Calculating Flow Volume: Real Analysis-Stokes or Complex Z' Technique:Preferred?

[tecumseh]

Calculating Flow Volume: Real Analysis-Stokes or Complex Z' Technique:Preferred??

Homework Statement

Which technique, Stokes (real analysis) or the complex analysis, is preferred or considered the better for use in calculating the volume of flow through a bounded surface (heat, electro, fluid) in physical problems?

Homework Equations

Stokes integral of (curl of field X dS/dZ) or absZ' = (dPhi/dX - dPhi/dy) = V and then use V to calculate volume of flow or flux, whichever.

The Attempt at a Solution

I have no solid experience working with the complex variable function here but I could; however, I have lots with the Green and Stokes theorems. Which way would you go if you were going to do extensive work with physical phenomena and why?

 

[mighty2000]

it is important to consider the strengths and limitations of both Stokes (real analysis) and complex analysis when determining which technique is preferred for calculating the volume of flow through a bounded surface in physical problems.

Stokes' theorem is a powerful tool in real analysis that relates the surface integral of a vector field to the line integral of its curl along the boundary of the surface. This is especially useful in physical problems involving fluid flow or electromagnetism, where the curl of a vector field represents the rotation or circulation of the flow. Stokes' theorem is a fundamental result in real analysis and is widely applicable in many physical problems.

On the other hand, complex analysis offers a different perspective and set of tools for solving physical problems involving bounded surfaces. The use of complex variable functions and the Cauchy-Riemann equations can simplify calculations and provide elegant solutions in certain cases. However, complex analysis may not always be applicable or necessary for physical problems involving flow through bounded surfaces.

In conclusion, both Stokes (real analysis) and complex analysis have their own strengths and limitations when it comes to calculating the volume of flow through a bounded surface in physical problems. In order to determine which technique is preferred, it is important to carefully consider the specific problem at hand and choose the approach that will provide the most accurate and efficient solution. Both techniques have their place in the study of physical phenomena, and it may be beneficial to have experience and understanding in both areas. Ultimately, the best approach will depend on the specific problem and the preferences of the scientist.