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| [[File:Gauss-Bonnet theorem.svg|thumb|300px|An example of complex region where Gauss-Bonnet theorem can apply. Shows the sign of geodesic curvature.]]
| | Many persons are pretty familiar with terms like "obesity" plus "overweight." What's less certain is how well you know precisely what each one signifies plus how various factors are included. Given the misunderstanding available, a expression like "regular weight obesity" may easily appear more confusing. This article might set the record straight.<br><br>There are certain factors to seriously consider in a workout regimen and you want to seriously consider the following regarding total body fat. The perfect body fat for a wellness guy adult is about 10 - 12%. The ideal body fat for a healthy female adult 14 16%. BMI is normally an indicator of wellness and is computed using charts that are based on age, height plus present body weight. Even when the body fat falls within your BMI or Body Mass Index it is nevertheless possible that you might not fall in the policies of body fat reported above. A really extreme athlete can not even apply to a [http://safedietplansforwomen.com/bmi-chart bmi chart] and it could furthermore rely on the sport. It actually is special to every individual. But if the close to a BMI then closer to desired body fat we are close to achieving the objective of flat hard washboard abs.<br><br>There appear to be a consistent preference among bmi chart men men for a woman BMI index about 20. With BMI above 25 being considered too excellent and below 15 being too skinny.<br><br>Lets state an adult guy is 60 and weighs 200 pounds. According to this chart, his BMI would be 27.1, that puts him into the overweight category. If he loses 17 pounds, the same man, now at 183 pounds, would have a BMI of 24.8, which would place him inside the regular weight category.<br><br>Besides the apparent fat plus BMI measurements, you could like to consider finding a model which has separate methods for guy plus female consumers. Weight distribution for guys and women are different plus scales which have a gender choice might provide more exact readings.<br><br>The waist-to-hip ratio is a useful measure for determining wellness bmi chart women risk due to the site of fat storage. It is calculated by dividing the ratio of ab girth by cool measuring.<br><br>Hydrostatic Weighing - This type of weighing is also known as because underwater weighing. In order to work this, a chair is placed over a scale in an underwater environment. The scale is then zeroed out and when performed, the individual is asked to sit found on the chair. The person should full exhale plus there is a have to bend down till the head is additionally underwater. Once the scale stabilizes, the reading is taken and then chosen to determine the body fat percentage.<br><br>What in the event you have eighty pounds to lose to move your BMI into a healthy range? Don't be frustrated. Remember that the goal is for weight loss you can maintain for a lengthy time. In the race between your tortoise plus the hare, the tortoise won. The consequences of being the hare, that is, finding a quick-fix crash diet, can eventually place you further behind inside the race. Think of the advantages of the goal described above. First, losing 10 % of the weight and keeping it off may enable you feel better and will decrease several of the wellness risks associated with being overweight. Second, we will have learned lifestyle behaviors which can become habit, plus the upcoming milestone won't appear as hard to reach. |
| The '''Gauss–Bonnet theorem''' or '''Gauss–Bonnet formula''' in [[differential geometry]] is an important statement about [[surface]]s which connects their geometry (in the sense of [[curvature]]) to their topology (in the sense of the [[Euler characteristic]]). It is named after [[Carl Friedrich Gauss]] who was aware of a version of the theorem but never published it, and [[Pierre Ossian Bonnet]] who published a special case in 1848.
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| == Statement of the theorem ==
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| Suppose <math>M</math> is a [[Compact space|compact]] two-dimensional [[Riemannian manifold]] with boundary <math>\partial M</math>. Let <math>K</math> be the [[Gaussian curvature]] of <math>M</math>, and let <math>k_g</math> be the [[geodesic curvature]] of <math>\partial M</math>. Then
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| :<math>\int_M K\;dA+\int_{\partial M}k_g\;ds=2\pi\chi(M), \, </math>
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| where ''dA'' is the [[volume element|element of area]] of the surface, and ''ds'' is the line element along the boundary of ''M''. Here, <math>\chi(M)</math> is the [[Euler characteristic]] of <math>M</math>.
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| If the boundary <math>\partial M</math> is [[piecewise smooth]], then we interpret the integral <math>\int_{\partial M}k_g\;ds</math> as the sum of the corresponding integrals along the smooth portions of the boundary, plus the sum of the [[angle]]s by which the smooth portions turn at the corners of the boundary.
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| ==Interpretation and significance==
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| The theorem applies in particular to compact surfaces without boundary, in which case the integral
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| :<math>\int_{\partial M}k_g\;ds</math>
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| can be omitted. It states that the total Gaussian curvature of such a closed surface is equal to 2π times the Euler characteristic of the surface. Note that for [[orientable manifold|orientable]] compact surfaces without boundary, the Euler characteristic equals <math>2-2g</math>, where <math>g</math> is the [[genus (mathematics)|genus]] of the surface: Any orientable compact surface without boundary is topologically equivalent to a sphere with some handles attached, and <math>g</math> counts the number of handles.
