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,

• "can one compute line integrals of scalar-valued functions in the real-variable setting" Yes, like this: $$\int\limits_C {f(x,y)ds} = \int_a^b {f\left( {x(t),y(t)} \right)\left\| {\vec r'(t)} \right\|} dt$$ – user204299 Aug 1 '15 at 22:18
• Hi @JakeLebovic, so the integrand is a scalar times the norm of r'(t), times the time-differential. What is the norm object? Is that distance traveled? So that the scalar* distance = "total distance" ... times the time-differential, which when integrated gives the total displacement. Do I have the right idea? Thanks, – user258314 Aug 1 '15 at 22:30
• @user258314 the integral Jake Lebovic wrote is the integral with respect to arclength. Physically, you can think of $f(x,y)$ as the mass density per unit arclength then the integral gives the net mass of the curve. Or, if you prefer, $f(x,y)>0$ is the height above the $xy$-plane and this integral is the area of the curvy curtain hung below the space curve $(x(t),y(t),f(x(t),y(t))$ where $(x(t),y(t))$ parametrizes the curve $C$ in the $xy$-plane. – James S. Cook Aug 1 '15 at 23:42

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.