Consciousness Studies/The Philosophical Problem/Appendix

Gauss's Analysis of Curved Surfaces - The Origin of the Metric Tensor
It became apparent at the start of the nineteenth century that issues such as Euclid's parallel postulate required the development of a new type of geometry that could deal with curved surfaces and real and imaginary planes. At the foundation of this approach is Gauss's analysis of curved surfaces which allows us to work with a variety of coordinate systems and displacements on any type of surface.

Suppose there is a line on a surface. The length of this line can be expressed in terms of a coordinate system. A short length of line Ds in a two dimensional space may be expressed in terms of Pythagoras' theorem as:

D s2 = D x2 + D y2 Suppose there is another coordinate system on the surface with two axes: x1, x2, how can the length of the line be expressed in terms of these coordinates? Gauss tackled this problem and his analysis is quite straightforward for two coordinate axes:

Figure 1:



It is possible to use elementary differential geometry to describe displacements along the plane in terms of displacements on the curved surfaces:

DY = Dx1dY/dx<SUB>1</SUB> + <FONT FACE="Symbol">D</FONT>x<SUB>2</SUB><FONT FACE="Symbol">d</FONT>Y/<FONT FACE="Symbol">d</FONT>x<SUB>2</SUB>

<FONT FACE="Symbol">D</FONT>Z = <FONT FACE="Symbol">D</FONT>x<SUB>1</SUB><FONT FACE="Symbol">d</FONT>Z/<FONT FACE="Symbol">d</FONT>x<SUB>1</SUB> + <FONT FACE="Symbol">D</FONT>x<SUB>2</SUB><FONT FACE="Symbol">d</FONT>Z/<FONT FACE="Symbol">d</FONT>x<SUB>2</SUB>

The displacement of a short line is then assumed to be given by a formula, called a metric, such as Pythagoras' theorem

<FONT FACE="Symbol">D</FONT>S<SUP>2</SUP> = <FONT FACE="Symbol">D</FONT>Y<SUP> 2</SUP> + <FONT FACE="Symbol">D</FONT>Z<SUP>2</SUP>

Or some other metric such as the metric of a 4D Minkowskian space:

<FONT FACE="Symbol">D</FONT>S<SUP>2</SUP> = -<FONT FACE="Symbol">D</FONT>T<SUP> 2</SUP> + <FONT FACE="Symbol">D</FONT>X<SUP>2</SUP> + <FONT FACE="Symbol">D</FONT>Y<SUP>2</SUP> + <FONT FACE="Symbol">D</FONT>Z<SUP>2</SUP>

This type of analysis can be extended to any number of dimensions. It is then possible to express the short length, <FONT FACE="Symbol">D</FONT>s, in terms of the coordinates. The full algebraic analysis is given at the end of this appendix. In 3D the expression for the length is:

<FONT FACE="Symbol">D</FONT>s<SUP>2</SUP> = <FONT FACE="Symbol">SS</FONT> (<FONT FACE="Symbol">d</FONT>X/<FONT FACE="Symbol">d</FONT>x<SUP>i </SUP><FONT FACE="Symbol">d</FONT>X/<FONT FACE="Symbol">d</FONT>x<SUP>k</SUP> + <FONT FACE="Symbol">d</FONT>Y/<FONT FACE="Symbol">d</FONT>x<SUP>i </SUP><FONT FACE="Symbol">d</FONT>Y/<FONT FACE="Symbol">d</FONT>x<SUP>k</SUP> + <FONT FACE="Symbol">d</FONT>Z/<FONT FACE="Symbol">d</FONT>x<SUP>i </SUP><FONT FACE="Symbol">d</FONT>Z/<FONT FACE="Symbol">d</FONT>x<SUP>k</SUP>) <FONT FACE="Symbol">D</FONT>x<SUP>i</SUP><FONT FACE="Symbol">D</FONT>x<SUP>k</SUP>

(for i=1 to 3 and k=1 to 3)

and so, using indicial notation:

<FONT FACE="Symbol">D</FONT>s<SUP>2</SUP> = g<SUB>ik</SUB><FONT FACE="Symbol">D</FONT>x<SUP>i</SUP><FONT FACE="Symbol">D</FONT>x<SUP>k</SUP>

Where

g<SUB>ik</SUB> = (<FONT FACE="Symbol">d</FONT>x/<FONT FACE="Symbol">d</FONT>x<SUP>i </SUP><FONT FACE="Symbol">d</FONT>x/<FONT FACE="Symbol">d</FONT>x<SUP>k</SUP> + <FONT FACE="Symbol">d</FONT>y/<FONT FACE="Symbol">d</FONT>x<SUP>i </SUP><FONT FACE="Symbol">d</FONT>y/<FONT FACE="Symbol">d</FONT>x<SUP>k</SUP> + <FONT FACE="Symbol">d</FONT>z/<FONT FACE="Symbol">d</FONT>x<SUP>i </SUP><FONT FACE="Symbol">d</FONT>z/<FONT FACE="Symbol">d</FONT>x<SUP>k</SUP>)

If the coordinates are not merged then <FONT FACE="Symbol">D</FONT>s is dependent on both sets of coordinates. In matrix notation:

<FONT FACE="Symbol">D</FONT>s<SUP>2</SUP> = g<FONT FACE="Symbol">D</FONT>x<FONT FACE="Symbol">D</FONT>x

becomes:

Where a, b, c, d stand for the values of g<SUB>ik</SUB>.

