) A(2,0,3), B(1,1,1), C(4, 6,6), D(-1,2,3) 2. Sum the expression with -symbol: 2.1 Al m mj n 2.2 A B l lm l n mp 2.4 l lj j 2.4 Cm ml 2.27 j m jm m n jn n 2.28 n m nm mm m nn n mn
11 All lm 3.6 B x x x 2 A x x m A x x m 2 Bm x x l 2 T x x l 3.8 B l 4.20 x m A m exp x r] (a, r) 5.5 grad 3 r 5.4 rot[ a, r] [ a, r] 5.6 rot 3 r 5.7 dv ar 5.8 rot ar 5.9 dv r ln 2 (a, r) 5.10 grad r ln 2 (a , r) ln(a, r) 5.12 (b,)[ a, r] 5.13 (r,)[ r, rb] 5.14 dv r ln r 2. define a tensor of the n-th rank 3. write down the rule for adding tensors 4. write down the rule for multiplying tensors 5. give a definition of a pseudotensor 6. give a definition of Levi Civita's pseudotensor. 7. indicate how the rank of the tensor changes when it is differentiated by a scalar argument 8. indicate how the rank of the tensor changes when it is differentiated by the coordinates of the radius vector m
12 9. write the operator in a Cartesian coordinate system 10. define the flow of a vector field 11. physical meaning of divergence 12. formulate the Stokes theorem for vector fields 13. physical meaning of the curl 14. calculate dv grad 15. calculate dv rota 16. calculate rot grad a , b, c 17. write in tensor form d) Approximate topics of abstracts. 1. Systems of curvilinear coordinates. 2. Toroidal coordinate system. Laplacian of a scalar function. 3. Three-dimensional parabolic coordinates. Laplacian of a scalar function. 4. Ellipsoidal coordinates. Laplacian of a scalar function. 5. Paraboloidal coordinates. Laplacian of a scalar function. 6. Bicylindrical coordinates. Laplacian of a scalar function. 7. Bipolar coordinates. Laplacian of a scalar function. 8. Parabolic coordinates. Laplacian of a scalar function. 9. Conic coordinates. Laplacian of a scalar function. 10. Coordinates of an elliptical cylinder. Laplacian of a scalar function. 11. Coordinates of a parabolic cylinder. Laplacian of a scalar function. 12. Toroidal coordinate system. Scalar function gradient. 13. Three-dimensional parabolic coordinates. Scalar function gradient. 14. Ellipsoidal coordinates. Scalar function gradient. 15. Paraboloidal coordinates. Scalar function gradient. 16. Bicylindrical coordinates. Scalar function gradient. 17. Bipolar coordinates. Scalar function gradient. 18. Parabolic coordinates. Scalar function gradient. 19. Conic coordinates. Scalar function gradient. 20. Coordinates of an elliptical cylinder. Laplacian of a scalar function. 21. Coordinates of a parabolic cylinder. Laplacian of a scalar function. 22. Permutation group. 23. Mathieu group. 24. Transformations of space. 25. Point symmetry groups. 26. Reducible and irreducible representations 27. Multiplication of symmetry operations 28. Generators of point groups.
MINISTRY OF EDUCATION AND SCIENCE OF THE RUSSIAN FEDERATION Federal State Budgetary Educational Institution of Higher Professional Education "Kemerovo State University" Department
I. Abstract. The work program is compiled on the basis of the state educational standard of higher professional education for the course "Vector and tensor analysis" and the curriculum for the specialty
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2 COMPILERS: N.G. Abrashina-Zhadayeva - Head of the Department of Higher Mathematics and Mathematical Physics of the Belarusian State University, Doctor of Physical and Mathematical Sciences of the Russian Federation,
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MINISTRY OF EDUCATION AND SCIENCE OF THE RUSSIAN FEDERATION Federal State Budgetary Educational Institution of Higher Professional Education "Kemerovo State University"
MINISTRY OF EDUCATION AND SCIENCE OF THE RUSSIAN FEDERATION Federal State Budgetary Educational Institution of Higher Professional Education "Kemerovo State University" Department
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Federal Agency for Fisheries Kamchatka State Technical University Faculty of Information Technology Department of Higher Mathematics "APPROVED" Dean of FEU Rychka I.A. "" 007 WORKING
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MINISTRY OF EDUCATION AND SCIENCE OF THE RUSSIAN FEDERATION Federal State Autonomous Institution of Higher Professional Education "Kazan (Volga Region) Federal University" Institute
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LA Svirkina Candidate of Physical and Mathematical Sciences, Associate Professor of the Department of Higher Mathematics, Director of the Center for Additional Educational Programs, St. Petersburg State University
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CONTENTS Preface .................................................................. ........... 5 1. Elements of linear algebra .............................. ........... 6 IDZ 1. Qualifiers .................................. ............
