The formula in complex analysis is
and the formula in the real variable setting, for a gradient field, is:
$$\int F\cdot dr$$ $$=\int f_x\,dx + f_y\,dy + f_z\,dz,$$
where the integrand is said to be an "exact differential" (or total differential.)
Are the formulas essentially the same thing, when we regard the complex function as a "vector field" mapping $C^2 \to C^2$?
Also, can one compute line integrals of scalar-valued functions in the real-variable setting -- or would this not make any physical sense?