Which is:

(<FONT FACE="Symbol">D</FONT>x<SUB>1</SUB>a +<FONT FACE="Symbol"> D</FONT>x<SUB>2</SUB>c)<FONT FACE="Symbol"> D</FONT>x<SUB>1</SUB> + (<FONT FACE="Symbol">D</FONT>x<SUB>1</SUB>b +<FONT FACE="Symbol"> D</FONT>x<SUB>2</SUB>d) <FONT FACE="Symbol">D</FONT>x<SUB>2</SUB> = <FONT FACE="Symbol">D</FONT>x<SUB>1</SUB><SUP>2</SUP>a + 2<FONT FACE="Symbol">D</FONT>x<SUB>1</SUB><FONT FACE="Symbol">D</FONT>x<SUB>2</SUB>(c<FONT FACE="Symbol"> + </FONT>b)<FONT FACE="Symbol"> + D</FONT>x<SUB>2</SUB><SUP>2</SUP>d

So:

<FONT FACE="Symbol">D</FONT>s<SUP>2</SUP> = <FONT FACE="Symbol">D</FONT>x<SUB>1</SUB><SUP>2</SUP>a + 2<FONT FACE="Symbol">D</FONT>x<SUB>1</SUB><FONT FACE="Symbol">D</FONT>x<SUB>2</SUB>(c<FONT FACE="Symbol"> + </FONT>b)<FONT FACE="Symbol"> + D</FONT>x<SUB>2</SUB><SUP>2</SUP>d

<FONT FACE="Symbol">D</FONT>s<SUP>2</SUP> is a bilinear form that depends on both <FONT FACE="Symbol">D</FONT>x<SUB>1</SUB> and <FONT FACE="Symbol">D</FONT>x<SUB>2</SUB>. It can be written in matrix notation as:

<FONT FACE="Symbol">D</FONT>s<SUP>2</SUP> = <FONT FACE="Symbol">D</FONT>x<SUP>T</SUP> A <FONT FACE="Symbol">D</FONT>x

Where A is the matrix containing the values in g<SUB>ik</SUB>. This is a special case of the bilinear form known as the quadratic form because the same matrix (<FONT FACE="Symbol">D</FONT>x) appears twice; in the generalised bilinear form B = x<SUP>T</SUP>Ay (the matrices x and y are different).

If the surface is a Euclidean plane then the values of g<SUB>ik</SUB> are:

Which become:

So the matrix A is the unit matrix I and:

<B><FONT FACE="Symbol">D</FONT>s<SUP>2</SUP> = <FONT FACE="Symbol">D</FONT>x<SUP>T</SUP> I <FONT FACE="Symbol">D</FONT>x</B>and:

<FONT FACE="Symbol">D</FONT>s<SUP>2</SUP> = <FONT FACE="Symbol">D</FONT>x<SUB>1</SUB><SUP>2</SUP><FONT FACE="Symbol"> + D</FONT>x<SUB>2</SUB><SUP>2</SUP>

Which recovers Pythagoras' theorem.

If the surface is derived from <FONT FACE="Symbol">D</FONT>s<SUP>2</SUP> = -<FONT FACE="Symbol">D</FONT>Y<SUP>2</SUP> + <FONT FACE="Symbol">D</FONT>Z<SUP>2</SUP> then the values of g<SUB>ik</SUB> are:

Which becomes:

Which allows the original 'rule' to be recovered i.e.: <FONT FACE="Symbol">D</FONT>s<SUP>2</SUP> = -<FONT FACE="Symbol">D</FONT>x<SUB>1</SUB><SUP>2</SUP><FONT FACE="Symbol"> + D</FONT>x<SUB>2</SUB><SUP>2</SUP>

The Space-Time Interval
The fundamental assumption of modern relativity theory is that the space-time interval is invariant. The space-time interval is given by the following equation rather than Pythagoras' theorem:

<FONT FACE="Symbol">D</FONT>s<SUP>2</SUP> = - <FONT FACE="Symbol">D</FONT>t<SUP>2</SUP> + <FONT FACE="Symbol">D</FONT>x<SUB>1</SUB><SUP>2</SUP> + <FONT FACE="Symbol">D</FONT>x<SUB>2</SUB><SUP>2</SUP> + <FONT FACE="Symbol">D</FONT>x<SUB>3</SUB><SUP>2</SUP>

The origin of the negative sign in front of <FONT FACE="Symbol">D</FONT>t is of considerable interest. It could originate from an assumption that time is imaginary, that time is real and the metric has a negative sign for time, or that time is mixed real and imaginary with a Pythagorean metric.