MINISTRY OF EDUCATION AND SCIENCE OF THE REPUBLIC OF KAZAKHSTAN West Kazakhstan State University named after m.utemisov WORKING CURRICULUM Topical issues of mathematical analysis 6M060100 Mathematics
MINISTRY OF EDUCATION AND SCIENCE OF THE RUSSIAN FEDERATION Federal State Budgetary Educational Institution of Higher Professional Education "Kemerovo State University" Mathematical
7. Covariant formulation of Maxwell's equations and dynamic equations for potentials. Dynamic (differential) equations for electromagnetic field potentials. We substitute the definition of potentials
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Federal Agency for Education
GOU VPO ½Syktyvkar State University\
Yu.N. BELYAEV
INTRODUCTION TO VECTOR
Tutorial
Syktyvkar 2008
ÓÄÊ 514.742.4(075) ÁÁÊ 22.14
Published by order of the editorial and publishing council of Syktyvkar State University
Ðå ö å í ç å í ò û:
Department of Mathematical Analysis, Komi State Pedagogical Institute,
G.V. Ufimtsev, Ph.D. Phys.-Math. Sciences, Associate Professor, Syktyvkar Forest Institute
Belyaev Yu.N.
Á 43 Introduction to Vector Analysis: Tutorial. Syktyvkar: Iz-vo SyktGU, 2008. 215 p.: ill.
ISBN 978-5-87237-601-1
This manual contains basic information from the algebra of vectors.
The rules for differentiating a vector function with respect to a scalar argument are demonstrated using examples from mechanics, in particular from the kinematics of a material point and an absolutely rigid body.
The main functions of the point are the gradient of the scalar field, the divergence and the vortex of the vector field are given in a form invariant with respect to the choice of the coordinate system. The integral representation of the vortex and the divergence of the vector field are used to prove the Ostrogradsky and Stokes theorems. A selection of formulas for the gradient, divergence, curl and Laplace operator in some orthogonal coordinate systems, as well as tasks for independent work of students with examples of solving typical problems used to control the assimilation of the material are given.
The book is intended for students of physical specialties.
c Belyaev Yu.N., 2008
c Syktyvkar State University, 2008
ISBN 978-5-87237-601-1
1.5. Multiplying a vector by a number. . . . . . . . . . . . . ten
1.6. Addition of vectors. . . . . . . . . . . . . . . . . . . eleven
1.7. Basic properties of vectors. . . . . . . . . . . . . . eleven
1.8. polygon rule. . . . . . . . . . . . . . . . thirteen
1.9. Difference of vectors. . . . . . . . . . . . . . . . . . . fourteen
Ÿ 2. Examples of vectors. . . . . . . . . . . . . . . . . . . . . . 17
2.1. The radius vector of a point. . . . . . . . . . . . . . . . . . 17
2.2. Movement, speed and acceleration. . . . . . . . . 22
2.3. The concept of strength. . . . . . . . . . . . . . . . . . . . . . 25
Tasks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Ÿ 3. Linear space. . . . . . . . . . . . . . . . . . . 28
3.1. Examples of linear spaces. . . . . . . . . . . 29
3.2. Dimension and basis of a linear space. . . . 34
4.1. Vector basis. . . . . . . . . . . . . . . . . . . . 38
4.2. Vector coordinate properties. . . . . . . . . . . . . . 39
4.3. The dimension of the vector set. . . . . . . . . . 40
Tasks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Ÿ 5. Vector projections. . . . . . . . . . . . . . . . . . . . . . 43
5.1. Projection of a vector onto a plane. . . . . . . . . . . . 43
5.2. Projection of a vector onto an axis. . . . . . . . . . . . . . . . 44
5.3. Properties of the projection of a vector onto an axis. . . . . . . . . . 45
Ÿ 6. Application to trigonometry. . . . . . . . . . . . . . . 46
6.1. Projections of a unit vector. . . . . . . . . . . . . 46
6.2. The trigonometric form of the projection. . . . 46
6.3. Basic trigonometric identity. . . . . . 47
6.4. Casting formulas. . . . . . . . . . . . . . . . . . 47
6.5. Sine theorem. . . . . . . . . . . . . . . . . . . . . 47
