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 $\mathbf {S}$ is a surface bounded by a simple closed contour $C$ , then

\mathbf {L} =\oint _{C}\mathbf {x} \times (d\mathbf {x} \times \mathbf {n} )=-\int _{\mathbf {S} }(\mathbf {x} \times \mathbf {n} )\nabla \cdot \mathbf {n} \ dS.

Here $\mathbf {x}$ is the position vector and $\mathbf {n}$ is the unit normal on the surface. An immediate consequence is that if $\mathbf {S}$ is a closed surface, then the line integral tends to zero, leading to the result,

\int _{\mathbf {S} }(\mathbf {x} \times \mathbf {n} )\nabla \cdot \mathbf {n} \ dS=0,

or, in index notation, we have

\int _{\mathbf {S} }x_{j}\nabla \cdot \mathbf {n} \ dS_{k}=\int _{\mathbf {S} }x_{k}\nabla \cdot \mathbf {n} \ dS_{j}.

That is to say the tensor

T_{ij}=\int _{\mathbf {S} }x_{j}\nabla \cdot \mathbf {n} \ dS_{i}

defined on a closed surface is always symmetric, i.e., $T_{ij}=T_{ji}$ .

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

L_{i}=\oint _{C}[dx_{i}(n_{i}x_{j}+n_{k}x_{k})+dx_{j}(-n_{i}x_{j})+dx_{k}(-n_{i}x_{k})].

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

L_{i}=\int _{\mathbf {S} }\left\{n_{i}\left[{\frac {\partial }{\partial x_{j}}}(-n_{i}x_{k})-{\frac {\partial }{\partial x_{k}}}(-n_{i}x_{j})\right]+n_{j}\left[{\frac {\partial }{\partial x_{k}}}(n_{j}x_{j}+n_{k}x_{k})-{\frac {\partial }{\partial x_{i}}}(-n_{i}x_{k})\right]+n_{k}\left[{\frac {\partial }{\partial x_{i}}}(-n_{i}x_{j})-{\frac {\partial }{\partial x_{j}}}(n_{j}x_{j}+n_{k}x_{k})\right]\right\}\ dS.

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

L_{i}=\int _{\mathbf {S} }\left[-{\frac {1}{2}}x_{k}{\frac {\partial }{\partial x_{j}}}(n_{i}^{2}+n_{k}^{2})+{\frac {1}{2}}x_{j}{\frac {\partial }{\partial x_{k}}}(n_{i}^{2}+n_{j}^{2})+n_{j}x_{k}\left({\frac {\partial n_{i}}{\partial x_{i}}}+{\frac {\partial n_{k}}{\partial x_{k}}}\right)-n_{k}x_{j}\left({\frac {\partial n_{i}}{\partial x_{i}}}+{\frac {\partial n_{j}}{\partial x_{j}}}\right)\right]\ dS,

or, in other words,

L_{i}=\int _{\mathbf {S} }\left[{\frac {1}{2}}\left(x_{j}{\frac {\partial }{\partial x_{k}}}-x_{k}{\frac {\partial }{\partial x_{j}}}\right)|\mathbf {n} |^{2}-(x_{j}n_{k}-x_{k}n_{j})\nabla \cdot \mathbf {n} \right]\ dS.

And since $|\mathbf {n} |^{2}=1$ , we have

L_{i}=-\int _{\mathbf {S} }(x_{j}n_{k}-x_{k}n_{j})\nabla \cdot \mathbf {n} \ dS,

thus proving the lemma.

References

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.
Chandrasekhar, S.; Wali, K. C. (2001). A Quest for Perspectives: Selected Works of S. Chandrasekhar: With Commentary. World Scientific.

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[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.