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}]

Existential logic got ya down?

Having just started a graduate applied mathematics program without a heavy math background, they're putting me through boot camp. Lots of the proofs require combining ands and ors with the existence quantifiers, and I invariably screw these up. Quick, does ? How about ?

One nice tool I've just found is Molle:

Molle is a cross-platform prover for modal logic, that exploits the modal semantic tableaux method. It features a very usable graphical interface, with interactive representation of generated models.

Particularly sweet is the Java Web Start version mollecino. You feed Molle first order logic statements and it will prove whether or not they are correct. It's a great tool, especially to convince yourself that things you think may be true actually are true.

To play with second order quantifiers, recast them as collections of statements. For example, to pretend that you've got say that is really a set of three separate conditions , , and . Then convert into and feed that into Molle. Similarly, change into .

To check feed Molle the input (A|B|C|D)&(Z|X|Y|W)=>(A&Z)|(B&X)|(C&Y)|(D&W). To check feed Molle the input (A&Z)|(B&X)|(C&Y)|(D&W)=>(A|B|C|D)&(Z|X|Y|W). The first statement isn't valid but the second one is.

Raw egg topping on ice cream

The Internet has failed me.

In a conversation at the dinner table, the subject of weird ice cream toppings came up. I volunteered that one particularly weird topping is raw egg.

My wife looked at me like I was crazy. Explaining that, when whisked and frozen by placing it atop very cold vanilla ice cream, raw egg tastes something like caramel didn't seem to convince her.

I declared that the Internet would confirm that I wasn't making it up. And I searched. Though I found lots of information about whether using raw egg in ice cream recipes was safe, I found zilch on using raw egg as a topping.

I heard about it when I was in high school; I've tried the concoction several times with successful results. Hints: lots of whisking, cold ice cream, drizzle so it is a thin shell over the ice cream's surface. Don't try it if you're very young, very old, have a weak immune system, or take those drugs listed in all of the nasty interaction parts of TV commercials.

Can anyone confirm that I didn't make this up, and that there's some sort of a source for this recipe?