Tasks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . | |
Ÿ 7. Vector in an orthonormal basis. . . . . . . . . . . |
7.1. Vector coordinates in an orthonormal basis. fifty
7.2. The length of the vector. . . . . . . . . . . . . . . . . . . . . 52
7.3. Direction cosines. . . . . . . . . . . . . . . . 52
7.4. Angle between directions. . . . . . . . . . . . . . 52
7.5. Radius vector in Cartesian coordinates. . 53
7.6. Determination of the vector sum by the projection method. 55
Tasks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Ÿ 8. Scalar product of vectors. . . . . . . . . . . . . 59
8.1. Properties of the scalar product. . . . . . . . . . 60
8.2. Euclidean space. . . . . . . . . . . . . . . 61
8.3. Cosine theorem. . . . . . . . . . . . . . . . . . . 61
8.4. The scalar product in an orthonormal basis. . . . . . . . . . . . . . . . . . . . . . . . 63
Tasks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Ÿ 9. Vector product of vectors. . . . . . . . . . . . . 68
9.1. Vector product properties. . . . . . . . . . 69
9.2. Vector product in an orthonormal basis. . . . . . . . . . . . . . . . . . . . . . . . 70
9.3. Cross product expression in terms of
determinants of the second and third orders. . . . . . | ||
Tasks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . | ||
Ÿ 10. Products of three vectors. . . . . . . . . . . . . . . . | ||
10.1. Mixed work. . . . . . . . . . . . . . . | ||
10.2. Double vector product. . . . . . . . . . | ||
2. Vector-function of a scalar argument | ||
Derivative of a vector function with respect to a scalar argument | ||
1.1. The geometric meaning of the derivative. . . . . . . . . 79
1.2. Basic properties of derivatives. . . . . . . . . . . 82
Ÿ 2. Integral of a vector. . . . . . . . . . . . . . . . . . . . . 89
Ÿ 3. Axes of a natural trihedron. . . . . . . . . . . . . . 91
3.1. Frenet formulas. . . . . . . . . . . . . . . . . . . . . 93
3.2. Velocity and acceleration in the axes of the natural trihedron. 96
3.3. Calculation of the curvature of a spatial curve. . 99
3.4. Torsion of a spatial curve. . . . . . . . . 103 Tasks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Ÿ 4. Curvilinear orthogonal coordinate systems. . . 104
4.1. Basis vectors and Lame coefficients. . . . . . 107
4.2. Velocity and acceleration of a material point in a curvilinear orthogonal coordinate system. 108
Tasks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Ÿ 5. Addition of movements. Application to kinematics. . . . . 112
5.1. Movement of the reference system. Angular velocity. 113
5.2. Velocities of points of a rigid body. . . . . . . . . . . . . 116
5.3. Rigid body accelerations. . . . . . . . . . . . . . . . 118
5.4. Absolute speed of movement of a material point. . . . . . . . . . . . . . . . . . . . . . . . . 120
5.5. Addition of accelerations. . . . . . . . . . . . . . . . . . 125 Tasks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
Chapter 3. Point functions |
Ÿ 1. Scalar field. . . . . . . . . . . . . . . . . . . . . . . . 133
1.1. Level surface of a scalar field. . . . . . . . 133
1.2. Differentiable scalar field. . . . . . . . . 134
1.3. Directional derivative. . . . . . . . . . . . . 135
1.4. The geometric meaning of the gradient. . . . . . . . . . . 136
1.5. sum gradient. . . . . . . . . . . . . . . . . . . . . 137
1.6. Complex function gradient. . . . . . . . . . . . . . 141
1.7. Gradient in orthogonal coordinate system. . . . 143 Tasks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
Ÿ 2. Vector field. . . . . . . . . . . . . . . . . . . . . . . . 148
2.1. Vector line equation. . . . . . . . . . . . . . 148
2.2. Curvilinear integral of a vector field. . . . 151
2.3. Calculation of the curvilinear integral. . . . . . . 153
2.4. Vortex of the vector field. . . . . . . . . . . . . . . . . 156
3.1. Speed flow. . . . . . . . . . . . . . . . . . . . . 164
