How to Evaluate an Integral of a Vector Field along a Curve?

[iScience]

if i have a function U..

U=∫F∙ds

where F=<ayz+bx+c , axz+bz , axy+by> , a,b,c are constants

so.. F=(ayz+bx+c)$\hat{x}$ + (axz+bz)$\hat{y}$ + (axy+by)$\hat{z}$

then how do i solve this integral? i have to either replace the x,y,z terms with something in terms of 's' (which is the displacement by the way, ie.. s= $\sqrt{x^2+y^2+z^2}$
or i have to replace ds with some parametric ..stuff... how do i evaluate something like this?

 

[tiny-tim]

Hi iScience!

U=∫F∙ds
…
or i have to replace ds with some parametric ..stuff... how do i evaluate something like this?

Let's write that out in full …

it's an integral over a curve C between two endpoints P₁ and P₂: $U = \int_C \mathbf{F}\cdot d\mathbf{s}$

so yes you have to use some parameter, which may be s itself, or may be something easier to use, eg θ with ds = (-rsinθdθ,rcosθdθ)