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Term in mathematics
integral curve is a parametric curve that represents a specific solution to an ordinary differential equation or system of equations. Integral curves
Integral_curve
Definite integral of a scalar or vector field along a path
mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms path integral, curve integral, and curvilinear
Line_integral
Operation in calculus
For example, a line integral is defined for functions of two or more variables, and the interval of integration is replaced by a curve connecting two points
Integral
Mathematical function
Gaussian is a characteristic symmetric "bell curve" shape. The parameter a is the height of the curve's peak, b is the horizontal position of the center
Gaussian_function
Vector calculus construction
is called nonautonomous by definition. An integral curve of the equation above (also called an integral curve of X) is a map α : I ⊂ R ⟶ M {\displaystyle
Time_dependent_vector_field
Distance along a curve
hyperbolic arc led to the development of the elliptic integrals. In most cases, including even simple curves, there are no closed-form solutions for arc length
Arc_length
Branch of mathematics
differential calculus and integral calculus. Differential calculus studies instantaneous rates of change and slopes of curves; integral calculus studies accumulation
Calculus
be explained by plotting the integral curve Z = Z ( V ) {\displaystyle Z=Z(V)} as shown in the figure as a solid curve. The point A {\displaystyle A}
Guderley–Landau–Stanyukovich problem
Guderley–Landau–Stanyukovich_problem
Curve for which the time to roll to the end is equal for all starting points
tautochrone curve or isochrone curve (from Ancient Greek ταὐτό (tauto-) 'same', ἴσος (isos-) 'equal' and χρόνος (chronos) 'time') is the curve for which
Tautochrone_curve
Theorem in complex analysis
In mathematics, the Cauchy integral theorem (also known as the Cauchy–Goursat theorem) in complex analysis, named after Augustin-Louis Cauchy (and Édouard
Cauchy's_integral_theorem
Curve used in computer graphics and related fields
A Bézier curve (/ˈbɛz.i.eɪ/ BEH-zee-ay, French pronunciation: [bezje]) is a parametric curve used in computer graphics and related fields. A sequence
Bézier_curve
Algebraic variety
modular group of integral 2×2 matrices SL(2, Z). The term modular curve can also be used to refer to the compactified modular curves X(Γ) which are compactifications
Modular_curve
Vector field on tangent bundle
Usually a spray is required to be homogeneous in the sense that its integral curves t→ΦHt(ξ)∈TM obey the rule ΦHt(λξ)=ΦHλt(ξ) in positive re-parameterizations
Spray_(mathematics)
Topics referred to by the same term
Path integral may refer to: Line integral, the integral of a function along a curve Contour integral, the integral of a complex function along a curve used
Path_integral
Method of evaluating certain integrals along paths in the complex plane
contour integral can be defined in analogy to the line integral by first defining the integral along a directed smooth curve in terms of an integral over
Contour_integration
Theorem in calculus relating line and double integrals
vector calculus, Green's theorem relates a line integral around a simple closed curve C to a double integral over the plane region D (surface in R 2 {\displaystyle
Green's_theorem
Coordinates to capture characteristics of rotating frames of reference
take an integral curve (blue helical curve in Fig. 1) of the spacelike basis vector p → 3 {\displaystyle {\vec {p}}_{3}} , we obtain a curve which we
Born_coordinates
On finding a maximal set of solutions of a system of first-order homogeneous linear PDEs
vector field always gives rise to integral curves; Frobenius gives compatibility conditions under which the integral curves of r vector fields mesh into coordinate
Frobenius theorem (differential topology)
Frobenius_theorem_(differential_topology)
Mechanical graphing calculator for solving differential equations
