What are properties of Line Integrals?
Channel: MechanicaLEi
Date: 2018-08-19

Welcome back MechanicaLEi, did you know that properties of line integrals help find the work done on a particle travelling on a curve inside a force field represented as a vector? This makes us wonder, what are properties of line integral? Before we jump in check out the previous part of this series to learn about what Cauchy's Theorem is? Now, the first property involves multiplication by a scalar. Let alpha be a scalar, F be a vector field and let C be an oriented curve. Since, alpha into F equals to each vector component being multiplied by alpha. We have a simple result of what happens to a line integral multiplied by a scalar alpha. Second, if F and G are two vector fields defined on the same curve C, then the sum of their line integral is given as: Third, If the curve C is an oriented curve, if you go the opposite direction on C you get the opposite answer for your line integral. Mathematically: Fourth and last, if the curve C is not a smooth curve, but is a piece-wise smooth curve, it can be thought of as the sum of finitely many smooth curves. The line integral around the total path is the sum of the line integrals around each of the smaller curves. A point Z not belonging to complex set is a singular point or a singularity of a function f, if f is not analytic at zo and every neighbourhood of z not contains a point at which f is analytic. A special case of singularity is called a pole and is defined as the limit from z to z not f of z equals to infinity. for example, z not is a pole for 1 upon z and 1 upon z squared. Hence, we first saw what properties of line integrals are and then went on to see what singularity and poles are?
In the next episode of MechanicaLEi find out what Taylor’s and Laurent’s series are?

Attributions:

Doh De Oh by Kevin MacLeod is licensed under a Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/)
Source: http://incompetech.com/music/royalty-free/index.html?isrc=USUAN1100255
Artist: http://incompetech.com/

Subtle Library by Fabian Measures (http://freemusicarchive.org/music/Fabian_Measures/) is licensed under a Creative Commons Attribution licence ( https://creativecommons.org/licenses/by/4.0/)
Source: http://freemusicarchive.org/music/Fabian_Measures/Emotion_Music/Subtle_Library