Viivaintegraaleja

Viivaintegraaleja, also known as line integrals, are a fundamental concept in multivariable calculus and vector calculus. They extend the idea of integration from curves to functions defined over curves in two or three-dimensional space. There are two primary types of line integrals: line integrals of a scalar function and line integrals of a vector field. A line integral of a scalar function, often denoted as ∫_C f(x, y, z) ds, calculates the "mass" or "area" of a curve C in space, where f(x, y, z) represents the density or height at each point along the curve. The parameter ds represents an infinitesimal arc length along the curve. To evaluate such an integral, the curve C is typically parameterized by a vector function r(t) = over an interval [a, b]. The integral then transforms into a standard definite integral: ∫_a^b f(x(t), y(t), z(t)) ||r'(t)|| dt, where ||r'(t)|| is the magnitude of the derivative of the parameterization, representing the arc length element. A line integral of a vector field, denoted as ∫_C F · dr or ∫_C P dx + Q dy + R dz, represents the work done by a force field F along a curve C. Here, F is a vector field, and dr is an infinitesimal displacement vector along the curve. If F = and the curve is parameterized by r(t), the integral becomes ∫_a^b F(r(t)) · r'(t) dt. This integral measures the component of the force field F acting in the direction of the curve's motion. If the vector field is conservative, meaning it can be expressed as the gradient of a scalar potential function, the line integral can be evaluated more easily using the fundamental theorem of line integrals.