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basic_tools:pythagorean_theorem [2018/05/06 14:01] ida [Abstract] |
basic_tools:pythagorean_theorem [2018/05/06 14:02] ida [Abstract] |
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- (ii) more than that, it was the shortest ‘distance’ between two points; | - (ii) more than that, it was the shortest ‘distance’ between two points; | ||
- (iii) most amazing of all, Pythagoras’s $(ds)^2$ was but one possibility - more generally, it could be written as: | - (iii) most amazing of all, Pythagoras’s $(ds)^2$ was but one possibility - more generally, it could be written as: | ||
- | $$(ds)2 = g_{xx}(dx)(dx) + +g_{xy}(dx)(dy) + g_{yx}(dy)(dx) + g_{yy}(dy)(dy) ,$$ | + | $$(ds)^2 = g_{xx}(dx)(dx) + +g_{xy}(dx)(dy) + g_{yx}(dy)(dx) + g_{yy}(dy)(dy) ,$$ |
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> |