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| If one bends and deforms the surface <math>M</math>, its Euler characteristic, being a topological invariant, will not change, while the curvatures at some points will. The theorem states, somewhat surprisingly, that the
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| total integral of all curvatures will remain the same, no matter how the deforming is done. So for instance if you have a sphere with a "dent", then its total curvature is 4π (the Euler characteristic of a sphere being 2), no matter how big or deep the dent.
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| Compactness of the surface is of crucial importance. Consider for instance the [[unit disc|open unit disc]], a non-compact Riemann surface without boundary, with curvature 0 and with Euler characteristic 1: the Gauss–Bonnet formula does not work. It holds true however for the compact closed unit disc, which also has Euler characteristic 1, because of the added boundary integral with value 2π.
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| As an application, a [[torus]] has Euler characteristic 0, so its total curvature must also be zero. If the torus carries the ordinary Riemannian metric from its embedding in '''R'''<sup>3</sup>, then the inside has negative Gaussian curvature, the outside has positive Gaussian curvature, and the total curvature is indeed 0. It is also possible to construct a torus by identifying opposite sides of a square, in which case the Riemannian metric on the torus is flat and has constant curvature 0, again resulting in total curvature 0. It is not possible to specify a Riemannian metric on the torus with everywhere positive or everywhere negative Gaussian curvature.
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| The theorem also has interesting consequences for triangles. Suppose ''M'' is some 2-dimensional Riemannian manifold (not necessarily compact), and we specify a "triangle" on ''M'' formed by three [[geodesic]]s. Then we can apply Gauss–Bonnet to the surface ''T'' formed by the inside of that triangle and the piecewise boundary given by the triangle itself. The geodesic curvature of geodesics being zero, and the Euler characteristic of ''T'' being 1, the theorem then states that the sum of the turning angles of the geodesic triangle is equal to 2π minus the total curvature within the triangle. Since the turning angle at a corner is equal to π minus the interior angle, we can rephrase this as follows: | |
| :The sum of interior angles of a geodesic triangle is equal to π plus the total curvature enclosed by the triangle.
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| In the case of the plane (where the Gaussian curvature is 0 and geodesics are straight lines), we recover the familiar formula for the sum of angles in an ordinary triangle. On the standard sphere, where the curvature is everywhere 1, we see that the angle sum of geodesic triangles is always bigger than π.
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| == Special cases ==
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| A number of earlier results in spherical geometry and hyperbolic geometry over the preceding centuries were subsumed as special cases of Gauss–Bonnet.
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| === Triangles ===
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| In [[spherical trigonometry]] and [[hyperbolic trigonometry]], the area of a triangle is proportional to the amount by which its interior angles fail to add up to 180°, or equivalently by the (inverse) amount by which its exterior angles fail to add up to 360°. | |
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| The area of a [[spherical triangle]] is proportional to its excess, by [[Girard's theorem]] – the amount by which its interior angles add up to more than 180°, which is equal to the amount by which its exterior angles add up to less than 360°. | |
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| The area of a [[hyperbolic triangle]], conversely is proportional to its ''defect,'' as established by [[Johann Heinrich Lambert]]. | |
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| === Polyhedra ===
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| {{main|Descartes' theorem on total angular defect}}
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| [[Descartes' theorem on total angular defect]] of a [[polyhedron]] is the polyhedral analog:
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| it states that the sum of the defect at all the vertices of a polyhedron which is [[homeomorphic]] to the sphere is 4π. More generally, if the polyhedron has [[Euler characteristic]] <math>\chi=2-2g</math> (where ''g'' is the genus, meaning "number of holes"), then the sum of the defect is <math>2\pi \chi.</math>
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| This is the special case of Gauss–Bonnet, where the curvature is concentrated at discrete points (the vertices).
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| Thinking of curvature as a [[Measure (mathematics)|measure]], rather than as a function, Descartes' theorem is Gauss–Bonnet where the curvature is a [[discrete measure]], and Gauss–Bonnet for measures generalizes both Gauss–Bonnet for smooth manifolds and Descartes' theorem.
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| ==Combinatorial analog==
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| There are several combinatorial analogs of the Gauss–Bonnet theorem. We state the following one. Let <math>M</math> be a finite 2-dimensional [[pseudo-manifold]]. Let <math>\chi(v)</math> denote the number of triangles containing the vertex <math>v</math>. Then
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| :<math> \sum_{v\in{\mathrm{int}}{M}}(6-\chi(v))+\sum_{v\in\partial M}(3-\chi(v))=6\chi(M),\ </math>
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| where the first sum ranges over the vertices in the interior of <math>M</math>, the second sum is over the boundary vertices, and <math>\chi(M)</math> is the Euler characteristic of <math>M</math>.
