In the exercise above, the path we took to get from one point to another did not matter. They can be used to calculate the work done on a mass mm traveling in a gravitational field in classical mechanics.\newcommand\).Let d be the displacement vector from the initial to the final position. They can be used in electromagnetics to compute the work done on a charged particle traveling along a curve in a force field represented by a vector field. constant and the particle move along a straight line.Line integrals around closed curves are related to double integrals or surface integrals by Green's theorem and Stokes' theorem.Line Integral is used to calculate the magnetic field surrounding a conductor in Ampere's Law.Line integrals are classified into two types: scalar line integrals and vector line integrals.Find the work done by force field F ((y/x)2 2(y/x) ) moving an object from (1,1) to (4,-2). \(\int C f(r) ds = \int_a^b f \mid r’(t) \mid dt \) Evaluate the line integral of the vector field. A line integral might be used to calculate how much radiation a pirate would be exposed to from a radiation source near his treasure.The line integral is useful for calculating the work done by a force on a moving object in a vector field.A line integral is used in Faraday's Law of Magnetic Induction to calculate the voltage created in a loop.The ideas of integration were separately developed by Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century.Area and volume calculation methods extend back to ancient Greek mathematics.Line integrals are used in chemistry to calculate reaction rates and to gather information on radioactive decay reactions.
Applications edit The line integral has many uses in physics.
If the currents in all places can be precisely anticipated, one may calculate how many calories a swimmer will burn swimming along a specific route.It's an extension of simple integrals that's best used on curved surfaces. The three-dimensional surface areas can be calculated using line integrals.Line integrals are used in classical mechanics to compute the work done by an object of mass m travelling in a gravitational field.Where F is the vector force field acting on the object and dr is the unit tangent vector. The line integral describes the work done W by a force F on an object travelling along a curve C, W= \(\int_C\)F.dr.When applying Ampere's law, the line integral of a magnetic field B around a closed channel 'C' equals the total current flowing through the area limited by boundary 'C'.The total mass of the wire is then given as the scalar function's line integral as m= \(\int_C\) ρ(x,y,z)ds.
The mass of the wire per unit length is a continuous function ρ (x,y,z). Assume that a three-dimensional curve C describes a piece of wire.
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