Imaginary Time
Suppose that Pythagoras theorem applied to the space-time interval and:

<FONT FACE="Symbol">D</FONT>s<SUP><FONT FACE="Symbol">2</FONT></SUP> = <FONT FACE="Symbol">D</FONT> t<SUP> 2</SUP> + <FONT FACE="Symbol">D</FONT>x<SUB>1</SUB><SUP>2</SUP> + <FONT FACE="Symbol">D</FONT>x<SUB>2</SUB><SUP>2</SUP> + <FONT FACE="Symbol">D</FONT>x<SUB>3</SUB><SUP>2</SUP>

g<SUB>ik</SUB> = (<FONT FACE="Symbol">d</FONT>t/<FONT FACE="Symbol">d</FONT>x<SUP>i </SUP><FONT FACE="Symbol">d</FONT>t/<FONT FACE="Symbol">d</FONT>x<SUP>k</SUP><FONT FACE="Symbol"> + d</FONT>x/<FONT FACE="Symbol">d</FONT>x<SUP>i </SUP><FONT FACE="Symbol">d</FONT>x/<FONT FACE="Symbol">d</FONT>x<SUP>k</SUP> + <FONT FACE="Symbol">d</FONT>y/<FONT FACE="Symbol">d</FONT>x<SUP>i </SUP><FONT FACE="Symbol">d</FONT>y/<FONT FACE="Symbol">d</FONT>x<SUP>k</SUP> + <FONT FACE="Symbol">d</FONT>z/<FONT FACE="Symbol">d</FONT>x<SUP>i </SUP><FONT FACE="Symbol">d</FONT>z/<FONT FACE="Symbol">d</FONT>x<SUP>k</SUP>)

For a flat surface <FONT FACE="Symbol">d</FONT>t/<FONT FACE="Symbol">d</FONT>x<SUP>0</SUP> = <FONT FACE="Symbol">d</FONT>x/<FONT FACE="Symbol">d</FONT>x<SUP>1</SUP> = <FONT FACE="Symbol">d</FONT>y/<FONT FACE="Symbol">d</FONT>x<SUP>2</SUP> = <FONT FACE="Symbol">d</FONT>z/<FONT FACE="Symbol">d</FONT>x<SUP>3</SUP> = 1 and all other coefficients are zero therefore:

'g' =

Which means that the time interval must be imaginary if the assumption of relativity is to be supported i.e.: <FONT FACE="Symbol">D</FONT>t<FONT FACE="Symbol"> </FONT><SUP>2</SUP> = (<FONT FACE="Symbol">DT</FONT> <FONT FACE="Symbol">Ö</FONT> -1)<SUP> 2</SUP>

So that <FONT FACE="Symbol">D</FONT>s<SUP><FONT FACE="Symbol">2</FONT> </SUP>= <FONT FACE="Symbol">D</FONT>t<SUP>2</SUP> + <FONT FACE="Symbol">D</FONT>x<SUB>1</SUB><SUP>2</SUP> + <FONT FACE="Symbol">D</FONT>x<SUB>2</SUB><SUP>2</SUP> + <FONT FACE="Symbol">D</FONT>x<SUB>3</SUB><SUP>2</SUP> becomes <FONT FACE="Symbol">D</FONT>s<SUP><FONT FACE="Symbol">2</FONT></SUP> = - <FONT FACE="Symbol">D</FONT>T<SUP>2</SUP> + <FONT FACE="Symbol">D</FONT>x<SUB>1</SUB><SUP>2</SUP> + <FONT FACE="Symbol">D</FONT>x<SUB>2</SUB><SUP>2</SUP> + <FONT FACE="Symbol">D</FONT>x<SUB>3</SUB><SUP>2</SUP>

This form of time is not supported in General Relativity Theory

Real Time
If real time is used then the expressions for each displacement along each coordinate axis remain the same e.g.:

<FONT FACE="Symbol">D</FONT>T = <FONT FACE="Symbol">D</FONT>x<SUB>1</SUB><FONT FACE="Symbol">d</FONT>T/<FONT FACE="Symbol">d</FONT>x<SUB>1</SUB> + <FONT FACE="Symbol">D</FONT>x<SUB>2</SUB><FONT FACE="Symbol">d</FONT>T/<FONT FACE="Symbol">d</FONT>x<SUB>2</SUB> + <FONT FACE="Symbol">D</FONT>x<SUB>3</SUB><FONT FACE="Symbol">d</FONT>T/<FONT FACE="Symbol">d</FONT>x<SUB>3</SUB> + <FONT FACE="Symbol">D</FONT>x<SUB>4</SUB><FONT FACE="Symbol">d</FONT>T/<FONT FACE="Symbol">d</FONT>x<SUB>4</SUB>

etc.

But when they are combined the formula <FONT FACE="Symbol">D</FONT>s<SUP><FONT FACE="Symbol">2</FONT></SUP> = - <FONT FACE="Symbol">D</FONT>T<SUP>2</SUP> + <FONT FACE="Symbol">D</FONT>x<SUB>1</SUB><SUP>2</SUP> + <FONT FACE="Symbol">D</FONT>x<SUB>2</SUB><SUP>2</SUP> + <FONT FACE="Symbol">D</FONT>x<SUB>3</SUB><SUP>2</SUP> is used instead of Pythagoras' theorem (see above for a fully worked example in 2D). This results in the following metric tensor:

'g' =

Where g<SUB>00</SUB> is given by -1 times (<FONT FACE="Symbol">d</FONT>t/<FONT FACE="Symbol">d</FONT>x<SUP>0</SUP>)<SUP> 2</SUP>.

