The proof uses index notation with the Levi-Civita symbol and Kronecker delta to transform the double curl, a complex vector operation, into a more manageable algebraic form. This process shows that a complicated combination of curls is equivalent to the difference between the gradient of the divergence and the Laplacian of the vector field. By proving the identity on a component-by-component basis, the entire vector identity is confirmed, demonstrating the power of this method for simplifying complex vector calculus problems.
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$\complement\cdots$Counselor
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The most important takeaway is the effectiveness of using index notation with the Levi-Civita symbol ( $\varepsilon_{i j k}$ ) and the Kronecker delta ( $\delta_{i j}$ ) to prove vector identities. This method transforms complex vector calculus operations into a series of algebraic manipulations, making the proof systematic and unambiguous.
The proof hinges on a single, powerful identity: $\varepsilon_{i j k} \varepsilon_{k l m}=\delta_{i l} \delta_{j m}-\delta_{i m} \delta_{j l}$. This identity allows you to get rid of the complex curl operations and replace them with simpler Kronecker delta terms. It acts as a bridge, connecting the "curl of a curl" to a more manageable form that can be simplified into a gradient and a Laplacian.
The proof is broken down by analyzing each component of the vector equation. By showing that the $i$-th component of the left side is equal to the $i$-th component of the right side, the identity is proven for the entire vector. This highlights a fundamental principle in vector analysis: if an equation holds true for every component, it holds true for the entire vector.
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