integral curve is a parametric curve that represents a specific solution to an ordinary differential equation or system of equations. Integral curves...

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...

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...

Gaussian is a characteristic symmetric "bell curve" shape. The parameter a is the height of the curve's peak, b is the position of the center of the peak...

Arc length is the distance between two points along a section of a curve. Development of a formulation of arc length suitable for applications to mathematics...

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...

contours provide a precise definition of the curves on which an integral may be suitably defined. A curve in the complex plane is defined as a continuous...

tautochrone curve or isochrone curve (from Ancient Greek ταὐτό (tauto-) 'same' ἴσος (isos-) 'equal' and χρόνος (chronos) 'time') is the curve for which...

In mathematics, the Cauchy integral theorem (also known as the Cauchy–Goursat theorem) in complex analysis, named after Augustin-Louis Cauchy (and Édouard...

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...

usually known as normalized Fresnel integrals. The Euler spiral, also known as a Cornu spiral or clothoid, is the curve generated by a parametric plot of...

which is what the integral represents. If the curve dips below the x-axis, the integral gives a signed area. This means the integral adds the part above...

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...

(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 ) ...

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...

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...

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...

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...

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...

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...

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...

manifestation of Hamilton's equations in classical mechanics. The integral curves of a Hamiltonian vector field represent solutions to the equations...

Riemann–Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes. The definition of this integral was...

change in θ is equal to the integral of dθ. We can therefore express the winding number of a differentiable curve as a line integral: wind ( γ , 0 ) = 1 2 π...

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...

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...

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...

differential calculus and integral calculus. The former concerns instantaneous rates of change, and the slopes of curves, while the latter concerns accumulation...

and dominated convergence theorem). While the Riemann integral considers the area under a curve as made out of vertical rectangles, the Lebesgue definition...

vector field always gives rise to integral curves; Frobenius gives compatibility conditions under which the integral curves of r vector fields mesh into coordinate...