12B (Vector Calculus)

Let the new orthonormal basis satisfy

e_{i}^{'} = R_{ij} e_{j}

. Then an

n

T_{i_{1} \dots i_{n}}^{'} = R_{i_{1} j_{1}} \dots R_{i_{n} j_{n}} T_{j_{1} \dots j_{n}} .

Since

\partial / \partial x_{i}^{'} = R_{ij} \partial / \partial x_{j}

, the Hessian of a scalar obeys this law with two factors of

R

An isotropic tensor is unchanged under every rotation. In three dimensions the most general isotropic tensors of ranks zero through three are respectively

a, 0, b δ_{ij}, c ϵ_{ijk} .

b

The measure, Gaussian factor,

r^{- 1}

, coordinates, and derivatives all transform covariantly under rotations, so

T_{ijk l}

β = γ

Contracting

i = j

and using

\nabla^{2} (1/ r) = 0

away from the origin gives

3 α + 2 β = 0.

Next contract

i = k

and

j = l

. Since

\sum_{i} x_{i} \partial_{i} \partial_{j} (r^{- 1}) = \frac{2 x _{j}}{r ^{3}},

one obtains

3 α + 12 β = 2 \int_{R^{3}} \frac{e ^{- r^{2}}}{r}, d V = 4 π .

Solving yields

α = - \frac{4 π}{15}, β = γ = \frac{2 π}{5} .

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