# line integral example

• Jan 13th 2013, 02:56 AM
Kvandesterren
line integral example
Could some offer an example of finding the line integral of a circle’s arc which is not centered at the origin and which passes through a simple 2-plane vector field?

I am having trouble parameterizing a circle’s path into terms of sine and cosine.

• Jan 13th 2013, 01:41 PM
chiro
Re: line integral example
Hey Kvandesterren.

Is this 2-plane vector field just an arbitrary plane embedded in R^3?
• Jan 13th 2013, 09:52 PM
Kvandesterren
Re: line integral example
yes
• Jan 13th 2013, 10:16 PM
chiro
Re: line integral example
Can you rotate the plane to get it axis aligned with the x-y plane and then carry out the line integral (via a linear transformation)?

Are you aware of the substitution theorem for integrals in multiple dimensions?
• Jan 14th 2013, 09:22 AM
HallsofIvy
Re: line integral example
I don't believe you are asking about a general plane in 3 dimensions. I interpret your question as just about integrating u(x,y)i+ v(x, y)j about an arbitrary circle, or arc of a circle, with center at some point other than (0, 0). That's much easier!

A circle, with radius R, with center at $\displaystyle (x_0, y_0)$ is given by $\displaystyle (x- x_0)^2+ (y- y_0)^2= R$ and standard parametric equations $\displaystyle x= x_0+ R cos(\theta)$, $\displaystyle y= y_0+ R sin(\theta)$.