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
![{\displaystyle \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.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e5e795156b7c6badf76cd6df657bbbc0158dad7)
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,
![{\displaystyle \int _{\mathbf {S} }(\mathbf {x} \times \mathbf {n} )\nabla \cdot \mathbf {n} \ dS=0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/781ccde781d82227f5923667e958a321e6bdb39f)
or, in index notation, we have
![{\displaystyle \int _{\mathbf {S} }x_{j}\nabla \cdot \mathbf {n} \ dS_{k}=\int _{\mathbf {S} }x_{k}\nabla \cdot \mathbf {n} \ dS_{j}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9751ef55af3b1859acf068b5018da36211092cd1)
That is to say the tensor
![{\displaystyle T_{ij}=\int _{\mathbf {S} }x_{j}\nabla \cdot \mathbf {n} \ dS_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d519a505a9871c223e5e22d58ae4a170040f7b8)
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
![{\displaystyle L_{i}=\oint _{C}[dx_{i}(n_{j}x_{j}+n_{k}x_{k})+dx_{j}(-n_{i}x_{j})+dx_{k}(-n_{i}x_{k})].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2adb9d8dd487114f45a5b1c71e29400f2b4f21dc)
Converting the line integral to surface integral using Stokes's theorem, we get
![{\displaystyle 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.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f21c635b5808af82e3f1f9b33415b223e496c488)
Carrying out the requisite differentiation and after some rearrangement, we get
![{\displaystyle 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,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f962a2a7c541656ccf0b64ca4b8149258a274364)
or, in other words,
![{\displaystyle 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.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ece10fac16e521152b3c4f36eb1b07e906279d2)
And since
, we have
![{\displaystyle L_{i}=-\int _{\mathbf {S} }(x_{j}n_{k}-x_{k}n_{j})\nabla \cdot \mathbf {n} \ dS,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ecef87fced9e478a8d25db8e2e928732311e8cde)
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.