basic_tools:pythagorean_theorem
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basic_tools:pythagorean_theorem [2018/05/06 14:01] ida [Abstract] |
basic_tools:pythagorean_theorem [2021/07/04 20:27] (current) 92.75.106.64 |
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| - | $p^2+q^2=r^2$ **if and only if** the triangle is a right triangle | + | $a^2+b^2=c^2$ **if and only if** the triangle is a right triangle |
| <tabbox Abstract> | <tabbox 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.1525608078.txt.gz · Last modified: 2018/05/06 12:01 (external edit)
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