Appendix

Notation¶

Some useful math symbols¶

• $\forall$: “For all”

• $\exists$: “There exists”

• $\ni$: “such that”

Derivatives and partial derivatives¶

I will often write

Some useful tensors¶

1. The Kronecker delta function. If $I,J = i,\ldots,N$ are some indices then

2. The totally antisymmetric tensor

In $d$ dimensions, the tensor $\epsilon_{i_1,\ldots, i_d}$ is totally antisymmetric in the exchange of all indices. Thus the only nonvanishing components are ones for which $i_1,\ldots,i_d$ are a permutation of $1,2,\ldots,d$. We fix the tensor completely by edemanding that $\epsilon_{1,2,\ldots,d} = 1$.

Einstein summation convention¶

The Einstein summation convention is that all repeated indices are summed over unless otherwise stated explicitly. Thus, for some $d$-dimensional vectors $V^i$, $W^i$

Technically speaking I am being a little careless. In curved spaces (for example, on the sphere), there are geometrically distinct objects $V^i$, $V_i$, and I should only be summing over pairs of indices in which one is raised and one is lowered. In Euclidean space, however, there is a standard map which states that numerically, $V^i = V_i$, so I will ignore this issue for the time being.

(We can think of $V^i, W^i$ as elements of a vector space, and $V_i, W_i$ their dual vectors; we will discuss this language when we get to linear algebra).

Derivatives of expressions involving the Einstein summation convention often confuse people. For a general vector field $V^I(x)$,

Now if $V^I = W^I = x^I$, then $\del_J x^I = \delta^I_J$. You can convince yourselves that

and therefore,

More on the totally antisymmetric tensor¶

1. Contraction identities in $d$ dimensions A. $\epsilon_{i_1,\ldots,i_d}\epsilon^{i_1,\ldots,i_d} = d!$. B. In $d - 2$, $\epsilon_{ij} \epsilon^{ik} = \delta^k_j$. C. In $d = 3$, $\epsilon_{kmn}\epsilon^{k i j} = \delta^i_m \delta^j_n - \delta^i_n \delta^j_m$. D. In $d = 3$, $\epsilon_{ijk}\epsilon^{ijl} = 2\delta^l_k$. E. There are similar identities in higher dimensions, which we will leave aside for now.

2. Definition of cross product.

For general $d$, the cross product of two vectors $U^i V^i$ is a $d-2$-rank tensor:

For $d = 2$, this is a scalar (actually a “pseudo-scalar”):

For $d = 3$,

Or alternatively,

Some useful functions¶

1. The Heaviside step function $\theta(x)$: