In mathematics, curve integral is a kind of integral.The value taking edge of integral function is notsectionIt is a specific curve, called the integral path.There are many kinds of curve integrals. When the integral path is a closed curve, it is called loop integral or contour integral.Curve integrals can be divided into: the first type of curve integrals and the second type of curve integrals.
Chinese name
curvilinear integral
Foreign name
Line Integral
Basic Introduction
ρ (x, y) ds is called curve integral of arc length
Definition
Arc length curve integral is also called the first kind of curve integral
Let's take a look at an example first: let a curved component occupy a curve on the xOy surface, let the density distribution function of the component be ρ (x, y), let ρ (x, y) be defined on L and continuous on L, and find the mass of the component.For objects with uniform density, the mass can be directly calculated by ρ V;For objects with uneven density, curve integration is needed, dm=ρ (x, y) ds; so m=πρ (x, y) ds; L is the integration path, and πρ (x, y) ds is called the curve integration of arc length.
definition
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Let L be a smooth simple curve arc on the xOy plane, f (x, y) is bounded on L, and a point sequence is inserted arbitrarily on L
Divide L into n small arc segments
The length of is ds, and
Is any point on L, multiply
, and sum
, λ=max (ds), if
The limit of exists when λ → 0, and the limit value and the division of L
If the method of L is independent, the limit value is called the curve integral of f (x, y) on L to the arc length, which is recorded as:
；Where f (x, y) is called the integrand, L is called the integral curve, and the curve integral of arc length is also called the first kind of curve integral.
(The above definition is not completely rigorous, and a new definition is given): in the vector field A, take any smooth curve c connecting points P0 and P1, where vector OP0 is recorded as R0, and vector OP1 is recorded as R1ΔRIt is located on the tangent line of curve C, with the tangent point as the starting point
(where Δ R is bold) is equal to the small vector of arc element Δ R, and is used as the scalar product
, A is the vector of Δ R starting point,
Is the tangent of A in the arc
Projection on.Add the scalar product of all arc elements Δ R, increase the number of arc elements without limit and make the length of each arc element approach 0, and find the limit of U, so
。U is called the curve integral of vector A along curve c.
The difference between the two kinds of curve integrals mainly lies in the difference of integral elements;The integral element of curve integral for arc length is the arc length element ds;For example, the curve integral of L is divided into f (x, y) * ds. The integral element of the curve integral of the coordinate axis is the coordinate element dx or dy, for example, the curve integral of L 'is divided into P (x, y) dx+Q (x, y) dy. But the curve integral of the arc length is generally positive because of its physical significance, and the curve integral of the coordinate axis can obtain different symbols according to the different paths[1]。
Related concepts
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Integral connection
The curve integral of arc length and the curve integral of coordinate axis can be converted to each otherArc differentiationformula
, or
；In this way, the curve integral of arc length can be converted into the curve integral of coordinate axis.
In curve integral, the integrand can bescalarFunction orVector function。The integral value isfunction valueMultiply by the correspondingweight(Generallyarc lengthWhen the integral function is a vector function, it is generally the function value and curveinfinitesimalVectorialScalar product）PostRiemann and。Curve integral with weight and generalsectionThe main differences between integrals on.Many simple formulas in physics (for example) appear in the form of curve integral after generalization（
quantum mechanicsIn“Curve integral form”It is different from curve integral, because the integral used in the form of curve integral isfunction space Onfunctional integrals, that is, about each path in the spaceprobabilityFunction.However, curve integrals still play an important role in quantum mechanics. For example, complex contour integrals are often used to calculateQuantum scatteringTheoreticalProbability amplitude。
In variousConservatismOfsiteAre path independent. A common example isfield of gravityorelectric field。When calculating the work done by this field, an appropriate path can be selected for integration, which makes the calculation simple.For example:
