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CM Course Notes - Revision history
2026-06-18T11:13:26Z
Revision history for this page on the wiki
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Shawn Tang: /* Gallery */
2015-11-11T07:35:24Z
<p><span dir="auto"><span class="autocomment">Gallery</span></span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''Lecture Notes'''</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''Lecture Notes'''</div></td></tr>
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<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>[http://www.pa.ucla.edu/sites/default/files/files/dhoker%20lecture%20notes/<del class="diffchange diffchange-inline">classical_mechanics</del>.pdf Classical Mechanics] by Prof. Eric D'Hoker<ref>http://www.pa.ucla.edu/content/eric-dhoker-lecture-notes, retrieved <del class="diffchange diffchange-inline">17th April 2014</del>. </ref>.   </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>[http://www.pa.ucla.edu/sites/default/files/files/dhoker%20lecture%20notes/<ins class="diffchange diffchange-inline">mechanics2015</ins>.pdf Classical Mechanics] by Prof. Eric D'Hoker<ref>http://www.pa.ucla.edu/content/eric-dhoker-lecture-notes, retrieved <ins class="diffchange diffchange-inline">10th November 2015</ins>. </ref>.   </div></td></tr>
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Shawn Tang
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Tom Neiser: /* 220 Notes */
2014-05-24T22:14:20Z
<p><span dir="auto"><span class="autocomment">220 Notes</span></span></p>
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<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Origin of Canonical Momentum:<div class="mw-collapsible mw-collapsed" style="width:750px"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">\[p = \left( p - \frac{e}{c} A \right), \text{ where p inside the brackets is the quantum mechanical momentum } p = -i \hbar \nabla \text{ and A is the gauge invariant vector potential.}\]</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Canonical momentum is not the momentum fired out of a cannon, but rather the momentum that satisfies $p = \frac{\partial \mathcal{L}}{ \partial \dot{q}}$, where $\mathcal{L}$ is the Lagrangian of the system. </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>H=\frac{1}{2m} \left(p-\frac{qA}{c}\right)^2 +q\phi</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"> </div></ins></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''Hamiltonian Mechanics'''</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''Hamiltonian Mechanics'''</div></td></tr>
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Tom Neiser
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Tom Neiser: /* Gallery */
2014-04-18T01:04:50Z
<p><span dir="auto"><span class="autocomment">Gallery</span></span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Gallery==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Gallery==</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">'''Lecture Notes'''</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">[http://www.pa.ucla.edu/sites/default/files/files/dhoker%20lecture%20notes/classical_mechanics.pdf Classical Mechanics] by Prof. Eric D'Hoker<ref>http://www.pa.ucla.edu/content/eric-dhoker-lecture-notes, retrieved 17th April 2014. </ref>.  </ins></div></td></tr>
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Tom Neiser
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Tom Neiser: /* 220 Notes */
2014-04-14T23:23:40Z
<p><span dir="auto"><span class="autocomment">220 Notes</span></span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==220 Notes==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==220 Notes==</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===''<small>Exercises</small>''===</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===''<small>Exercises</small>''===</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==General Identities==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==General Identities==</div></td></tr>
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Tom Neiser
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Tom Neiser at 23:19, 14 April 2014
2014-04-14T23:19:00Z
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==220 Notes==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==220 Notes==</div></td></tr>
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<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">  \[</ins></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>--></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>--></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">'''Lagrangian Mechanics'''</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Method for finding normal modes of system <div class="mw-collapsible mw-collapsed" style="width:750px"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"> </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Since we expect oscillatory motion of a normal mode (where ω is the same for both masses), we try:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">\[</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">x_1(t) = A_1 e^{i \omega t} \\</ins></div></td></tr>
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<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">  \]</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"> </div></ins></div></td></tr>
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<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Euler-Lagrange Equation of system without constraints: <div class="mw-collapsible mw-collapsed" style="width:750px"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>\frac{\mathrm{d}}{\mathrm{d}t} \left ( \frac {\partial  L}{\partial \dot{q}_j} \right ) =  \frac {\partial L}{\partial q_j} </math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></div></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">'''Hamiltonian Mechanics'''</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Hamiltonian i.t.o. Lagrangian:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"><div class="mw-collapsible mw-collapsed" style="width:750px"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>H = \dot{\mathbf{q}}  \frac{\partial \mathcal{L}}{\partial \dot{\mathbf{q}}} - \mathcal{L} </math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></div></ins></div></td></tr>
