Curvature

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	riemannian geometry
	multivariable calculus
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The primordial example of extrinsic curvature is that of a Circle which has curvature equal to the inverse of its Radius everywhere. Smaller circles bend more sharply, and hence have higher curvature. Further, the curvature of a smooth curve is defined as the curvature of its Osculating Circle at each point. In a plane, this is a Scalar quantity, but in three or more dimensions it is described by a Curvature Vector that takes into account direction of the bend as well as its sharpness. The curvature of more complex objects (such as Surface s or even curved ''n''-dimensional Space s) are described by more complex objects from Linear Algebra , such as the general Riemann Curvature Tensor . The remainder of this article discusses, from a mathematical perspective, some geometric examples of curvature: the curvature of a curve embedded in a plane and the curvature of a surface in Euclidean space. See the links below for further reading. CURVATURE OF PLANE CURVES For a Plane Curve ''C'', the curvature at a given point ''P'' has a Magnitude equal to the '' Reciprocal '' of the Radius of an Osculating Circle (a circle that "kisses" or closely touches the curve at the given point, its center shaping the curve's Evolute ), and is a vector pointing in the direction of that circle's center. The smaller the radius ''r'' of the osculating circle, the larger the magnitude of the curvature (1/''r'') will be; so that where a curve is "nearly straight", the curvature will be close to zero, and where the curve undergoes a tight turn, the curvature will be large in magnitude. The magnitude of curvature at points on physical curves can be measured in Diopter s (also spelled dioptre); a diopter has the dimension ''length-1''. A straight line has curvature 0 everywhere; a circle of radius ''r'' has curvature 1/''r'' everywhere. Local expressions For a plane curve given parametrically as $c(t)\; =\; (x(t),y(t)).$ the curvature is :$F${Link without Title} = rac{x'y''-y'x''}{(x'^2+y'^2)^{3/2}}. For the less general case of a plane curve given explicitly as $y=f(x)$ the curvature is : This form is widely used in Engineering , for example; to derive the equations of Bending of beams, deriving approximations to the fluid flow around surfaces (in aeronautics) and the free surface boundary conditions in ocean waves. In all such applications, the assumption is made that the Slope is small compared with unity, so that the approximation: : may be used. This approximation yields a straightforward linear equation describing the phenomenon, which would otherwise remain intractable. If a curve is defined in polar coordinates as $r(\; heta)$, then its curvature is where here the prime refers to differentiation with respect to $heta$. Example Consider the Parabola $y\; =\; x^2$. We can parametrize the curve simply as $c(t)\; =\; (t,\; t^2)\; =\; (x,\; y)$, : $\backslash dot\{x\}=\; 1,\backslash quad\backslash ddot\{x\}=0,\backslash quad\; \backslash dot\{y\}=\; 2t,\backslash quad\backslash ddot\{y\}=2.$ Substituting

Given a Function ''r''(''t'') with values in R³ , the curvature at a given value of $t$ is