Einstein notation for common vector operators

I keep getting tripped up by Einstein summation convention when combined with common vector operators like grad, div, and curl. So I put together a cheatsheet:

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Vector calculus with Einstein notation quick reference Kronecker Delta and Levi-Civita relations: δii = 3 δi j δ jk = δik εi jk εi jm = 2δkm εi jk εimn = δ jm δkn − δ jn δkm εi jk εi jk = 6 Scalar, vector, and 2-tensor operations αv + βu = (αvi + βui ) ei αA + βB = αAi j + βBi j ei ⊗ e j vA = (vl el ) Ai j ei ⊗ e j = e j δil vl Ai j = e j vi Ai j Av = Ai j ei ⊗ e j (vl el ) = ei δ jl Ai j vl = ei Ai j v j AB = Ai j ei ⊗ e j Bi j ei ⊗ e j = Ail Bl j ei ⊗ e j tr (AB) = Ai j B ji A : B = tr AT B = Ai j Bi j Inner, outer, and cross products: a · b = ai bi a ⊗ b = ai b j ei ⊗ e j a × b = εi jk ai b j ek a × b × c = ai b j ci e j − ai bi c j e j = (a · c) b − (a · b) c Basic translations into summation convention: = ei ∂i ∆ = ∂i ∂i ϕ = ei ∂i ϕ ϕ = ∂i ∂ j ϕ ei ⊗ e j u = ∂ j ui ei ⊗ e j · u = ∂i u i × u = εi jk ∂i u j ek Tautologies using summation convention: × ( ϕ) = 0 ⇐⇒ εi jk ∂i ∂ j ϕek = 0 · ( × u) = 0 ⇐⇒ εi jk ∂i ∂ j uk = 0 curl gradient scalar divergence curl vector Del operator Laplacian operator, 2 Page 1 of 1 = · gradient : scalar → vector gradient gradient : scalar → tensor gradient : vector → tensor divergence: vector → scalar curl: vector → vector

Please let me know if you find it useful, or if you find any mistakes.

I posted the cheatsheet using Scribd following some hints from Peter Chen. Within the I finally found a way to show Page n of m. I also found I could use Mathematica's Signature command to doublecheck the two identities

Sum[Signature[{i,j,k}] d[i] d[j] f e[k], {i, 1,3}, {j,1,3}, {k,1,3}]
Sum[Signature[{i,j,k}] d[i] d[j] f[k], {i,1,3}, {j,1,3}, {k,1,3}]