Why is preferable the line integral than the area integral over C plan

[Jhenrique]

Why is preferable to use the line integral than the area integral over the complex plane?

 

[HallsofIvy]

I don't know what you mean by "preferable" here. Both are used for different purposes.

 

[WWGD]

Line integrals are used when you ( or anyone! ) integrates over contours/curves. When you integrate
over a surface, you ( or anyone!) integrates over a region , you use surface integrals, tho you may be able
to use Stokes' thm. to integrate over a curve if the conditions are right.

 

[platetheduke]

When you do a line integral in real euclidean space you it is defined using real number multiplication, when you do a line integral in the complex case it uses complex multiplication, which takes account of angles as well as magnitudes. So there is a different geometric interpretation.

Furthermore, just as in the real case, complex line integrals let you create local inverses of your functions. For example, $e^z$ has period $2\pi i$, so it is not 1-1. If it has an inverse $f(x)$ near $0$, then $f(1)=0$ and $e^{f(z)}=z$ near $z=0$. Taking derivatives gives $1=f^{\prime}(z)e^{f(z)}=f^{\prime}(z)z$. So $f^{\prime}(z)=\frac{1}{z}$. The analogue of the fundamental theorem of calculus shows that $f(z)=\int_{1}^{z}\frac{1}{t}dt$ no matter which path you take in $\mathbb{C}$ from $1$ to $z$ as long as your path doesn't go "too far" away from $1$. In this case, $f$ is one of the branches of the complex logarithm.

As final example, line integrals in $\mathbb{C}$ allow you to extract information about an analytic function (including its derivatives) due to my first point. This is not possible in $\mathbb{R}$.

In summary, analytic functions are very rigid, and continuous paths allow you to take data about them at one location and propagate it elsewhere. This is not possible in the reals, since knowing information about a smooth real function at one location can give you no information about how it behaves elsewhere. Just as a path let's you carry the information, an integral can allow you to see its cumulative effect or average it out.

 

[blue_raver22]

The line integral is preferable over the area integral when dealing with complex planes because it allows for a more precise calculation of the desired quantity. The complex plane is a two-dimensional space, and the area integral would require integration over a region in this space. This can be challenging and time-consuming, especially when the region is complex and irregularly shaped.

On the other hand, the line integral only requires integration along a specific curve or path, which can be easily defined and calculated. This makes it a more efficient and accurate method for determining quantities such as work, flux, or circulation in the complex plane.

Additionally, the line integral takes into account the direction of the path, which is essential in the complex plane where the path can intersect itself or have multiple branches. The area integral, on the other hand, does not consider direction and would give the same result regardless of the path taken.

Furthermore, the line integral can be generalized to higher dimensions, making it a more versatile tool for complex calculations in multi-dimensional spaces. The area integral, on the other hand, is limited to two dimensions.

In summary, the line integral is preferable over the area integral in the complex plane due to its efficiency, accuracy, consideration of direction, and generalizability to higher dimensions.