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import mmf_setup;mmf_setup.nbinit()
import logging;logging.getLogger('matplotlib').setLevel(logging.CRITICAL)
%matplotlib inline
import numpy as np, matplotlib.pyplot as plt

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Assignment 4: Tensors

Due Mon 22 Sep 2025 at the start of class

1. Antisymmetric Tensors

Recall that

\[\begin{gather*} \varepsilon_{abk}\varepsilon^{cdk} = \delta^{c}_{a}\delta^{d}_{b} - \delta^{c}_{b}\delta^{d}_{a}. \end{gather*}\]

Use this to derive an expression for \(\varepsilon_{aij}\varepsilon^{bij}\) to help you.

Consider a real antisymmetric tensor \(B^{ij} = -B^{ji}\) in 3D. Count the degrees of freedom, and show that any such tensor can be written

\[\begin{gather*} B^{mn} = \varepsilon^{mnk}A_k. \end{gather*}\]

Write out \(B^{mn}\) explicitly as a matrix showing where the three components \(A_{1}\), \(A_{2}\), and \(A_{3}\) go, then use the properties of the Levi-Civita symbol to show that

\[\begin{gather*} A_k = \tfrac{1}{2}\varepsilon_{ijk}B^{ij}. \end{gather*}\]

This is important for embedding the electromagnetic field \(\vect{B}\) into the electromagnetic field tensor \(F^{\mu\nu}\).

2. Diagonal Metric

If the covariant basis vectors \(\boldsymbol{\varepsilon_{i}}\) are orthogonal, show that:

• \(g_{ij}\) is diagonal.

• \(g^{ii} = 1/g_{ii}\) (no summation).

• \(\norm{\boldsymbol{\varepsilon^{i}}} = 1/\norm{\boldsymbol{\varepsilon_{i}}}\).

Note

The notation here is from Arfken, but differs slightly from what I have been using in class and in these notes in 4. Tensors and Manifolds. In these notes, I have used the notation \(\ket{X_{\alpha}} \equiv \boldsymbol{\varepsilon_{i}}\) for the basis vectors and \(\ket{X^{\alpha}}\equiv \boldsymbol{\varepsilon^{i}}\) for the contravariant basis vectors, though, as I argue there, the latter are quite unnatural and should only be used when absolutely needed.

3. Covariant Derivative

Verify that \(V_{i;j} = g_{ik}V^{k}_{;j}\) by showing that

\[\begin{gather*} \pdiff{V_{i}}{q^j} - V_k\Gamma^{k}_{ij} = g_{ik}\left[ \pdiff{V^{k}}{q^j} + V^{m}\Gamma^{k}_{mj} \right]. \end{gather*}\]

I.e., show that the correction piece in the covariant derivative changes sign when differentiating vectors (positive sign) vs covectors (negative sign).

4. Christoffel Symbols

Consider a two-dimensional space defined by the surface of a sphere of radius \(r\). The square of the line element is given by

\[\begin{gather*} \d{s}^2 = r^2\;\d\theta^2 + r^2\sin^2\theta\;\d\phi^2. \end{gather*}\]

1. Determine the coefficients of the metric tensor in both covariant form, \(g_{ij}\) and contravariant form \(g^{ij}\).

2. Evaluate all nonzero Christoffel symbols of the first and second kind.

3. Use these to show that

   \[\begin{gather*} L^2 = \vect{L}\cdot\vect{L} = -\frac{1}{\sin\theta}\pdiff{}{\theta}\left(\sin\theta \pdiff{}{\theta}\right) - \frac{1}{\sin^2\theta}\pdiff[2]{}{\phi} \end{gather*}\]

   from last week by directly acting \(\vect{L}\) on a test function twice:

   \[\begin{gather*} \vect{L} = \I\left( \ket{\hat{e}_{\theta}}\frac{1}{\sin\theta}\pdiff{}{\phi} - \ket{\hat{e}_{\phi}}\pdiff{}{\theta} \right). \end{gather*}\]

5. Jacobians

For the transformation (with \(x\geq 0\) and \(y \geq 0\))

\[\begin{gather*} u = x + y, \qquad v = x/y, \end{gather*}\]

compute the following Jacobian in two ways:

\[\begin{gather*} \mat{J} = \pdiff{(x, y)}{(u, v)} \end{gather*}\]

1. By direct computation.

2. By first computing \(\mat{J}^{-1}\).

3. Compute the induced metric \(g_{ij}\) in the coordinates \((u, v)\).

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L = 5
dx = 0.2
Nx = 100
N = (L//dx)*Nx
x = np.arange(N+1) * dx / Nx + 0.1
x, y = np.meshgrid(x, x, indexing='ij', sparse=True)
z = x+1j*y
w = x + y + 1j*x/y

plt.clf();plt.close('all')
fig, ax = plt.subplots()

def plot(z, ax, fmt='-k', **kw):
    ax.plot(z[::Nx,:].T.real, z[::Nx,:].T.imag, fmt, **kw)
    ax.plot(z[:, ::Nx].real, z[:, ::Nx].imag, fmt, **kw)

plot(z, ax, '-k', lw=1)
plot(w, ax, '-C0', lw=1)
ax.set(aspect=1, xlim=(0,5), ylim=(0,5));