Why doesn't ds appear in the statement of Green's Theorem?

You're right about what $\mathbf{r}$ is, but not about the structure of the integral itself.

The function we're actually integrating in the line integral side of Green's theorem is a dot product: $$\int_a^b \mathbf{F}(\mathbf{r}(t))\cdot \mathbf{r}'(t)\,dt$$ where $\mathbf{F}(x,y)=(L(x,y),M(x,y))$. By standard theory, this doesn't depend on exactly how we parametrize the curve.

That's the one-dimensional integral. You didn't say anything about the two-dimensional integral on the other side of the integral, so I'll assume you understand that part.

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Differential Geometry

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