(traces) the integral curve Y = F ( x ) = ∫ f ( x ) d x , {\displaystyle Y=F(x)=\int f(x)dx,} when we are given the differential curve, y = f ( x )
Integraph
Vector field defined for any energy function
manifestation of Hamilton's equations in classical mechanics. The integral curves of a Hamiltonian vector field represent solutions to the equations
Hamiltonian_vector_field
Basic integral in elementary calculus
Lebesgue integral. Consider a curve on a graph which stays above the x-axis, beginning at x = a and ending at x = b. The area under that curve, from a
Riemann_integral
Set of integral curves of a vector field
relativity, a congruence (more properly, a congruence of curves) is the set of integral curves of a (nowhere vanishing) vector field in a four-dimensional
Congruence (general relativity)
Congruence_(general_relativity)
Representation of a curve by a function of a parameter
the parametric form can be straightforwardly deduced. Bézier curve Curve Integral curve Parametrization by arc length Parametric derivative Parametric
Parametric_equation
Visual aid to depiction of a vector field
visual aid for visualizing vector fields. It consists of an imaginary integral curve which is tangent to the field vector at each point along its length
Field_line
Method of mathematical integration
and dominated convergence theorem). While the Riemann integral considers the area under a curve as made out of vertical rectangles, the Lebesgue definition
Lebesgue_integral
Concepts in mathematics
the flow domain. For each p ∈ M the map Dp → M is the unique maximal integral curve of V starting at p. A global flow is one whose flow domain is all of
Vector_flow
Algebraic curve in mathematics
mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point O. An elliptic curve is defined over
Elliptic_curve
Study of curves from a differential point of view
geometry of curves is the branch of geometry that deals with smooth curves in the plane and the Euclidean space by methods of differential and integral calculus
Differentiable_curve
Special function defined by an integral
ComplexPlot3D Mathematics portal Böhmer integral Fresnel zone Track transition curve Euler spiral Zone plate Dirichlet integral Abramowitz & Stegun 1983, eqn 7
Fresnel_integral
Lie group homomorphism from the real numbers
one parameter group of local diffeomorphisms, sending points along integral curves of the vector field. The local flow of a vector field is used to define
One-parameter_group
Swedish mathematician (1861–1935)
investigation of integral curves to first-order differential equations, in particular he was intrigued by the complicated behaviour of the integral curves in the
Ivar_Otto_Bendixson
Mathematical function having a characteristic S-shaped curve or sigmoid curve
neurons. Sigmoid curves are also common in statistics as cumulative distribution functions (which go from 0 to 1), such as the integrals of the logistic
Sigmoid_function
Study of sudden qualitative behavior changes caused by small parameter changes
qualitative or topological structure of a given family of curves, such as the integral curves of a family of vector fields, and the solutions of a family
Bifurcation_theory
Differentiation under the integral sign formula
The double integrals are surface integrals over the surface Σ, and the line integral is over the bounding curve ∂Σ. The Leibniz integral rule can be
Leibniz_integral_rule
How many times curves wind around each other
linking of two closed curves in three-dimensional space. Intuitively, the linking number represents the number of times that each curve winds around the other
Linking_number
Visual representation of solutions to a differential equation
of the tangent to the graph of the differential equation's solution (integral curve) at each point (x, y) as a function of the point coordinates. It can
Slope_field
Extends the Jordan curve theorem to characterize the inner and outer regions
Cauchy integral formula for the winding number, it can be seen that the winding number is constant on connected components of the complement of the curve, is
Schoenflies_problem
Concept of complex analysis
powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals and infinite series as well
Residue_theorem
Finitely many for a smooth algebraic curve of genus > 0 defined over a number field
mathematics, Siegel's theorem on integral points states that a curve of genus greater than zero has only finitely many integral points over any given number
Siegel's theorem on integral points
Siegel's_theorem_on_integral_points
Vector field on a pseudo-Riemannian manifold that preserves the metric tensor
{\displaystyle \nabla _{\partial _{x}}g} transports the metric tensor along an integral curve generated by the vector field (whose image is parallel to the x-axis)
Killing_vector_field
Map from a Lie algebra to its Lie group
constructed as the integral curve of either the right- or left-invariant vector field associated with X {\displaystyle X} . That the integral curve exists for
Exponential_map_(Lie_theory)
then θ defined as above is 0, and hence the curve is easily verified to be a solution (i.e. an integral curve) for the above Pfaffian system for any nonzero
Integrability conditions for differential systems
Integrability_conditions_for_differential_systems
Part of mathematics that addresses the stability of solutions
A)^{2}=-(a-b)^{2}\leq 0\\\det A=ab>0\end{cases}}} . The origin is a source, with integral curves of form y = c x b / a {\displaystyle y=cx^{b/a}} Similarly for a ,
Stability_theory
Special function defined by an integral
In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied
Elliptic_integral
Integral of drug concentration in blood plasma over time
In the field of pharmacokinetics, the area under the curve (AUC) is the definite integral of the concentration of a drug in blood plasma as a function
Area under the curve (pharmacokinetics)
Area_under_the_curve_(pharmacokinetics)
Assignment of a vector to each point in a subset of Euclidean space
in each point along the curve. The line integral is constructed analogously to the Riemann integral and it exists if the curve is rectifiable (has finite
Vector_field
Curve whose curvature changes linearly
curve whose curvature changes linearly with its curve length (the curvature of a circular curve is equal to the reciprocal of the radius). This curve
Euler_spiral
In mathematics, singular integral operators on closed curves arise in problems in analysis, in particular complex analysis and harmonic analysis. The
Singular integral operators on closed curves
Singular_integral_operators_on_closed_curves
Generalization of elliptic integrals
e. an elliptic curve, such functions are the elliptic integrals. Such functions were first introduced to study hyperelliptic integrals, i.e., for the
Abelian_integral
(mathematics) jet bundle Frobenius theorem (differential topology) Integral curve Diffeomorphism Large diffeomorphism Orientability characteristic class
List of differential geometry topics
List_of_differential_geometry_topics
Technique for solving hyperbolic partial differential equations
solution to (1) is the union of integral curves of the vector field (2). Each integral curve is called a characteristic curve of the PDE (1) equation and
Method_of_characteristics
determine when, given two periodic functions with the same period, the integral curve is closed. The filling area conjecture, that a hemisphere has the minimum
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Differential equation containing derivatives with respect to only one variable
where I {\displaystyle I} is an interval, is called a solution or integral curve for F {\displaystyle F} , if u {\displaystyle u} is n {\displaystyle
Ordinary differential equation
Ordinary_differential_equation
Topics referred to by the same term
the curve is a common application of definite integrals. It may also refer to: Area under the curve (pharmacokinetics), a pharmacology integral Receiver
Area under the curve (disambiguation)
Area_under_the_curve_(disambiguation)
Evaluates a line integral through a gradient field using the original scalar field
integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve.
