Chandrasekhar–Wentzel lemma

In vector calculus, Chandrasekhar–Wentzel lemma was derived by Subrahmanyan Chandrasekhar and Gregor Wentzel in 1965, while studying the stability of rotating liquid drop.[1][2] The lemma states that if is a surface bounded by a simple closed contour , then

Here is the position vector and is the unit normal on the surface. An immediate consequence is that if is a closed surface, then the line integral tends to zero, leading to the result,

or, in index notation, we have

That is to say the tensor

defined on a closed surface is always symmetric, i.e., .

Proof

Let us write the vector in index notation, but summation convention will be avoided throughout the proof. Then the left hand side can be written as

Converting the line integral to surface integral using Stokes's theorem, we get

Carrying out the requisite differentiation and after some rearrangement, we get

or, in other words,

And since , we have

thus proving the lemma.

References

  1. Chandrasekhar, S. (1965). "The Stability of a Rotating Liquid Drop". Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences. 286 (1404): 1–26. doi:10.1098/rspa.1965.0127.
  2. Chandrasekhar, S.; Wali, K. C. (2001). A Quest for Perspectives: Selected Works of S. Chandrasekhar: With Commentary. World Scientific.
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