Mixed Real and Imaginary Time
There is a third possibility that is not generally discussed. The 'plane' in figure 1 is a plane in the observer's coordinate system and the surface has its own coordinate system. If the time coordinate on the surface were 'imaginary' and that on the plane were real (or vice versa) then using Pythagoras' theorem:

<FONT FACE="Symbol">D</FONT>s<SUP><FONT FACE="Symbol">2</FONT></SUP> = <FONT FACE="Symbol">Dt</FONT><SUP>2</SUP> + <FONT FACE="Symbol">D</FONT>x<SUB>1</SUB><SUP>2</SUP> + <FONT FACE="Symbol">D</FONT>x<SUB>2</SUB><SUP>2</SUP> + <FONT FACE="Symbol">D</FONT>x<SUB>3</SUB><SUP>2</SUP>

where <FONT FACE="Symbol">t</FONT> equals (kt), k being a constant that is yet to be determined. In flat space-time g<SUB>00</SUB> is given by (<FONT FACE="Symbol">d</FONT>t/<FONT FACE="Symbol">d</FONT>x<SUP>0</SUP>)<SUP> 2</SUP>. But t is imaginary so g<SUB>00</SUB> equals -1. This then gives exactly the same metric tensor as the assumption of real time.

'g' =

The Modern Formulation of the Metric Tensor
The modern formulation uses the following mathematical expression for the space-time interval:

s<SUP><FONT FACE="Symbol">2</FONT></SUP> = g<SUB><FONT FACE="Symbol">mn</FONT> </SUB>x<SUP><FONT FACE="Symbol">n</FONT></SUP>x<SUP><FONT FACE="Symbol">m</FONT></SUP>

where the values of s, x represent tiny displacements in each of the four coordinate axes and 'g' is the metric of the space. It is assumed that g<SUB>00</SUB> is opposite in sign to the other constants on the principle diagonal (i.e.: real or mixed real and imaginary time are assumed). With this assumption the expression becomes:

s<SUP>2</SUP> = x<SUB>1</SUB><SUP>2</SUP> + x<SUB>2</SUB><SUP>2</SUP> + x<SUB>3</SUB><SUP>2</SUP> - x<SUB>4</SUB><SUP>2</SUP>

The expansion is shown below. In matrix notation this is:

(Where t is time in metres, i.e.:c times time in secs). The numbers -1,1,1,1 are the values of the combinations of differential coefficients that were described above.

Evaluating the first matrix multiplication this becomes:

Which resolves to: s<SUP><FONT FACE="Symbol">2</FONT></SUP> = x<SUB>1</SUB><SUP>2</SUP> + x<SUB>2</SUB><SUP>2</SUP> + x<SUB>3</SUB><SUP>2</SUP> - t<SUP>2</SUP>

Which is the metric of space-time and applies to quite large values of s,x and t in the absence of accelerations and strong gravitational fields. Notice how the computation is more like a squared norm than a simple square and carries with it the physical implication of a product of a vector with its reflection (!).

The metric is normally expressed in differential form so that it can be used with a curved space-time and with displacements that are not measured relative to the origin.

ds<SUP><FONT FACE="Symbol">2</FONT></SUP> = dx<SUB>1</SUB><SUP>2</SUP> + dx<SUB>2</SUB><SUP>2</SUP> + dx<SUB>3</SUB><SUP>2</SUP> - dt<SUP>2</SUP>

Or, equivalently:

ds<SUP><FONT FACE="Symbol">2</FONT></SUP> = dt<SUP>2</SUP> - dx<SUB>1</SUB><SUP>2</SUP> - dx<SUB>2</SUB><SUP>2</SUP> - dx<SUB>3</SUB><SUP>2</SUP>

Full analysis of the constants in Gauss' analysis
<FONT FACE="Symbol">D</FONT>Y = <FONT FACE="Symbol">D</FONT>x<SUB>1</SUB><FONT FACE="Symbol">d</FONT>Y/<FONT FACE="Symbol">d</FONT>x<SUB>1</SUB> + <FONT FACE="Symbol">D</FONT>x<SUB>2</SUB><FONT FACE="Symbol">d</FONT>Y/<FONT FACE="Symbol">d</FONT>x<SUB>2</SUB>

<FONT FACE="Symbol">D</FONT>Y<SUP>2</SUP> = (<FONT FACE="Symbol">D</FONT>x<SUB>1</SUB><FONT FACE="Symbol">d</FONT>Y/<FONT FACE="Symbol">d</FONT>x<SUB>1</SUB> + <FONT FACE="Symbol">D</FONT>x<SUB>2</SUB><FONT FACE="Symbol">d</FONT>Y/<FONT FACE="Symbol">d</FONT>x<SUB>2</SUB>)<SUP>2</SUP>