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<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Transform from rotating to fixed frame:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"><div class="mw-collapsible mw-collapsed" style="width:750px"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">\[</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">\vec{v}' = \vec{v}  - \vec{\omega} \times \vec{r}</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">\]</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></div></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">'''Relativity'''</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Write the product $\gamma\beta$ in terms of only $\gamma$:<div class="mw-collapsible mw-collapsed" style="width:750px"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>\gamma \beta = \sqrt{\frac{\beta^2}{1- \beta^2}}= \sqrt{\frac{A}{1-\beta^2} + B} \\</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">\text{In the words of the notorious J.D. Jackson, ''we see'' that }A = 1 \text{ and }B = -1 \text{ , such that } \\ \gamma \beta = \sqrt{\frac{1}{1-\beta^2} - 1}=\sqrt{\gamma^2 - 1}  </math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">E.g.: Lim#3021</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></div></ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"><br/></ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===''<small>Exercises</small>''===</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===''<small>Exercises</small>''===</div></td></tr>
</table>
Tom Neiser
https://physwiki.com/w/index.php?title=CM_Course_Notes&diff=10395&oldid=prev
Tom Neiser at 23:12, 14 April 2014
2014-04-14T23:12:44Z
<p></p>
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<td colspan='2' style="background-color: white; color:black;">← Older revision</td>
<td colspan='2' style="background-color: white; color:black;">Revision as of 23:12, 14 April 2014</td>
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<td colspan="2" class="diff-lineno">Line 1:</td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Here is a conglomerate of notes gathered for the graduate CM class<del class="diffchange diffchange-inline">. Compare with the [[Formula Sheet]] for the comprehensive exams</del>.  </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Here is a conglomerate of notes gathered for the graduate CM class.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><div class="fb-share-button" data-href="http://developers.facebook.com/docs/plugins/" data-type="button_count"></div></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><div class="fb-share-button" data-href="http://developers.facebook.com/docs/plugins/" data-type="button_count"></div></div></td></tr>
</table>
Tom Neiser
https://physwiki.com/w/index.php?title=CM_Course_Notes&diff=10348&oldid=prev
Tom Neiser at 22:21, 9 April 2014
2014-04-09T22:21:29Z
<p></p>
<table class='diff diff-contentalign-left'>
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<td colspan='2' style="background-color: white; color:black;">Revision as of 22:21, 9 April 2014</td>
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<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Here is a conglomerate of notes gathered for the graduate <del class="diffchange diffchange-inline">EM classes 210A and 210B</del>. Compare with the [[Formula Sheet]] for the comprehensive exams.  </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Here is a conglomerate of notes gathered for the graduate <ins class="diffchange diffchange-inline">CM class</ins>. Compare with the [[Formula Sheet]] for the comprehensive exams.  </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><div class="fb-share-button" data-href="http://developers.facebook.com/docs/plugins/" data-type="button_count"></div></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><div class="fb-share-button" data-href="http://developers.facebook.com/docs/plugins/" data-type="button_count"></div></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
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<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>==<del class="diffchange diffchange-inline">221A </del>Notes==</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>==<ins class="diffchange diffchange-inline">220 </ins>Notes==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><!--   </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><!--   </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>  <div class="mw-collapsible mw-collapsed" style="width:750px"></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>  <div class="mw-collapsible mw-collapsed" style="width:750px"></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===''<small>Exercises</small>''===</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===''<small>Exercises</small>''===</div></td></tr>
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<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">==210B Notes==</del></div></td><td colspan="2"> </td></tr>
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<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">  </del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"> </div></del></div></td><td colspan="2"> </td></tr>
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<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">===''<small>Exercises</small>''===</del></div></td><td colspan="2"> </td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==General Identities==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==General Identities==</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>__NOCACHE__</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>__NOCACHE__</div></td></tr>
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Tom Neiser
https://physwiki.com/w/index.php?title=CM_Course_Notes&diff=10347&oldid=prev
Tom Neiser: Created page with "Here is a conglomerate of notes gathered for the graduate EM classes 210A and 210B. Compare with the Formula Sheet for the comprehensive exams. <div class="fb-share-butt..."
2014-04-09T22:20:43Z
<p>Created page with "Here is a conglomerate of notes gathered for the graduate EM classes 210A and 210B. Compare with the <a href="/w/index.php/Formula_Sheet" title="Formula Sheet">Formula Sheet</a> for the comprehensive exams. <div class="fb-share-butt..."</p>
<p><b>New page</b></p><div>Here is a conglomerate of notes gathered for the graduate EM classes 210A and 210B. Compare with the [[Formula Sheet]] for the comprehensive exams. <br />
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==221A Notes==<br />
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===''<small>Exercises</small>''===<br />
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===''<small>Exercises</small>''===<br />
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Tom Neiser