3.2. Vector field flow. . . . . . . . . . . . . . . . . 166
3.3. Normal to the surface. . . . . . . . . . . . . . . . . 167
3.4. Flow calculation. . . . . . . . . . . . . . . . . . . 168
3.5. flow through a closed surface. . . . . . . . . 170 Tasks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
Ÿ 4. Divergence of the vector field. . . . . . . . . . . . . . . 171
4.1. Discrepancy in orthogonal coordinate system. 172
4.2. Solenoid vector field. Vector potential. . . . . . . . . . . . . . . . . . . . . . . . 175
4.3. Laplace vector field. . . . . . . . . . . . . . . 175 Tasks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
Ÿ 5. Symbolic designations of the main differentials
social operations. . . . . . . . . . . . . . . . . . . . . 177
5.1. Symbol vector nabla. . . . . . . . . . . . . 177
5.2. Laplace operator, . . . . . . . . . . . . . . . . . . . 178
5.3. The derivative of a vector with respect to another vector. . . . . 179
5.4. Differential operations from products of functions. . . . . . . . . . . . . . . . . . . . . . . 179
5.5. Differential operations of the second order. . 183 Tasks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
Ÿ 6. Some orthogonal coordinate systems. . . . . . 184
6.1. System of cylindrical coordinates. . . . . . . . . 185
6.2. Spherical coordinate system. . . . . . . . . . . 186
6.3. System of parabolic cylindrical coordinates. . . . . . . . . . . . . . . . . . . . . . . . . 187
6.4. System of paraboloidal coordinates. . . . . . . . 188
6.5. System of elliptic cylindrical coordinates. 189
6.6. System of prolate ellipsoidal coordinates. 190 Tasks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
Ÿ 7. Stokes' theorem. . . . . . . . . . . . . . . . . . . . . . . . 192 Tasks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
Ÿ 8. Ostrogradsky's theorem and related formulas. 195
8.1. Ostrogradsky's theorem. . . . . . . . . . . . . . . . 195
8.2. The formula for the gradient. . . . . . . . . . . . . . . . 201
8.3. The formula for the vortex. . . . . . . . . . . . . . . . . . . 201
8.4. Green's formulas. . . . . . . . . . . . . . . . . . . . . 202 Tasks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
Bibliographic list. . . . . . . . . . . . . . . . . . . 207 Answers and solutions. . . . . . . . . . . . . . . . . . . . . . . . 209
BASIC INFORMATION FROM THE ALGEBRA OF VECTORS
Ÿ 1. The geometric concept of a vector
1.1. Introduction. One of the basic geometric concepts, vector, arose as a mathematical abstraction of objects characterized by magnitude and direction in the works of several scientists almost simultaneously in the middle of the 19th century. The first vector calculus on a plane was developed in 1835 by the Italian scientist Bellavitis (Guito Bellavitis, 1835-1880). Around the same time, the work of Argan (Jean Robert Argand, 1768-1822) and Wessel (Caspar Wessel, 17451818) on the geometric interpretation of complex numbers gained fame. The final formulation of vector algebra was carried out by Hermann Grassmann (18091877), William Rowen Hamilton (18051865) and J.W. Gibbs (Josiah Willard Gibbs, 1839-1903).
The concept of a vector plays an important role in modern mathematics and its applications, for example, in mechanics, the theory of relativity, electrodynamics, quantum physics, and other branches of natural science.
1.2. Scalars and vectors. Quantities are called scalar (scalars) if, after choosing a unit of measure, they are completely characterized by one number. Examples of scalars are time t, volume V, mass m, temperature T, work done by a force A, electric charge q, etc.
Two scalars of the same dimension are equal if, when measured with the same unit of measure, the same
bad numbers.
Quantities such as speed ~v, acceleration ~a, force F, on-
electric field strength E, requiring for their
giving not only an indication of a numerical value, but also a direction in space, are called vector quantities, or
vectors.