<FONT FACE="Symbol">D</FONT>Y<SUP>2</SUP> = <FONT FACE="Symbol">D</FONT>x<SUB>1</SUB><FONT FACE="Symbol">d</FONT>Y/<FONT FACE="Symbol">d</FONT>x<SUB>1</SUB> * <FONT FACE="Symbol">D</FONT>x<SUB>1</SUB><FONT FACE="Symbol">d</FONT>Y/<FONT FACE="Symbol">d</FONT>x<SUB>1</SUB> + <FONT FACE="Symbol">D</FONT>x<SUB>1</SUB><FONT FACE="Symbol">d</FONT>Y/<FONT FACE="Symbol">d</FONT>x<SUB>1</SUB> * <FONT FACE="Symbol">D</FONT>x<SUB>2</SUB><FONT FACE="Symbol">d</FONT>Y/<FONT FACE="Symbol">d</FONT>x<SUB>2</SUB> + <FONT FACE="Symbol">D</FONT>x<SUB>1</SUB><FONT FACE="Symbol">d</FONT>Y/<FONT FACE="Symbol">d</FONT>x<SUB>1</SUB> * <FONT FACE="Symbol">D</FONT>x<SUB>2</SUB><FONT FACE="Symbol">d</FONT>Y/<FONT FACE="Symbol">d</FONT>x<SUB>2</SUB> + <FONT FACE="Symbol">D</FONT>x<SUB>2</SUB><FONT FACE="Symbol">d</FONT>Y/<FONT FACE="Symbol">d</FONT>x<SUB>2</SUB> * <FONT FACE="Symbol">D</FONT>x<SUB>2</SUB><FONT FACE="Symbol">d</FONT>Y/<FONT FACE="Symbol">d</FONT>x<SUB>2</SUB>

<FONT FACE="Symbol">D</FONT>Y<SUP>2</SUP> = <FONT FACE="Symbol">D</FONT>x<SUB>1</SUB><FONT FACE="Symbol">D</FONT>x<SUB>1</SUB><FONT FACE="Symbol">d</FONT>Y/<FONT FACE="Symbol">d</FONT>x<SUB>1</SUB> <FONT FACE="Symbol">d</FONT>Y/<FONT FACE="Symbol">d</FONT>x<SUB>1</SUB> + <FONT FACE="Symbol">D</FONT>x<SUB>1</SUB><FONT FACE="Symbol">D</FONT>x<SUB>2</SUB><FONT FACE="Symbol">d</FONT>Y/<FONT FACE="Symbol">d</FONT>x<SUB>1</SUB><FONT FACE="Symbol">d</FONT>Y/<FONT FACE="Symbol">d</FONT>x<SUB>2</SUB> + <FONT FACE="Symbol">D</FONT>x<SUB>1</SUB><FONT FACE="Symbol">D</FONT>x<SUB>2</SUB><FONT FACE="Symbol">d</FONT>Y/<FONT FACE="Symbol">d</FONT>x<SUB>1</SUB><FONT FACE="Symbol">d</FONT>Y/<FONT FACE="Symbol">d</FONT>x<SUB>2</SUB> + <FONT FACE="Symbol">D</FONT>x<SUB>2</SUB><FONT FACE="Symbol">D</FONT>x<SUB>2</SUB><FONT FACE="Symbol">d</FONT>Y/<FONT FACE="Symbol">d</FONT>x<SUB>2</SUB><FONT FACE="Symbol">d</FONT>Y/<FONT FACE="Symbol">d</FONT>x<SUB>2</SUB>

And

<FONT FACE="Symbol">D</FONT>Z = <FONT FACE="Symbol">D</FONT>x<SUB>1</SUB><FONT FACE="Symbol">d</FONT>Z/<FONT FACE="Symbol">d</FONT>x<SUB>1</SUB> + <FONT FACE="Symbol">D</FONT>x<SUB>2</SUB><FONT FACE="Symbol">d</FONT>Z/<FONT FACE="Symbol">d</FONT>x<SUB>2</SUB>

<FONT FACE="Symbol">D</FONT>Z<SUP>2</SUP> = (<FONT FACE="Symbol">D</FONT>x<SUB>1</SUB><FONT FACE="Symbol">d</FONT>Z/<FONT FACE="Symbol">d</FONT>x<SUB>1</SUB> + <FONT FACE="Symbol">D</FONT>x<SUB>2</SUB><FONT FACE="Symbol">d</FONT>Z/<FONT FACE="Symbol">d</FONT>x<SUB>2</SUB>)<SUP>2</SUP>

<FONT FACE="Symbol">D</FONT>Z<SUP>2</SUP> = <FONT FACE="Symbol">D</FONT>x<SUB>1</SUB><FONT FACE="Symbol">d</FONT>Z/<FONT FACE="Symbol">d</FONT>x<SUB>1</SUB> * <FONT FACE="Symbol">D</FONT>x<SUB>1</SUB><FONT FACE="Symbol">d</FONT>Z/<FONT FACE="Symbol">d</FONT>x<SUB>1</SUB> + <FONT FACE="Symbol">D</FONT>x<SUB>1</SUB><FONT FACE="Symbol">d</FONT>Z/<FONT FACE="Symbol">d</FONT>x<SUB>1</SUB> * <FONT FACE="Symbol">D</FONT>x<SUB>2</SUB><FONT FACE="Symbol">d</FONT>Z/<FONT FACE="Symbol">d</FONT>x<SUB>2</SUB> + <FONT FACE="Symbol">D</FONT>x<SUB>1</SUB><FONT FACE="Symbol">d</FONT>Z/<FONT FACE="Symbol">d</FONT>x<SUB>1</SUB> * <FONT FACE="Symbol">D</FONT>x<SUB>2</SUB><FONT FACE="Symbol">d</FONT>Z/<FONT FACE="Symbol">d</FONT>x<SUB>2</SUB> + <FONT FACE="Symbol">D</FONT>x<SUB>2</SUB><FONT FACE="Symbol">d</FONT>Z/<FONT FACE="Symbol">d</FONT>x<SUB>2</SUB> * <FONT FACE="Symbol">D</FONT>x<SUB>2</SUB><FONT FACE="Symbol">d</FONT>Z/<FONT FACE="Symbol">d</FONT>x<SUB>2</SUB>

