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basic_tools:pythagorean_theorem [2018/05/06 12:01]
ida [Abstract]
basic_tools:pythagorean_theorem [2018/05/06 12:02] (current)
ida [Abstract]
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 where the $g_{ij}$ are coefficients to be determined. In Pythagoras’s Theorem, it just so happens that we have the values $g_{xx}=1$, $g_{xy}=0$, $g_{yx}=0$, and $g_{yy}=1$. where the $g_{ij}$ are coefficients to be determined. In Pythagoras’s Theorem, it just so happens that we have the values $g_{xx}=1$, $g_{xy}=0$, $g_{yx}=0$, and $g_{yy}=1$.
  
-In other cases (other spaces),the $g_{ij}$ need not have these particular values, and need not be constants - they could even be functions of x and y. This discussion has, so far, been in a space of only two dimensions, but Riemann generalized his ‘distance function’ still further to n dimensions.+In other cases (other spaces),the $g_{ij}$ need not have these particular values, and need not be constants - they could even be functions of x and y. This discussion has, so far, been in a space of only two dimensions, but Riemann generalized his ‘distance function’ still further to n dimensions. [...] 
 +Most remarkable of all, Riemann showed that all the geometric properties of an n-dimensional space can be completely determined by just this ‘distance’,​ $ds$ (the square root of the above equation. For example, if any of the coefficients,​ $g_{ij}$ , are not constant but are functions of the coordinates,​ then the corresponding space is not ‘flat’ (Euclidean) but ‘curved’. This, finally, is what ‘curved’ means: it is a measure of the departure from Euclidean space, and is determined by certain functions (Riemann’s ‘curvature functions’) of the $g_{ij}$ -coefficients. While the ‘distance’ is specific to the given space, its value is invariant as regards which coordinate representation has been adopted.
  
 <​cite>​The Lazy Universe by Coopersmith</​cite>​ <​cite>​The Lazy Universe by Coopersmith</​cite>​
basic_tools/pythagorean_theorem.txt · Last modified: 2018/05/06 12:02 by ida