The terms scalar (1843) and vector (1845) were coined by Hamilton, who derived them respectively from the Latin words scale and vector bearing.
The simplest scalar is an abstract number, and the simplest vector is a straight line segment having a certain length and a certain direction from the starting point of the segment to its end point.
1.3. Image and notation of vector quantities. There are several different forms of notation for vector quantities. One of the oldest dashes above the letter. This is how Argan designated the directed segment. Maxwell (James Clerk Maxwell, 1831-1879) denoted vectors in Gothic letters, Hamilton and Gibbs in Greek letters. The designation of vectors in bold letters was proposed by Oliver Heaviside, 1850-1925).
In this paper, geometric vectors are denoted by the letters
you with an arrow at the top: ~a, b, ~c, etc. Sometimes we will
be a vector whose start point is A and end point
B, symbol AB. In the figures, vectors are depicted as straight line segments having not only a certain length, but also a certain direction, indicated by an arrow at the end of the segment.
The length of a vector, otherwise known as the modulus of the vector, is denoted by the same letter as the vector, but without the arrow. Sometimes, to denote the module of a vector, the designation of the vector itself is taken, placed in straight brackets. For example, p = jp~j is the modulus of the vector p~.
Zero-vector vector 0, whose length is zero, can have any direction in space.
The angle between the vectors p~ and q~ is the smallest angle by which one vector must be rotated so that it coincides in direction with the other (Fig. 1). We will denote this angle as
ox (p;~ q~).
Ÿ 1. The geometric concept of a vector |
1.4. Vector equality. When we compare vector physical quantities, it is assumed that they have the same physical dimension.
There are three different types of vectors. Each of them combines a set of vectors with the same properties.
Free vectors are determined by the direction of the line of action and the modulus. Such vectors are equal if they are equal in magnitude.
f = g and are equally directed, i.e. (f; ~g) = 0: Other
In other words, we do not distinguish between two free vectors f and ~g, which have different application points and are obtained from one another by parallel translation.
The equality of two vectors f and ~g is symbolically written as follows:
Linked vector. To determine the associated vector AB, you need to specify its line of action (in Fig. 2a this is the line xx0 ), the direction along this line (from x to x0 ), its origin (A) and the length of the vector. Linked vectors are vectors whose equivalence requires that they be equal in length, have the same direction, and have a common origin. An example of such a vector is the force applied to some point of the elastic
) (p;~ q~) | |||||
Sliding vector. The definition remains the same as in the previous case, if we exclude the requirement to fix the beginning of the vector. It can be located anywhere on the xx0 axis. Sliding vectors are those that are considered
are equivalent if they are equal in absolute value, equally
directed and lie on the same straight line (for example, AB = A B on (Fig. 2b)). Examples of such vectors are forces spreading
viewed in static mechanics.
Since the direction of the null vector is not specified, all null vectors are assumed to be equal.
All the following rules, in particular the multiplication of a vector by scalars and the rule of vector addition, will be given in relation to free vectors. It is not difficult to extend these definitions to coupled and sliding vectors.
1.5. Multiplying a vector by a number. When vector ~a is multiplied by a real number, a vector ~c is obtained, such that its modulus is equal to j ja, and directed in the same direction as the vector ~a for > 0, and in the opposite direction if< 0. Умножение любого вектора ~a на нуль дает нулевой вектор. Таким образом,
~c; c = a; (~c;~a) = 0; If > 0; | |||||
0; If = 0:d | |||||
a; (~c;~a) = ; If< 0; |
|||||
The vectors ~c and ~a related by equality (1.1) are parallel to each other or lie on the same straight line. Such vectors are called collinear 1 .
On fig. As an example, Figure 3 shows the vector ~a and the vectors 2~a and 0:5~a resulting from multiplication by the numbers 2 and 0:5.
In accordance with the given definition of multiplication of a vector by a number, any vector ~a can be represented as a product
~a = a~ea ; |
1 The term is derived from the Latin co together èlinearis linear and literally means ½colinearity\. Hamilton in his vector calculus
In addition, he introduced the name termino-collinear for vectors that have a common beginning and whose ends lie on the same straight line. This name was simplified by Gibbs, thanks to whom the term ½collinearity \ entered the vector
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