Brief on Line Integrals in Conservative fields

A conservative vector field is defined as one that meets any one of the following three properties (all of which are defined inside the article):

 F(x,y)The route independence of the line integrals of F.

Line integrals of F are always 0 over closed loops. 

F is  irrotational, which means their curl is zero everywhere (with a slight caveat).

Line integrals in Conservative fields

A conservative vector field is a vector field that is the gradient of a function in vector calculus. The line integral in conservative vector fields is route independent, which means that any path between two points has the same value as the line integral. The line integral’s path independence is equivalent to the vector field’s conservatism. A conservative vector field is also irrotational, which means it has a vanishing curl in three dimensions. If the domain is simply connected, an irrotational vector field is always conservative. Conservative vector fields are vector fields that represent forces in physical systems where energy is conserved. They emerge naturally in mechanics. Because the effort required to move along a path in configuration space depends only on the path’s endpoints for a conservative system, a potential energy that is independent of the actual path followed can be defined.

Arc length

• The distance between two places along a portion of a curve is called arc length. Rectification of a curve is the process of determining the length of an irregular arc segment by approximating it as connected (straight) line segments. A curve is considered to be rectifiable if its rectification yields a finite number (and hence the curve has a finite length). A polygonal path can be created by connecting a finite number of points on a curve with (straight) line segments to mimic a curve in the plane. Because the length of each linear segment is easily calculated (for example, using the Pythagorean theorem in Euclidean space), the entire length of the approximation may be computed by adding the lengths of each linear segment; this approximation is known as the (cumulative) chordal distance. 
• If the curve isn’t already a polygonal path, increasing the number of line segments with smaller lengths will improve the curve length approximations. Rectification of a curve is the process of determining the length of a curve by approximating it as connected (straight) line segments. The lengths of subsequent approximations will not decrease and may continue to increase forever, but as the lengths of the segments approach arbitrarily tiny, they will tend to a finite limit for smooth curves.

Line integrals

The integration of a function along a curve is represented by a line integral in calculus. Line integrals are also known as route integrals, curvilinear integrals, and curve integrals. Line integrals are useful for estimating work done by a force on an item travelling in a vector field, the mass of a wire, reaction rates, the position of a celestial body, the centre of mass of a wire, moments of inertia of a wire, the magnetic field that encircles a conductor (Ampere’s law), and voltage created in a loop, Faraday’s Law of Magnetic Induction. A line integral is a type of integral that involves integrating a function along a curve in the coordinate system. To represent the function to be integrated, a scalar field or a vector field might be utilised. Both scalar and vector-valued functions can be combined along a curve. The vector line integral is calculated by adding all of the values of the points on the vector field.

Conclusion

In vector calculus, a conservative vector field is the function’s gradient. In conservative vector fields, the line integral is route independent, meaning it has the same value regardless of which path between two points is taken. Electrical engineers utilise line integrals to calculate the exact length of power cable needed to connect two substations that are thousands of miles apart. For long journeys, space flight engineers typically use line integrals. The path of the various circling velocities of Earth and the planet the probe is directed at is taken into account while launching exploration satellites. Line integrals are also used to compute an object’s velocity and trajectory, estimate planet locations, and learn more about electromagnetic.