Mathematical Physics Vol 1

Chapter 4. Field theory

110

To each point of space three coordinate surfaces correspond, i.e. each point represents an intersection of three coordinate surfaces. A coordinate line is located at the intersection of two coordinate surfaces (Fig. 4.17). As three such, in general, curved lines, pass through each point, they form a curvilinear coordinate system. For each point in space tangents to coordinate lines at that point can be drawn, so that they are oriented in the direction in which the values of q i grow. The base vectors of this coordinate system are the tangent vectors.

R Note that in the case of a curvilinear coordinate system, the base vectors change from one point to another, unlike in rectilinear coordinate systems.

Coordinate systems, in relation to which the position of points in space is determined, are also called reference systems. Depending on the type of the coordinate line, coordinates can be - rectilinear (Fig. 4.18) and - curvilinear (Fig. 4.16). If the coordinate surfaces, i.e. the coordinate lines, are mutually perpendicular at all points in space, they form an orthogonal (rectangular) coordinate system. In the special case of the Cartesian coordinate system, the position of a point in space is determined by the position vector

r = x i + y j + z k .

(4.129)

It is obvious that

∂ r ∂ x

∂ r ∂ y

∂ r ∂ z

i =

, j =

, k =

(4.130)

.

Thus, unit vectors (orts) i , j and k can be obtained as partial derivatives of the position vector by point coordinates. In the case of generalized coordinates, the position of the same point is determined by the same position vector r , but expressed in relation to the curvilinear coordinates q i (see 4.126). Then the base vectors of that coordinate system are determined (as in the case of Cartesian coordinates (4.130)) by g i = ∂ r ∂ q i , ( i = 1 , 2 , 3 ) . (4.131) Thus, the vectors g i are the base vectors of the system of generalized coordinates q i and they have the directions of the tangents to the coordinate lines. That they are base vectors is concluded on the basis of their linear independence. We notice that the indices that denote the base vectors g i and e i are at the bottom position. If coordinates of the basic vectors are expressed in relation to Cartesian rectangular coordi nates, the magnitudes of these vectors are determined by the following expressions | g i | = s ∂ x ∂ q i 2 + ∂ y ∂ q i 2 + ∂ z ∂ q i 2 ≡ h i , ( i = 1 , 2 , 3 ) , (4.132) where h i are the so called Lamé 17 ormetric coefficients . If we denote the unit vectors in the directions of these vectors by e i , we obtain

g i | g i | ⇒

e i =

g i = h i e i .

(4.133)

17 Gabriel Lamé (1795-1870), French mathematician and engineer. He made important contributions to analytical geometry and analytical mechanics, as well as to the theory of elasticity and the theory of heat conduction.