Why is there a need for $\Delta S$ in the curl definition?

Question

According to what I read in Sadiku, the curl definition requires this surface delta s to tend to zero for the line integral to approach the curl . But why can't we just take the line integral as curl ? Isn't it a good measure of the tendency of a vector field to rotate? Is removing dependency of the arbitrary curve along which the line integral is taken, the only reason for introducing delta s or is there some other reason?

Answer 1

$L$ in the integral is the boundary of $S$. You want the definition be independent of $S$.