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~fdz？dx；dy . (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) ss？1~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.
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where the summation with respect to s has been replaced by the summation with respect to m.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
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.)
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.