What is the significance of the negative direction in Green's Theorem?

[cersepn]

After much trouble with using the in-built engine to display all the mathematical formula, i thought scanning the question was the only way i could get my point across

For (1) is C also defined by the parametric representation of $$\widetilde{}C$$ which is why the direction of C is negative?
Then for (3), i don't get where the '2t dt' term comes from

Help would be much appreciated.. I've been searching online for some good green theorem lessons but I'm still kinda confused after reading them all.. thanks

 

[lippyka]

again!For (1), yes, C is also defined by the parametric representation of \widetilde{}C. The direction of C is negative because the parametric representation is given as (x, y) = (t, -t). This means that, as t increases, the x-coordinate increases in the positive direction, while the y-coordinate decreases in the negative direction. For (3), the '2t dt' term comes from the integration of the function f(t) = 2t. The integral of f(t) can be written as \int f(t) dt = \int 2t dt = 2t^2/2 + c, where c is a constant. To simplify this, we can drop the constant term, leaving us with 2t dt.