Difference between line integrals in complex analysis and real analysis, The formula in complex analysis is 
$$\int f(\gamma(t))\cdot(\gamma'(t)dt$$
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?
Thanks,
 A: In the complex domain the theory of integration is quite different. In large part, this is due to Cauchy's integral formula. Long story short, there are many theorems in complex analysis which are not replicated in the real case. So, I would not agree these are the "same" thing. 
That said, there is a fundamental theorem of calculus for both. As you say, the gradient vector field $\vec{F} = \nabla f$ has $\int_C \vec{F} \cdot d\vec{r} = f(B)-f(A)$ if the curve $C$ goes from $A$ to $B$ in a domain where the $f$ is defined near to $C$. Likewise, if $f(z) = g'(z)$ then $\int_C f(z) dz = g(z_1)-g(z_o)$ where $C$ goes from $z_o$ to $z_1$ again supposing $g$ is defined near $C$.
You mention a vector field from $\mathbb{C}^2$ to $\mathbb{C}^2$. However, the integral you wrote I think is just for functions from $\mathbb{C}$ to $\mathbb{C}$. These correspond to certain integrals of vector fields on $\mathbb{R}^2$. And, there are applications to fluid-flow or two-dimensional electrostatics. That said, I suspect the most natural justification for $\int_C f(z) dz$ is simply that it is the natural generalization of the Riemann integral if we replace real numbers and functions with their complex analogs.