Gradient_theorem
Curve defined as zeros of polynomials
mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective
Algebraic_curve
{\displaystyle \nabla _{{\vec {e}}_{0}}{\vec {e}}_{0}=0} which means that the integral curves of the timelike unit vector field e0 are timelike geodesics, and also
Monochromatic electromagnetic plane wave
Monochromatic_electromagnetic_plane_wave
Relationship between derivatives and integrals
has a holomorphic antiderivative F on U. Then for every curve γ : [a, b] → U, the curve integral can be computed as ∫ γ f ( z ) d z = F ( γ ( b ) ) − F
Fundamental theorem of calculus
Fundamental_theorem_of_calculus
Generalization of definite integrals to functions of multiple variables
calculus), a multiple integral is a definite integral of a function of several real variables, for instance, f(x, y) or f(x, y, z). Integrals of a function of
Multiple_integral
Type of calculus problem
authors.[citation needed] Boundary value problem Constant of integration Integral curve Norton's dome Coddington, Earl A.; Levinson, Norman (1955). Theory of
Initial_value_problem
Generalization of the Riemann integral
Riemann–Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes. The definition of this integral was
Riemann–Stieltjes_integral
Fastest curve descent without friction
mathematics, a brachistochrone curve (from Ancient Greek βράχιστος χρόνος (brákhistos khrónos) 'shortest time'), or curve of fastest descent, is the one
Brachistochrone_curve
Result in integral geometry
(also Cauchy-Crofton formula) is a classic result of integral geometry relating the length of a curve to the expected number of times a "random" line intersects
Crofton_formula
Theorem in topology
Lipschitz continuous. The following conditions are equivalent: Every integral curve of X starting in F remains in F. (X(m),v) ≤ 0 for every exterior normal
Bony–Brezis_theorem
Generalization of Riemannian manifolds
is a geodesic of (M, F) if and only if its tangent curve γ': [a, b] → TM∖{0} is an integral curve of the smooth vector field H on TM∖{0} locally defined
Finsler_manifold
Diagnostic plot of binary classifier ability
A receiver operating characteristic curve, or ROC curve, is a graphical plot that illustrates the performance of a binary classifier model (although it
Receiver operating characteristic
Receiver_operating_characteristic
Curve with axes measuring the area of another curve
quadrator 'squarer') is a curve that can be used for quadrature, constructing the area under another curve. For instance, in integral calculus as developed
Quadratrix
Line integral of the fluid velocity around a closed curve
In physics, circulation is the line integral of a vector field around a closed curve embedded in the field. In fluid dynamics, the field is the fluid velocity
Circulation_(physics)
parametrized by pseudoholomorphic curves, and so the path integrals reduce to integrals over moduli spaces of pseudoholomorphic curves (or rather stable maps),
Pseudoholomorphic_curve
Theorem in mathematics
to the curve at the point ( x , f ( x ) ) {\displaystyle (x,f(x))} . Thus the mean value theorem says that given any chord of a smooth curve, we can
Mean_value_theorem
Special function defined by an integral
mathematics, trigonometric integrals are a family of nonelementary integrals involving trigonometric functions. The different sine integral definitions are Si
Trigonometric_integral
Formulation of quantum mechanics
The path-integral formulation of quantum mechanics generalizes the action principle of classical mechanics. It replaces the classical notion of a single
Path-integral_formulation
Number of times a curve wraps around a point in the plane
{\displaystyle \gamma } is a closed curve, the total change in ln ( r ) {\displaystyle \ln(r)} is zero, and thus the integral of d z z {\textstyle {\frac {dz}{z}}}
Winding_number
Instrument in differential geometry
interested in finding integral curves to X {\displaystyle X} . More precisely, given p ∈ M {\displaystyle p\in M} one is interested in curves γ p : R → M {\displaystyle
Fundamental_vector_field
Approximation technique in integral calculus
right Riemann sum with the integral of x ↦ x2 over [ 0 , 2 ] {\textstyle [0,2]} . Taking an example, the area under the curve y = x2 over [0, 2] can be
Riemann_sum
Topics referred to by the same term
Congruence (manifolds), in the theory of smooth manifolds, the set of integral curves defined by a nonvanishing vector field defined on the manifold Congruence
Congruence
In the theory of smooth manifolds, a congruence is the set of integral curves defined by a nonvanishing vector field defined on the manifold. Congruences
Congruence_(manifolds)
Type of spline curve
In mathematics, a Pythagorean hodograph curve or PH curve is a curve defined by a polynomial parametric equation for which the speed (the derivative of
Pythagorean_hodograph_curve
Intrinsic geometric structures in mathematics
