Search results
Results from the WOW.Com Content Network
Before the full formal development of calculus, the basis for the modern integral form for arc length was independently discovered by Hendrik van Heuraet and Pierre de Fermat. In 1659 van Heuraet published a construction showing that the problem of determining arc length could be transformed into the problem of determining the area under a ...
Animation of the torsion and the corresponding rotation of the binormal vector. Let r be a space curve parametrized by arc length s and with the unit tangent vector T.If the curvature κ of r at a certain point is not zero then the principal normal vector and the binormal vector at that point are the unit vectors
Surfaces that occur in two of the main theorems of vector calculus, Stokes' theorem and the divergence theorem, are frequently given in a parametric form. The curvature and arc length of curves on the surface, surface area, differential geometric invariants such as the first and second fundamental forms, Gaussian, mean, and principal curvatures ...
In all these formulae (h, k) are the center coordinates of the hyperbola, a is the length of the semi-major axis, and b is the length of the semi-minor axis. Note that in the rational forms of these formulae, the points ( −a , 0) and (0 , −a ) , respectively, are not represented by a real value of t , but are the limit of x and y as t tends ...
The coordinate-independent definition of the square of the line element ds in an n-dimensional Riemannian or Pseudo Riemannian manifold (in physics usually a Lorentzian manifold) is the "square of the length" of an infinitesimal displacement [2] (in pseudo Riemannian manifolds possibly negative) whose square root should be used for computing curve length: = = (,) where g is the metric tensor ...
A parametric C r-curve or a C r-parametrization is a vector-valued function: that is r-times continuously differentiable (that is, the component functions of γ are continuously differentiable), where , {}, and I is a non-empty interval of real numbers.
The above condition on the parametrisation imply that the arc length s is a differentiable monotonic function of the parameter t, and conversely that t is a monotonic function of s. Moreover, by changing, if needed, s to – s , one may suppose that these functions are increasing and have a positive derivative.
AP Calculus BC includes all of the topics covered in AP Calculus AB, as well as the following: Convergence tests for series; Taylor series; Parametric equations; Polar functions (including arc length in polar coordinates and calculating area) Arc length calculations using integration; Integration by parts; Improper integrals