Re: line integral example

Hey Kvandesterren.

Is this 2-plane vector field just an arbitrary plane embedded in R^3?

Re: line integral example

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?

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 $\displaystyle x= x_0+ R cos(\theta)$, $\displaystyle y= y_0+ R sin(\theta)$.