corresponding to x in F, travel s {\displaystyle {\sqrt {s}}} along the integral curve for U1 to the point B at α s ( x ) {\displaystyle \alpha _{\sqrt {s}}(x)}
Riemannian connection on a surface
Riemannian_connection_on_a_surface
conductor of an elliptic curve over the field of rational numbers (or more generally a local or global field) is an integral ideal, which is analogous
Conductor of an elliptic curve
Conductor_of_an_elliptic_curve
Curved stands of seating located at sports stadiums
particularly in Italy; so named, originally, due to their curved or bending shape. The curva plays an integral part in the culture of Ultras and European football
Curva
Result in general relativity
interpreted as a family or congruence of nonintersecting world lines via the integral curve, not necessarily geodesics), Raychaudhuri's equation in D {\displaystyle
Raychaudhuri_equation
Equations with an unknown function under an integral sign
analysis, integral equations are equations in which an unknown function appears under an integral sign. In mathematical notation, integral equations may
Integral_equation
Geometric model of the planar projection of the physical universe
bijective parametrization of the curve C such that r(a) and r(b) give the endpoints of C. A double integral refers to an integral within a region D in R2 of
Euclidean_plane
Fractal curve resembling a blancmange pudding
mathematics, the blancmange curve is a self-affine fractal curve constructible by midpoint subdivision. It is also known as the Takagi curve, after Teiji Takagi
Blancmange_curve
American mathematician
that every non-vanishing vector field on the 3-sphere has a closed integral curve. In 1995, he demonstrated that Sergei Novikov's compact leaf theorem
Paul_A._Schweitzer
French mathematician (born 1956)
Picard–Lindelöf theorem deals with integral curves of Lipschitz-continuous vector fields. By viewing integral curves as characteristic curves for a transport equation
Pierre-Louis_Lions
Cubic plane curve
\\y&=2a\cos ^{2}\theta .\end{aligned}}} The main properties of this curve can be derived from integral calculus. The area between the witch and its asymptotic line
Witch_of_Agnesi
Mathematical element
said to be integral over a subring A of B if b is a root of some monic polynomial over A. If A, B are fields, then the notions of "integral over" and of
Integral_element
Integration over a non-flat region in 3D space
calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the
Surface_integral
Geometric model of the physical space
infinitesimal increment of that curve. For some scalar field f : U ⊆ Rn → R, the line integral along a piecewise smooth curve C ⊂ U is defined as ∫ C f d
Three-dimensional_space
Integral transform
In mathematics, the Riemann–Liouville integral associates with a real function f : R → R {\displaystyle f:\mathbb {R} \rightarrow \mathbb {R} } another
Riemann–Liouville_integral
Algebraic curve
this is a curve) by the normalization (integral closure) process. It turns out that after doing this, there is an open cover of the curve by two affine
Hyperelliptic_curve
Lebesgue-Stieltjes integration
general measure-theoretic framework. The Lebesgue–Stieltjes integral is the ordinary Lebesgue integral with respect to a measure known as the Lebesgue–Stieltjes
Lebesgue–Stieltjes integration
Lebesgue–Stieltjes_integration
French mathematician, physicist and engineer (1854–1912)
particular, Poincaré investigated the nature of the trajectories of the integral curves in the plane, gave a classification of singular points (saddle, focus
Henri_Poincaré
Mathematically-calculated curve in which a straight section changes into a curve
A transition curve (also, spiral easement or, simply, spiral) is a spiral-shaped length of highway or railroad track that is used between sections having
Track_transition_curve
Curve whose range contains the unit square
In mathematical analysis, a space-filling curve is a curve whose range reaches every point in a higher dimensional region, typically the unit square (or
Space-filling_curve
Special mathematical function
closed form of integrals of the Fermi–Dirac distribution and the Bose–Einstein distribution, and is also known as the Fermi–Dirac integral or the Bose–Einstein
Polylogarithm
Mathematical measure of how much a curve or surface deviates from flatness
the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or surface is contained
Curvature
Mathematical method in calculus
partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative
Integration_by_parts
INTEGRAL CURVE
INTEGRAL CURVE
Boy/Male
Arabic, Muslim
Glitter; Curve; Shine; Brightness
Surname or Lastname
English
English : topographic name for someone who lived on a curved or irregularly shaped piece of land, from Old English wÅh ‘curved’, ‘crooked’ + land ‘land’, ‘estate’, or a habitational name from Woolland in Dorset, named from an Old English winn, wynn ‘meadow’, ‘pasture’ + land ‘land’, ‘estate’.
