Appendix

Appendix#

Notation#

Some useful math symbols#

  • \(\forall\): “For all”

  • \(\exists\): “There exists”

  • \(\ni\): “such that”

Derivatives and partial derivatives#

I will often write

(311)#\[\del_i \equiv \frac{\partial}{\partial x^i}\]

Some useful tensors#

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

(312)#\[\begin{split}\delta^I_J = \begin{cases} 1 & I = J \\ 0 & I \neq J \end{cases}\end{split}\]

  1. 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 \(V^i\), \(W^i\)

(313)#\[V^i W^i \equiv \sum_{j = 1}^d V^j W^j\]

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 \(V^I(x)\),

\[ \frac{\del}{\del x^J} V^I W_I = W^I \del_J V^I + V^I \del_J W^I\]

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

\[\delta^I_J V^I = V^J\]

and therefore,

\[\del_J (x^I x^I) = 2 x^J\]

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:

(314)#\[({\vec U} \times {\vec V})_{i_1,\ldots,i_{d-2}} = \epsilon_{i_1,\ldots,i_{d-2}, i_{d--1}, i_d} U^{i_{d-1}} V^{i_d}\]

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

(315)#\[({\vec U}\times {\vec V}) = \epsilon_{ij} U^i V^j = A^1 V^2 - U^2 V^1\]

For \(d = 3\),

(316)#\[({\vec U}\times {\vec V})^i = \epsilon_{ijk} U^j V^k\]

Or alternatively,

(317)#\[U^i V^j - U^j V^i = \epsilon_{ijk} ({\vec U}\times {\vec B})^k\]

Some useful functions#

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

(318)#\[\begin{split}\theta(x) = \begin{cases} 0, & \text{if}\ x < 0 \\ 1, & \text{if} x > 0 \end{cases}\end{split}\]
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