DOC

tensor theory

By Roy Smith,2014-03-15 22:18
11 views 0
in spacetime in relativity theory, may take on negative values, so that itself is not necessarily real. If ds = 0 for not all zero, the displacement...

    Tensor Theory

Introduction and definitions

In n-dimensional space V (called a "manifold" in mathematics), points are specified by n

    n12xassigning values to a set of n continuous real variables ,x.....x called the coordinates.

    In many cases these will run from -? to +?, but the range of some or all of these can be finite.

Examples: In Euclidean space in three dimensions, we can use cartesian coordinates x, y and z,

    each of which runs from -? to +?. For a two dimensional Euclidean plane, Cartesians may again be employed, or we can use plane polar coordinates r, ( whose ranges are 0 to ? and 0

    to 2 respectively.

    Coordinate transformations. The coordinates of points in the manifold may be assigned in a

    n12x,x.....xnumber of different ways. If we select two different sets of coordinates, and n12 x ,;,,x ,;,.....x ,;there will obviously be a connection between them of the form

    rrn12x ,;f(x,x....x) r = 1, 2........n. (1)

    where the f's are assumed here to be well behaved functions. Another way of expressing the same relationship is

    rrn12x ,;?,;x (x,x....x) r = 1, 2........n. (2)

    rrnn1212x ,;(xf(x,x....x),x....x)where denotes the n functions , r = 1, 2......n.

Recall that if a variable z is a function of two variables x and y, i.e. z = f (x, y), then the

    connection between the differentials dx, dy and dz is

    ~f~fdzdxdy . (3) xy~~

    Extending this to several variables therefore, for each one of the new coordinates we have

    nr~x ,;rsdx ,;dx? . r=1, 2........n. (4) ss1~x

    The transformation of the differentials of the coordinates is therefore linear and homogeneous, which is not necessarily the case for the transformation of the coordinates themselves.

    1

    Range and Summation Conventions. Equations such as (4) may be simplified by the use of two conventions:

    Range Convention: When a suffix is unrepeated in a term, it is understood to take all values in the range 1, 2, 3.....n.

    Summation Convention: When a suffix is repeated in a term, summation with respect to that suffix is understood, the range of summation being 1, 2, 3.....n.

    With these two conventions applying, equation (4) may be written as

    r~x ,;rsdx ,;dx. (5) s~x

    Note that a repeated suffix is a "dummy" suffix, and can be replaced by any convenient alternative. For example, equation (5) could have been written as

    r~x ,;rmdx ,;dx . (6) m~x

    where the summation with respect to s has been replaced by the summation with respect to m.

Contravariant vectors and tensors. Consider two neighbouring points P and Q in the

    rrrP Qmanifold whose coordinates are x and x + dx respectively. The vector

    ris then described by the quantities dx which are the components of the vector in this

    P Qcoordinate system. In the dashed coordinates, the vector is described by the components

    rrdx ,; which are related to dx by equation (5), the differential coefficients being evaluated at P.

    rrdx ,; The infinitesimal displacement represented by dx or is an example of a contravariant

    vector.

    rDefn. A set of n quantities T associated with a point P are said to be the components of a contravariant vector if they transform, on change of coordinates, according to the equation

    r~x ,;rsT ,;T . (7) s~x

    where the partial derivatives are evaluated at the point P. (Note that there is no requirement that the components of a contravariant tensor should be infinitesimal.)

    2

    2rs quantities T associated with a point P are said to be the components of Defn. A set of n

    a contravariant tensor of the second order if they transform, on change of coordinates, according to the equation

    rs~x ,;~x ,;rsmnT ,;T . (8) mn~x~x

    Obviously the definition can be extended to tensors of higher order. A contravariant vector is the same as a contravariant tensor of first order.