<FONT FACE="Symbol">D</FONT>Z<SUP>2</SUP> = <FONT FACE="Symbol">D</FONT>x<SUB>1</SUB><FONT FACE="Symbol">D</FONT>x<SUB>1</SUB><FONT FACE="Symbol">d</FONT>Z/<FONT FACE="Symbol">d</FONT>x<SUB>1</SUB> <FONT FACE="Symbol">d</FONT>Z/<FONT FACE="Symbol">d</FONT>x<SUB>1</SUB> + <FONT FACE="Symbol">D</FONT>x<SUB>1</SUB><FONT FACE="Symbol">D</FONT>x<SUB>2</SUB><FONT FACE="Symbol">d</FONT>Z/<FONT FACE="Symbol">d</FONT>x<SUB>1</SUB><FONT FACE="Symbol">d</FONT>Z/<FONT FACE="Symbol">d</FONT>x<SUB>2</SUB> + <FONT FACE="Symbol">D</FONT>x<SUB>1</SUB><FONT FACE="Symbol">D</FONT>x<SUB>2</SUB><FONT FACE="Symbol">d</FONT>Z/<FONT FACE="Symbol">d</FONT>x<SUB>1</SUB><FONT FACE="Symbol">d</FONT>Z/<FONT FACE="Symbol">d</FONT>x<SUB>2</SUB> + <FONT FACE="Symbol">D</FONT>x<SUB>2</SUB><FONT FACE="Symbol">D</FONT>x<SUB>2</SUB><FONT FACE="Symbol">d</FONT>Z/<FONT FACE="Symbol">d</FONT>x<SUB>2</SUB><FONT FACE="Symbol">d</FONT>Z/<FONT FACE="Symbol">d</FONT>x<SUB>2</SUB>

Therefore:

<FONT FACE="Symbol">D</FONT>Y<SUP>2</SUP> + <FONT FACE="Symbol">D</FONT>Z<SUP>2</SUP> =

<FONT FACE="Symbol">(d</FONT>Y/<FONT FACE="Symbol">d</FONT>x<SUB>1</SUB><FONT FACE="Symbol">d</FONT>Y/<FONT FACE="Symbol">d</FONT>x<SUB>1</SUB><FONT FACE="Symbol"> + dZ</FONT>/<FONT FACE="Symbol">d</FONT>x<SUB>1</SUB><FONT FACE="Symbol">dZ</FONT>/<FONT FACE="Symbol">d</FONT>x<SUB>1</SUB><FONT FACE="Symbol">)D</FONT>x<SUB>1</SUB><FONT FACE="Symbol"> D</FONT>x<SUB>1</SUB>

<FONT FACE="Symbol">+ (d</FONT>Y/<FONT FACE="Symbol">d</FONT>x<SUB>2</SUB>Y/<FONT FACE="Symbol">d</FONT>x<SUB>1</SUB><FONT FACE="Symbol"> + dZ</FONT>/<FONT FACE="Symbol">d</FONT>x<SUB>2</SUB><FONT FACE="Symbol">dZ</FONT>/<FONT FACE="Symbol">d</FONT>x<SUB>1</SUB><FONT FACE="Symbol">)D</FONT>x<SUB>2</SUB><FONT FACE="Symbol">D</FONT>x<SUB>1</SUB>

<FONT FACE="Symbol">+ (d</FONT>Y/<FONT FACE="Symbol">d</FONT>x<SUB>1</SUB><FONT FACE="Symbol">d</FONT>Y/<FONT FACE="Symbol">d</FONT>x<SUB>2</SUB><FONT FACE="Symbol"> + dZ</FONT>/<FONT FACE="Symbol">d</FONT>x<SUB>1</SUB><FONT FACE="Symbol">dZ</FONT>/<FONT FACE="Symbol">d</FONT>x<SUB>2</SUB><FONT FACE="Symbol">)D</FONT>x<SUB>1</SUB><FONT FACE="Symbol"> D</FONT>x<SUB>2</SUB>

<FONT FACE="Symbol">+ (d</FONT>Y/<FONT FACE="Symbol">d</FONT>x<SUB>2</SUB><FONT FACE="Symbol">d</FONT>Y/<FONT FACE="Symbol">d</FONT>x<SUB>2</SUB><FONT FACE="Symbol"> + dZ</FONT>/<FONT FACE="Symbol">d</FONT>x<SUB>2</SUB><FONT FACE="Symbol">dZ</FONT>/<FONT FACE="Symbol">d</FONT>x<SUB>2</SUB><FONT FACE="Symbol">)D</FONT>x<SUB>2</SUB><FONT FACE="Symbol"> D</FONT>x<SUB>2</SUB>

