11C (Vector Calculus)

For a Cartesian change of basis represented by an orthogonal matrix

Q

, vectors transform as

E_{i}^{'} = Q_{ia} E_{a}

and

B_{i}^{'} = Q_{ia} B_{a}

. Their squared norms are invariant and

Q_{ia} Q_{jb} δ_{ab} = δ_{ij}

. Substitution into the given expression therefore gives

T_{ij}^{'} = Q_{ia} Q_{jb} T_{ab},

the transformation law of a Cartesian second-rank tensor. The magnetic field's extra axial sign under an improper physical reflection, if included, appears twice and cancels in its quadratic contribution.

T = EE + BB - \frac{1}{2} (E^{2} + B^{2}) I

M = (E \cdot \nabla) E + E \nabla \cdot E + (B \cdot \nabla) B + B \nabla \cdot B - \frac{1}{2} \nabla (E^{2} + B^{2}) .

The identity

(A \cdot \nabla) A - \frac{1}{2} \nabla A^{2} = - A \times (\nabla \times A)

follows by contracting two Levi-Civita symbols, or directly by differentiating components. Using Maxwell's equations consequently yields

M = ρ E - E \times (\nabla \times E) - B \times (\nabla \times B) = ρ E + E \times B_{t} - B \times (J + E_{t}) = ρ E + J \times B + \partial_{t} (E \times B) .

\partial_{t} (E \times B) = M - ρ E - J \times B, M_{i} = \partial_{j} T_{ij} .

The two subtracted terms are the Lorentz force density, while

E \times B

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