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| Similar formulas can be obtained for 2-dimensional pseudo-manifold when we replace triangles with higher polygons. For polygons of n vertices, we must replace 3 and 6 in the formula above with n/(n-2) and 2n/(n-2), respectively.
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| For example, for [[quadrilateral]]s we must replace 3 and 6 in the formula above with 2 and 4, respectively. More specifically, if <math>M</math> is a closed 2-dimensional [[digital manifold]], the genus turns out <ref>Chen L and Rong Y, Linear Time Recognition Algorithms for Topological Invariants in 3D, arXiv:0804.1982, ICPR 2008</ref>
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| :<math> g = 1+(M_{5} +2 M_{6}-M_{3})/8, \ </math>
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| where <math>M_{i}</math> indicates the number of surface-points each of which has <math>i</math> adjacent points on the surface.
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| ==Generalizations==
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| Generalizations of the Gauss–Bonnet theorem to ''n''-dimensional Riemannian manifolds were found in the 1940s, by [[Carl B. Allendoerfer|Allendoerfer]], [[André Weil|Weil]], and [[Shiing-Shen Chern|Chern]]; see [[generalized Gauss–Bonnet theorem]] and [[Chern–Weil homomorphism]]. The [[Riemann–Roch theorem]] can also be seen as a generalization of Gauss–Bonnet.
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| An extremely far-reaching generalization of all the above-mentioned theorems is the [[Atiyah–Singer index theorem]].
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| A generalization to 2-manifolds that need not be compact is [[Cohn-Vossen's inequality]].
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| ==References==
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| {{reflist}}
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| ==External links==
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| * {{springer|title=Gauss-Bonnet theorem|id=p/g043410}}
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| *[http://mathworld.wolfram.com/Gauss-BonnetFormula.html Gauss–Bonnet Theorem] at Wolfram Mathworld
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| {{DEFAULTSORT:Gauss-Bonnet theorem}}
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| [[Category:Theorems in differential geometry]]
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| [[Category:Riemann surfaces]]
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Many persons are pretty familiar with terms like "obesity" plus "overweight." What's less certain is how well you know precisely what each one signifies plus how various factors are included. Given the misunderstanding available, a expression like "regular weight obesity" may easily appear more confusing. This article might set the record straight.
There are certain factors to seriously consider in a workout regimen and you want to seriously consider the following regarding total body fat. The perfect body fat for a wellness guy adult is about 10 - 12%. The ideal body fat for a healthy female adult 14 16%. BMI is normally an indicator of wellness and is computed using charts that are based on age, height plus present body weight. Even when the body fat falls within your BMI or Body Mass Index it is nevertheless possible that you might not fall in the policies of body fat reported above. A really extreme athlete can not even apply to a bmi chart and it could furthermore rely on the sport. It actually is special to every individual. But if the close to a BMI then closer to desired body fat we are close to achieving the objective of flat hard washboard abs.
There appear to be a consistent preference among bmi chart men men for a woman BMI index about 20. With BMI above 25 being considered too excellent and below 15 being too skinny.
Lets state an adult guy is 60 and weighs 200 pounds. According to this chart, his BMI would be 27.1, that puts him into the overweight category. If he loses 17 pounds, the same man, now at 183 pounds, would have a BMI of 24.8, which would place him inside the regular weight category.
Besides the apparent fat plus BMI measurements, you could like to consider finding a model which has separate methods for guy plus female consumers. Weight distribution for guys and women are different plus scales which have a gender choice might provide more exact readings.
The waist-to-hip ratio is a useful measure for determining wellness bmi chart women risk due to the site of fat storage. It is calculated by dividing the ratio of ab girth by cool measuring.
Hydrostatic Weighing - This type of weighing is also known as because underwater weighing. In order to work this, a chair is placed over a scale in an underwater environment. The scale is then zeroed out and when performed, the individual is asked to sit found on the chair. The person should full exhale plus there is a have to bend down till the head is additionally underwater. Once the scale stabilizes, the reading is taken and then chosen to determine the body fat percentage.
What in the event you have eighty pounds to lose to move your BMI into a healthy range? Don't be frustrated. Remember that the goal is for weight loss you can maintain for a lengthy time. In the race between your tortoise plus the hare, the tortoise won. The consequences of being the hare, that is, finding a quick-fix crash diet, can eventually place you further behind inside the race. Think of the advantages of the goal described above. First, losing 10 % of the weight and keeping it off may enable you feel better and will decrease several of the wellness risks associated with being overweight. Second, we will have learned lifestyle behaviors which can become habit, plus the upcoming milestone won't appear as hard to reach.