Chandrasekhar–Wentzel lemma explained

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

S is a surface bounded by a simple closed contour C , thenL = \oint _{C} x x (d x x n) = -\int _{S} ( x x n)\nabla ⋅ n dS.

Here

x is the position vector and n is the unit normal on the surface. An immediate consequence is that if S is a closed surface, then the line integral tends to zero, leading to the result,\int _{S} ( x x n)\nabla ⋅ n dS =0,

or, in index notation, we have

\int _{S} x _{j\nabla ⋅ n dS} _{k} = \int _{S} x _{k} \nabla ⋅ n dS _{j.}

That is to say the tensor

T _{ij} = \int _{S} x _{j\nabla ⋅ 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} _{jx} _{j+n} _{kx} _{k)} + dx _{j(-n} _{ix} _{j)+dx} _{k(-n} _{ix} _{k)].}

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

L _{i} = \int _{S} \left\{n _{i\left[ (\partial)/(\partial x _{j})} ( - n _{ix} _{k)} - (\partial)/(\partial x _{k}) ( - n _{ix} _{j)\right]} + n _{j} \left[ (\partial)/(\partial x _{k}) (n _{jx} _{j+n} _{kx} _{k)} - (\partial)/(\partial x _{i}) ( - n _{ix} _{k)\right]} + n _{k\left[ (\partial)/(\partial x _{i})} ( - n _{ix} _{j)} - (\partial)/(\partial x _{j}) (n _{jx} _{j+n} _{kx} _{k)\right]\right\} dS.}

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

L _{i} = \int _{S} \left[ - (1)/(2) x _{k (\partial)/(\partial x _{j})} (n ^{2 )} _{k} + (1)/(2) x _{j (\partial)/(\partial x _{k})} (n ^{2 ) + n} _{jx} ^{k\left( (\partial n _{i})/(\partial x _{i})} + (\partial n _{k})/(\partial x _{k}) \right) - n _{kx} _{j} \left( (\partial n _{i})/(\partial x _{i}) + (\partial n _{j})/(\partial x _{j}) \right)\right] dS,

or, in other words,

L _{i} = \int _{S} \left[ (1)/(2) \left(x _{j (\partial)/(\partial x _{k})} - x _{k (\partial)/(\partial x _{j})} \right) | n| ^{2} - (x _{jn} _{k-x} _{kn} _{j)\nabla ⋅ n\right] dS.}

And since

| n| ^{2=1} , we haveL _{i} = - \int _{S} (x _{jn} _{k-x} _{kn} _{j)\nabla ⋅ n dS,}

thus proving the lemma.

Notes and 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 . 10.1098/rspa.1965.0127 .
2. Book: Chandrasekhar, S. . Wali . K. C. . 2001 . A Quest for Perspectives: Selected Works of S. Chandrasekhar: With Commentary . World Scientific .