(8.1) Proof of the BAC–CAB Rule

Prove the identity:

\[ \mathbf{A} \times (\mathbf{B} \times \mathbf{C}) = \mathbf{B} (\mathbf{A} \cdot \mathbf{C}) - \mathbf{C} (\mathbf{A} \cdot \mathbf{B}) \]

by writing it out in the summation convention, and use it to show that:

\[ \nabla \times (\nabla \times \mathbf{E}) = \nabla (\nabla \cdot \mathbf{E}) - \nabla^2 \mathbf{E} \]

We aim to prove the vector identity:

\[ \mathbf{A} \times (\mathbf{B} \times \mathbf{C}) = \mathbf{B} (\mathbf{A} \cdot \mathbf{C}) - \mathbf{C} (\mathbf{A} \cdot \mathbf{B}) \]

using the summation convention.

Step 1: Expressing Cross Products Using the Levi-Civita Symbol

We start with the definition of the cross product:

\[ (\mathbf{B} \times \mathbf{C})_k = \epsilon_{klm} B_l C_m \]

Taking the cross product with \( \mathbf{A} \):

\[ (\mathbf{A} \times (\mathbf{B} \times \mathbf{C}))_i = \epsilon_{ijk} A_j (\mathbf{B} \times \mathbf{C})_k \]

Substituting the previous expression:

\[ (\mathbf{A} \times (\mathbf{B} \times \mathbf{C}))_i = \epsilon_{ijk} A_j \epsilon_{klm} B_l C_m \]

Step 2: Using the Levi-Civita Identity

From tensor algebra, we use the identity:

\[ \epsilon_{ijk} \epsilon_{klm} = \delta_{il} \delta_{jm} - \delta_{im} \delta_{jl} \]

Applying this to our equation:

\[ (\mathbf{A} \times (\mathbf{B} \times \mathbf{C}))_i = (\delta_{il} \delta_{jm} - \delta_{im} \delta_{jl}) A_j B_l C_m \]

Step 3: Applying the Kronecker Delta

The Kronecker delta acts as an identity filter, allowing us to replace indices:

\[ A_j B_l C_m \delta_{il} \delta_{jm} = A_j B_i C_j \] \[ A_j B_l C_m \delta_{im} \delta_{jl} = A_j B_j C_i \]

Thus, simplifying the expression:

\[ (\mathbf{A} \times (\mathbf{B} \times \mathbf{C}))_i = A_j B_i C_j - A_j B_j C_i \]

Step 4: Recognizing Dot Products

Noticing that:

\[ A_j C_j = \mathbf{A} \cdot \mathbf{C}, \quad A_j B_j = \mathbf{A} \cdot \mathbf{B} \]

we rewrite:

\[ (\mathbf{A} \times (\mathbf{B} \times \mathbf{C}))_i = B_i (\mathbf{A} \cdot \mathbf{C}) - C_i (\mathbf{A} \cdot \mathbf{B}) \]

which, in vector notation, gives the final result:

\[ \mathbf{A} \times (\mathbf{B} \times \mathbf{C}) = \mathbf{B} (\mathbf{A} \cdot \mathbf{C}) - \mathbf{C} (\mathbf{A} \cdot \mathbf{B}) \]

Step 5: Application to Curl Operators

Substituting \( \mathbf{A} = \nabla \), \( \mathbf{B} = \nabla \), and \( \mathbf{C} = \mathbf{E} \), we immediately obtain:

\[ \nabla \times (\nabla \times \mathbf{E}) = \nabla (\nabla \cdot \mathbf{E}) - \nabla^2 \mathbf{E} \]

This is a fundamental identity in electromagnetism, appearing in Maxwell’s equations.