# Vector Calculus

• I. N. Bronshtein
• K. A. Semendyayev

## Abstract

Scalar and vector quantities. Quantities whose values can be expressed by means of positive or negative numbers (scalars) are called scalar quantities (e.g., mass, temperature, work and so on). Quantities whose specification requires both the magnitude and the direction in the space (e.g., force, velocity, acceleration, magnetic field intensity and so on) are called vector quantities and can be expressed by means of vectors.

## Keywords

Vector Field Scalar Field Radius Vector Field Versus Reciprocal System
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Notes

1. (1).
According to this definition, a vector remains unchanged by a parallel displacement of its initial point to an arbitrary point of the space; such vectors are called free. In some problems of mechanics we consider vectors whose initial point is fixed in the space (so-called bound vectors) or vectors which can be displaced only along a straight line (sliding vectors).Google Scholar
2. (1).
In this chapter the coordinate system is assumed to be right-handed (see p. 256).Google Scholar
3. (1).
The upper indices should not be confused with the exponents of powers. Such a notation is convenient, for the scalars a1, a2, a3 are contravariant coordinates of the vector a (see p. 621).Google Scholar
4. (1).
We consider here only non-zero vectors.Google Scholar
5. (1).
The last member of equation (1″) is written as a summation convention used in the tensor calculus: instead of the whole sum, only its general term is written. The same index denoted by a Greek letter (the summation index) which appears in the general term as a subscript and a superscript assumes the values 1, 2, 3, so that this term stands for the sum of the terms obtained by giving the summation index each of the values 1, 2, 3. Thus $$\begin{array}{*{20}{c}} \hfill {{{g}_{{{{\alpha }_{\beta }}}}}{{a}^{\alpha }}{{a}^{\beta }} = {{g}_{{11}}}{{a}^{1}}{{b}^{1}} + {{g}_{{12}}}{{a}^{1}}{{b}^{2}} + {{g}_{{13}}}{{a}^{1}}{{b}^{3}} + {{g}_{{21}}}{{a}^{2}}{{b}^{1}} + {{g}_{{22}}}{{a}^{2}}{{b}^{2}} + {{g}_{{22}}}{{a}^{2}}{{b}^{3}} + } \\ \hfill { + {{g}_{{31}}}{{a}^{2}}{{b}^{1}} + {{g}_{{22}}}{{a}^{3}}{{b}^{2}} + {{g}_{{22}}}{{a}^{2}}{{b}^{3}}.} \\ \end{array}$$.Google Scholar
6. (1).
See footnote on p. 620.Google Scholar
7. (1).
Sometimes the term plane field is used to mean a field defined for points of the space such that it is constant at all points of a straight line parallel to a given direction. Such a field should rather be culled a plane-parallel field; its investigation reduces to the investigation of the field in a plane perpendicular to the given fixed direction.Google Scholar
8. (1).
9. (1).
See footnote on p. 625. An analogous remark is true for a vector field.Google Scholar
10. (1).
For solution of such differential equations see pp. 517, 529.Google Scholar
11. (2).
For the operator ∇ (nabla) see p. 643.Google Scholar
12. (1).
We assume here and in the following that c and c are constants.Google Scholar
13. (2).
For the expressions (V grad) W and rot V see pp. 644, 642.Google Scholar
14. (1).
In this section an exposition of the line integral of the second type in the general form is given.Google Scholar
15. (1).
The vector function V is continuous if all the coefficients of its resolution into three base vectors e1,e2,e3 are continuous.Google Scholar
16. (1).
This is a condition for integrability (see p. 492).Google Scholar
17. (2).
This is a primitive function (see p. 493). In physics, the integral with the opposite sign is sometimes called a potential at the point $$r:\;=\int\limits_{r_0}^{r} V(r)dr$$.Google Scholar
18. (1).
Or else a potential with the opposite sign, see the previous footnote.Google Scholar
19. (2).
In this section we give a vectorial treatment of the theory of the surface integral of the second type in the general form.Google Scholar
20. (1).
Each elementary region dSi is contracted to a point in the sense of the footnote on p. 496.Google Scholar
21. (1).
Furnished with the signs “+” or “−” (see p. 508).Google Scholar
22. (1).
For the symbol ∇ (nabla), see p. 643.Google Scholar
23. (1).
For the symbol ∇ (nabla) see p. 643.Google Scholar
24. (2).
We can remove the sign “ − ” by interchanging the factors under the integral sign: ∫dS×V (see p. 617).Google Scholar
25. (1).
26. (1).
See pp. 512 and 513.Google Scholar
27. (1).
For a more accurate formulation, see p. 512.Google Scholar
28. (1).
Or the negative of the potential, see footnote (2) on p. 636.Google Scholar
29. (2).
Formula (1) is true provided that the divergence of V is a differentiable function and sufficiently rapidly decreases to zero, when r increases infinitely.Google Scholar
30. (3).
Formula (2) is true provided that the rotation of the field V is a differentiable function and sufficiently rapidly decreases to zero, if r increases infinitely.Google Scholar
31. (4).
Or + e/r, see footnote (2) on p. 636.Google Scholar