Velocity and Flow Analysis: Complex or Real-Which is Best?

[tecumseh]

I need some comments related to the accuracy of my thinking on velocity and flow analysis.

(1) As I see it now, complex analysis of velocity and flow in all kinds of physics problems can deal only with 2-D analysis and that its advantage over real analysis for this is that complex maps conformally, functions onto the uv plane which facilitates or makes the analysis easier than with real Green or Stokes' theory. Also complex cannot make use of the variety of contours that real can.

(2) Complex analysis of flow: Z' = d(phi)/dx - d(phi)/dy
and then the conjugate of the above is d(phi)/x + d(phi)/dy which is equal to the velocity or flow. Then the absolute value of V, the velocity in the above equals the speed of the flow at a specified point.

(3) Conclusion: Real analysis is used here in 3-D problems while complex analysis cannot do this although complex facilitates the analysis of such as airfoils by mapping the Z onto the uv plane.

Any comments are much appreciated. I have a PhD in an area other than math but I have 18 grad hours in math and am going for the MA. Math is also my hobby.

tecumseh@nc.rr.com

Don Wire (North Carolina)

Campbell University (government, history and mathematics instruction)

Fayetteville Technical Community College (critical thinking and cultural studies)

 

[fresh_42]

You cannot divide this the way you did. It is the problem which defines whether a real or a complex manifold is considered, not the dimension. Complex differentiation is different from real differentiation due to the Cauchy - Riemann conditions, which results in a different kind of theorems. This means that w.r.t. differentiation $\mathbb{C}\neq \mathbb{R}^2\,.$

For a general walkthrough differentiation, see
https://www.physicsforums.com/insights/the-pantheon-of-derivatives-i/