For a flat surface

<FONT FACE="Symbol">d</FONT>Y= <FONT FACE="Symbol">d</FONT>x<SUB>2</SUB> and <FONT FACE="Symbol">dZ</FONT>= <FONT FACE="Symbol">d</FONT>x<SUB>1</SUB> so <FONT FACE="Symbol">d</FONT>Y/<FONT FACE="Symbol">d</FONT>x<SUB>2</SUB> = 1 and <FONT FACE="Symbol">dZ</FONT>/<FONT FACE="Symbol">d</FONT>x<SUB>1</SUB> = 1 also <FONT FACE="Symbol">d</FONT>Y/<FONT FACE="Symbol">d</FONT>x<SUB>1</SUB> = 0 and <FONT FACE="Symbol">d</FONT>Z/<FONT FACE="Symbol">d</FONT>x<SUB>2</SUB> = 0.

<FONT FACE="Symbol">D</FONT>s<SUP>2</SUP> = <FONT FACE="Symbol">D</FONT>Y<SUP>2</SUP> + <FONT FACE="Symbol">D</FONT>Z<SUP>2</SUP> = <FONT FACE="Symbol">(0 + 1)D</FONT>x<SUB>1</SUB><FONT FACE="Symbol"> D</FONT>x<SUB>1</SUB><FONT FACE="Symbol">+ (0 + 0)D</FONT>x<SUB>2</SUB><FONT FACE="Symbol">D</FONT>x<SUB>1</SUB><FONT FACE="Symbol"> + (0 + 0)D</FONT>x<SUB>1</SUB><FONT FACE="Symbol"> D</FONT>x<SUB>2</SUB><FONT FACE="Symbol">+ (1 + 0)D</FONT>x<SUB>2</SUB><FONT FACE="Symbol"> D</FONT>x<SUB>2</SUB>

so <FONT FACE="Symbol">D</FONT>s<SUP>2</SUP> = <FONT FACE="Symbol">D</FONT>Y<SUP>2</SUP> + <FONT FACE="Symbol">D</FONT>Z<SUP>2</SUP> = <FONT FACE="Symbol">D</FONT>x<SUB>1</SUB><SUP>2</SUP> + <FONT FACE="Symbol">D</FONT>x<SUB>2</SUB><SUP>2</SUP>

Which recovers Pythagoras' theorem.

However in the most general case the small intervals may not be related by Pythagoras' theorem:

Suppose

<FONT FACE="Symbol">D</FONT>s<SUP>2</SUP> = -<FONT FACE="Symbol">D</FONT>Y<SUP>2</SUP> + <FONT FACE="Symbol">D</FONT>Z<SUP>2</SUP>

So, as before:

<FONT FACE="Symbol" SIZE=1>D</FONT><FONT SIZE=1>Y = </FONT><FONT FACE="Symbol" SIZE=1>D</FONT><FONT SIZE=1>x<SUB>1</SUB></FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>Y/</FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>x<SUB>1</SUB> + </FONT><FONT FACE="Symbol" SIZE=1>D</FONT><FONT SIZE=1>x<SUB>2</SUB></FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>Y/</FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>x<SUB>2</SUB></FONT>

<FONT FACE="Symbol" SIZE=1>D</FONT><FONT SIZE=1>Y<SUP>2</SUP> = </FONT><FONT FACE="Symbol" SIZE=1>D</FONT><FONT SIZE=1>x<SUB>1</SUB></FONT><FONT FACE="Symbol" SIZE=1>D</FONT><FONT SIZE=1>x<SUB>1</SUB></FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>Y/</FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>x<SUB>1</SUB> </FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>Y/</FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>x<SUB>1</SUB> + </FONT><FONT FACE="Symbol" SIZE=1>D</FONT><FONT SIZE=1>x<SUB>1</SUB></FONT><FONT FACE="Symbol" SIZE=1>D</FONT><FONT SIZE=1>x<SUB>2</SUB></FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>Y/</FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>x<SUB>1</SUB></FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>Y/</FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>x<SUB>2</SUB> + </FONT><FONT FACE="Symbol" SIZE=1>D</FONT><FONT SIZE=1>x<SUB>1</SUB></FONT><FONT FACE="Symbol" SIZE=1>D</FONT><FONT SIZE=1>x<SUB>2</SUB></FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>Y/</FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>x<SUB>1</SUB></FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>Y/</FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>x<SUB>2</SUB> + </FONT><FONT FACE="Symbol" SIZE=1>D</FONT><FONT SIZE=1>x<SUB>2</SUB></FONT><FONT FACE="Symbol" SIZE=1>D</FONT><FONT SIZE=1>x<SUB>2</SUB></FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>Y/</FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>x<SUB>2</SUB></FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>Y/</FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>x<SUB>2</SUB></FONT>

<FONT FACE="Symbol" SIZE=1>D</FONT><FONT SIZE=1>Z = </FONT><FONT FACE="Symbol" SIZE=1>D</FONT><FONT SIZE=1>x<SUB>1</SUB></FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>Z/</FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>x<SUB>1</SUB> + </FONT><FONT FACE="Symbol" SIZE=1>D</FONT><FONT SIZE=1>x<SUB>2</SUB></FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>Z/</FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>x<SUB>2</SUB></FONT>