Surname or Lastname
Irish
Irish : reduced Anglicized form of either of two Gaelic names, Ó DuibhÃn ‘descendant of DuibhÃn’, a byname meaning ‘little black one’, or Ó DaimhÃn ‘descendant of DaimhÃn’, a byname meaning ‘fawn’, ‘little stag’. These are attenuated versions of Ó Dubháin and Ó Damháin, and are the phonetic origin of Anglicizations with an internal v (as opposed to w, as in Dewan, or monosyllabic forms with an o or u) (see Doane).English and French : nickname, of literal or ironic application, from Middle English, Old French devin, divin ‘excellent’, ‘perfect’ (Latin divinus ‘divine’).
Boy/Male
Indian
Internal Cleanliness
Boy/Male
Tamil
Vakratunda | வகà¯à®°à®¤à¯à®¨à¯à®Ÿà®¾Â
Curved trunk Lord, Lord Ganesh
Vakratunda | வகà¯à®°à®¤à¯à®¨à¯à®Ÿà®¾Â
Boy/Male
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Oriya, Sanskrit, Telugu
Curved
Boy/Male
Muslim
Glitter, Curve, Shine
Girl/Female
Arabic, Bengali, Gujarati, Hindu, Indian, Kannada, Marathi, Muslim, Punjabi, Sikh, Sindhi, Telugu
Heart; Inner Beauty; Fame; Internal Nature; Wisdom
Boy/Male
Indian
Glitter, Curve, Shine
Boy/Male
Native American
Curved bear claw.
Boy/Male
Tamil
Curved, Lord Krishna
Boy/Male
Hindu
Curved, Lord Krishna
Surname or Lastname
English and French
English and French : nickname for a handsome man (perhaps also ironically for an ugly one), from Old French beu, bel ‘fair’, ‘lovely’ (Late Latin bellus).Hungarian (Bél) : from the old secular Hungarian name Bél, or alternatively from bél ‘internal part’, probably an occupational name for a servant who worked in the household.Czech (BÄ›l) from Czech bÃlý ‘white’.
Boy/Male
Native American
Curve like foxtail grass.
Boy/Male
Hindu
Curved trunk Lord, Lord Ganesh
Girl/Female
American, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sindhi, Tamil, Telugu
Plucked Flower; Voice of Heart; Woman; Intellect; Behold of Any Beautiful Scene; Internal Beauty
Surname or Lastname
English
English : unexplained.
Boy/Male
Tamil
Harakodhandarama | ஹராகோதாநà¯à®¤à®¾à®°à®®à®¾à®‚
Armed with the curved kodhanda bow
Harakodhandarama | ஹராகோதாநà¯à®¤à®¾à®°à®®à®¾à®‚
Boy/Male
Hindu
Armed with the curved kodhanda bow
Surname or Lastname
English
English : occupational name for a gamekeeper or warden, from Middle English ranger, an agent derivative of range(n) ‘to arrange or dispose’.German : variant of Rang 2, 3.German : habitational name for someone from any of the places named Rangen, in Alsace, Bavaria, and Hesse.French : from a Germanic personal name formed with rang, rank ‘curved’, ‘bent’; ‘slender’.A person called Ranger from La Rochelle, France, is documented in Quebec City in 1684 with the secondary surname
INTEGRAL CURVE
INTEGRAL CURVE
INTEGRAL CURVE
INTEGRAL CURVE
INTEGRAL CURVE
INTEGRAL CURVE
INTEGRAL CURVE