 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 represented as a vector? This makes us wonder, what are properties of line integral? Before we jump in, check 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 the 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 piecewise smooth curve, it can be thought of as a sum of finitely many smooth curves. The line integral around the total path is the sum of line integrals around each of the smaller curves. A point Z0 belonging to a complex Z is a singular point or a singularity of a function F if F is not analytic at Z0 and every neighbourhood of Z0 contains a point at which F is analytic. A special case of singularity is called a pole and is defined as a limit from Z to Z0, F of Z equals to infinity. For example, Z is not 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. Mechanically, find out what Taylor's and Lorentz series are.