<FONT FACE="Symbol" SIZE=1>D</FONT><FONT SIZE=1>Z<SUP>2</SUP> = </FONT><FONT FACE="Symbol" SIZE=1>D</FONT><FONT SIZE=1>x<SUB>1</SUB></FONT><FONT FACE="Symbol" SIZE=1>D</FONT><FONT SIZE=1>x<SUB>1</SUB></FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>Z/</FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>x<SUB>1</SUB> </FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>Z/</FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>x<SUB>1</SUB> + </FONT><FONT FACE="Symbol" SIZE=1>D</FONT><FONT SIZE=1>x<SUB>1</SUB></FONT><FONT FACE="Symbol" SIZE=1>D</FONT><FONT SIZE=1>x<SUB>2</SUB></FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>Z/</FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>x<SUB>1</SUB></FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>Z/</FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>x<SUB>2</SUB> + </FONT><FONT FACE="Symbol" SIZE=1>D</FONT><FONT SIZE=1>x<SUB>1</SUB></FONT><FONT FACE="Symbol" SIZE=1>D</FONT><FONT SIZE=1>x<SUB>2</SUB></FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>Z/</FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>x<SUB>1</SUB></FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>Z/</FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>x<SUB>2</SUB> + </FONT><FONT FACE="Symbol" SIZE=1>D</FONT><FONT SIZE=1>x<SUB>2</SUB></FONT><FONT FACE="Symbol" SIZE=1>D</FONT><FONT SIZE=1>x<SUB>2</SUB></FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>Z/</FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>x<SUB>2</SUB></FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>Z/</FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>x<SUB>2</SUB></FONT>

So:

<FONT FACE="Symbol" SIZE=1>-D</FONT><FONT SIZE=1>Y<SUP>2</SUP> + </FONT><FONT FACE="Symbol" SIZE=1>D</FONT><FONT SIZE=1>Z<SUP>2</SUP> = </FONT>

<FONT FACE="Symbol" SIZE=1>(-(d</FONT><FONT SIZE=1>Y/</FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>x<SUB>1</SUB></FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>Y/</FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>x<SUB>1</SUB></FONT><FONT FACE="Symbol" SIZE=1>) + dZ</FONT><FONT SIZE=1>/</FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>x<SUB>1</SUB></FONT><FONT FACE="Symbol" SIZE=1>dZ</FONT><FONT SIZE=1>/</FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>x<SUB>1</SUB></FONT><FONT FACE="Symbol" SIZE=1>)D</FONT><FONT SIZE=1>x<SUB>1</SUB></FONT><FONT FACE="Symbol" SIZE=1> D</FONT><FONT SIZE=1>x<SUB>1</SUB></FONT>

<FONT FACE="Symbol" SIZE=1>+ (-(d</FONT><FONT SIZE=1>Y/</FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>x<SUB>2</SUB>Y/</FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>x</FONT><FONT FACE="Symbol" SIZE=1>) + dZ</FONT><FONT SIZE=1>/</FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>x<SUB>2</SUB></FONT><FONT FACE="Symbol" SIZE=1>dZ</FONT><FONT SIZE=1>/</FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>x<SUB>1</SUB></FONT><FONT FACE="Symbol" SIZE=1>)D</FONT><FONT SIZE=1>x<SUB>2</SUB></FONT><FONT FACE="Symbol" SIZE=1>D</FONT><FONT SIZE=1>x<SUB>1</SUB></FONT>

<FONT FACE="Symbol" SIZE=1>+ (-(d</FONT><FONT SIZE=1>Y/</FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>x<SUB>1</SUB></FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>Y/</FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>x<SUB>2</SUB></FONT><FONT FACE="Symbol" SIZE=1>) - dZ</FONT><FONT SIZE=1>/</FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>x<SUB>1</SUB></FONT><FONT FACE="Symbol" SIZE=1>dZ</FONT><FONT SIZE=1>/</FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>x<SUB>2</SUB></FONT><FONT FACE="Symbol" SIZE=1>)D</FONT><FONT SIZE=1>x<SUB>1</SUB></FONT><FONT FACE="Symbol" SIZE=1> D</FONT><FONT SIZE=1>x<SUB>2</SUB></FONT>

<FONT FACE="Symbol" SIZE=1>+ (-(d</FONT><FONT SIZE=1>Y/</FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>x<SUB>2</SUB></FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>Y/</FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>x<SUB>2</SUB></FONT><FONT FACE="Symbol" SIZE=1>) - dZ</FONT><FONT SIZE=1>/</FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>x<SUB>2</SUB></FONT><FONT FACE="Symbol" SIZE=1>dZ</FONT><FONT SIZE=1>/</FONT><FONT FACE="Symbol" SIZE=1>d</FONT><FONT SIZE=1>x<SUB>2</SUB></FONT><FONT FACE="Symbol" SIZE=1>)D</FONT><FONT SIZE=1>x<SUB>2</SUB></FONT><FONT FACE="Symbol" SIZE=1> D</FONT><FONT SIZE=1>x<SUB>2</SUB></FONT>