Tensor Notation∗

March 2, 2010

De�nitions

Summation Notation (A.K.A Einstein Notation)

Whenever an index appears more than once in a single term, this term is intended to be summed over allvalues of the repeated index, as though there were a sigma out front:

AiBi =3∑

n=1

AiBi

Kronecker Delta

δij ={

1 i = j0 i 6= j

The Kronecker delta can be used to change the index of some factor if no other factor has the same indexin that term. This occurs via the following reasoning:

δijAj = δ11A1 = δ22A2 = δ33A3 = A1 = A2 = A3 = ~A

Then, we can do things like this:

δijAi = Aj

You can move the Kronecker delta to the left of anything in a single term.

Levi-Civita Symbol (A.K.A. permutation symbol)

εijk =

1 ijk, jki, kij−1 jik, ikj, kji0 any repeated

You can move the Levi-Civita symbol to the left of anything in a single term.

Partial Symbol

∂i = ∇i

Used to denote a derivative, i. e., a nabla or del symbol in vector notation. You can't move any factorfrom the right side of the partial to the left side if the factor is a function of the partial operator. Also, whena partial is to the left of two or more factors that are a function of the partial, you must apply the productrule, like this:

∂i(AiBj) = Ai∂iBj +Bj∂iAi = ( ~A · ~∇) ~B + ~B(~∇ · ~A)

Notice how the factors with matching indices that were next to each other formed dot products.

∗These notes compiled from lecture notes. Thanks to lucid explanations by: P. Arendt, D. Raymond and J. Starrett.

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Useful Identities

Not sure if it has a name

εijkεpqk = δipδjq − δiqδjp

This can be used whenever you have a common last index between two Levi-Civita symbols. If the lastindices don't exactly match, you can cyclically permute indices until they do (or switch two indices and pullout a minus sign).

Cross Product

εijkAjBk = ( ~A× ~B)i

This is the ith component of A cross B.

Scalar Triple Product

εijkAiBjCk = ~A · ( ~B × ~C)

This is the volume of a parallelepiped.

Dot Product

δijAiBj = δiiAiBi = AiBi = ~A · ~B

This is a dot product. See how the Kronecker delta forces A and B to have the same indices (otherwise,the term is zero). After they have the same indices (by summation notation) this is just:

AxBx +AyBy +AzBz = ~A · ~B

Gradient

∂i f = ~∇f

If a factor has no index, that means it's a scalar. Thus, when a partial operator precedes it, it becomesa gradient, provided the factor is a function of one of the partial indices (x y z, etc.).

Laplacian

∂i∂if = ∇2f

Divergence

∂iAi = ∇iAi = ~∇ · ~A

Strategies in Working with Tensor Notation

• Look for matching indices on pairs of factors to form dot products.

• Look for Kronecker delta, vector pairs. You might be able to change the index on the vector so that itmatches the index on another vector, thus forming a dot product.

• Whenever you have multiple Levi-Civita symbols in the same term, try to match their last indices, soyou can use the useful identity shown above.

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A Detailed Example

Suppose you want to �nd the curl of the vorticity crossed with the velocity:

~∇× (~ω × ~v)

First, add a subscript to the expression to indicate that it is a vector with 3 components:

[~∇× (~ω × ~v)]i

Next, expand the dot products into Levi-Civita symbol groups, as shown above:

= εijk∂j(~ω × ~v)k

Note that the original index becomes the �rst one in the Levi-Civita symbol. Now, the other cross product:

= εijk∂j(εklmωlvm)k

Now, if we had two Levi-Civita symbols with the same last index1, we could use one of the identities,above. So, rotate the indices on the second symbol from klm to lmk, and pull the symbol over to the left ofthe term, which is always ok:

= εijkεlmk∂j(ωlvm)k

Then, we can apply the identity, like this, and break into two terms:

= (δilδjm − δimδjl)∂j(ωlvm)

= δilδjm∂jωlvm − δimδjl∂jωlvm

Notice that each factor (the partials and two vectors in each term) has a unique index. We want to matchindices if possible, so we'll use the Kronecker deltas to do this. The �rst term is only nonzero when i = l andj = m. Since we have j and m in the term, let's let the delta act on ∂j to change its index from j to m, soit will match the index on vm. We'll do a similar thing with the second term:

= δilδmm∂mωlvm − δimδjj∂jωjvm

We can do the same thing to get the �rst ω and second v to have the same index i:

= δiiδmm∂mωivm − δiiδjj∂jωjvi

The deltas are all equal to 1, and the partials must act in a product-rule fashion in each term:

= vm∂mωi + ωi∂mvm − vi∂jωj − ωj∂jvi

Now, we can pick out matching indices. The �rst term is a dot product between v and ∂, times ω. Howto see this? It is because v and ∂ are next to each other and share the same index. We can now go term byterm, and recognize di�erent products:

= (~v · ~∇) ~ωi + ~ωi(~∇ · ~v)− ~vi(~∇ · ~ω)− (~ω · ~∇)~vi

Notice that the matching indices are gone, since they make dot products (scalars) but we have retainedthe original index, i. This too disappears on conversion back to full vector notation:

= (~v · ~∇)~ω + ~ω(~∇ · ~v)− ~v(~∇ · ~ω)− (~ω · ~∇)~v.

1In this case, we must have two matching subscripts since the Levi-Civita symbol inside the parentheses, being that it

represents the kth component of the �rst one, must also have a k index.

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