Insight into determinants and certain line integrals

[kostoglotov]

I just did this following exercise in my text

If $C$ is the line segment connecting the point $(x_1,y_1)$ to $(x_2,y_2)$, show that

$$\int_C xdy - ydx = x_1y_2 - x_2y_1$$

I did, and I also noticed that if we put those points into a matrix with the first column $(x_1,y_1)$ and the second column $(x_2,y_2)$, then the answer is also the determinant of that matrix.

I also noticed that this will be true for the case where the vector field flows CCW along a set of concentric circles centered at the origin, growing larger in magnitude with distance from the origin.

Why is that? What does this mean? What is the connection that's happening here between the line integral and the determinant? I know that determinants are difficult to thoroughly explain in terms of how to interpret them, but what does this exercise say about the meaning of a determinant?

 

[theodoros.mihos]

Start by: $y=\lambda{x}+\beta$

 

[mathwonk]

do you know green's theorem? and the connection between determinants and area?