Why the weird $\partial{Q}$ notation for the integral region for Green's Theorem?
$$\int_{\partial{Q}} W \cdot ds = \iint_Q \frac{\partial{g}}{\partial{x}} - \frac{\partial{f}}{\partial{y}} dx\ dy$$
Why not just plain "Q" instead:
$$\int_{Q} W \cdot ds = \iint_Q \frac{\partial{g}}{\partial{x}} - \frac{\partial{f}}{\partial{y}} dx\ dy$$
If I define Q to be a rectangle region, then what's the difference?
Book says: " The sides Right, Left, Top, and Bottom of Q, with the orientations as indicated... when taken together are referred to as a "boundary". The usual notation for this is a $\partial{Q}$".
I still say, who the heck cares... just call it Q. What hair am I splitting if I remove the $\partial$ character from my notes?
Because the double integral is over some region $Q$ (e.g. a disc), while the line integral is over the boundary of $Q$ (e.g. the circle bounding that disc): they're not the same.
You can give it another name if you like, such as "$C$, the boundary of $Q$", but I wouldn't call it $Q$ again since $Q$ is already used for the entire region!
For this boundary of $Q$, the notation $\partial Q$ is common.
Referring to my example above, you could have the unit disc $Q$: $$Q=\left\{\left(x,y\right)\in\mathbb{R}^2 \;\vert\; x^2+y^2 \le 1\right\}$$ and its boundary, the unit circle $\partial Q$ (possibly with a chosen orientation): $$\partial Q=\left\{\left(x,y\right)\in\mathbb{R}^2 \;\vert\; x^2+y^2 = 1\right\}$$
Related: Meaning of partial differential in limits of integration?