Hello, there!
My problem is the following: What is the relationship between Cauchy's integral formula in Complex analysis, and real vector fields? I'm particularly interested in whether there is an analog formula, or theorem in a made up vector field with the right properties, that if we integrate on a simple closed curve, we get the value of the vector field at a point which is inside of the said closed curve? I'm aware of the mean-value theorem of harmonic functions, but it is only true for a circle/sphere, not an arbitrarily shaped curve/surface, so it's similar, but not the same.

Question

Hello, there!
My problem is the following: What is the relationship  between Cauchy's integral formula in Complex analysis, and real vector fields? I'm particularly interested in whether there is an analog formula, or theorem in a made up vector field with the right properties, that if we integrate on a simple closed curve, we get the value of the vector field at a point which is inside of the said closed curve? I'm aware of the mean-value theorem of harmonic functions, but it is only true for a circle/sphere, not an arbitrarily shaped curve/surface, so it's similar, but not the same.

anonymous · Answer

It is connected , as far as I know to Killing fields

anonymous · Answer

Read - as a starting point to a whole area of ideas, not simple answer http://en.wikipedia.org/wiki/Killing_vector_field

anonymous · Answer

@edyacc