Curve integration of arc length (the first curve integration): Let L be a smooth and simple curve arc on xOy plane, f(x, y) is bounded on L, and a series of points M 1, m2, m3 … are arbitrarily inserted on L, and Mn divides L into n small arc segments. The length of δ Li is ds, and Mi(x, y) is any point on L. Let λ=max(ds). If the limit of σ f(x, y) i * ds exists when λ→0, and the limit value has nothing to do with the division of L and the method of taking Mi in L, it is called the curve integral of F (x, y) to the arc length on L, and it is recorded as ∫f(x, y) * ds; Where f(x, y) is called integrand, l is called integral curve, and the curve integral of arc length is also called the first kind of curve integral.

Curve integral of coordinate axis (curve integral of the second kind)

The difference between two kinds of curve integrals mainly lies in the difference of integral elements; The integral element of arc-length curve integral is arc-length element ds; For example, the curve integral of l ∫f(x, y)*ds. The integral element of the curve integral of the coordinate axis is the coordinate element dx or dy, such as the curve integral of l' ∫P(x, y)dx+Q(x, y)dy. But the curve integral of arc length is usually positive because of its physical meaning, while the curve integral of coordinate axis can get different symbols according to different paths.

The curve integral of arc length and the curve integral of coordinate axis can be transformed into each other by using the arc differential formula DS = √ [1+(dy/dx) 2] * dx;

Or ds = √ [1+(dx/dy) 2] * dy; In this way, the curve integral of arc length can be transformed into the curve integral of coordinate axis.

In curve integration, the integrand can be a scalar function or a vector function. The value of the integral is the riemann sum of the function value of each point on the path multiplied by the corresponding weight (generally arc length, when the integral function is a vector function, it is generally the scalar product of the function value and the infinitesimal vector of the curve). Weighting is the main difference between curve integral and general interval integral. Many simple formulas in physics (for example) appear in the form of curve integral after popularization (). Curve integration is a very important tool in physics, such as calculating the work done in electric field or gravity field, or calculating the probability of particles appearing in quantum mechanics.