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For describing functions of three dimensions, away from the coordinate origin, the coefficients of the multipole expansion can be written as functions of the distance to the origin, <math>r</math>—most frequently, as a [[Laurent series]] in powers of <math>r</math>. For example, to describe the electromagnetic potential, <math>V</math>, from a source in a small region near the origin, the coefficients may be written as: |
For describing functions of three dimensions, away from the coordinate origin, the coefficients of the multipole expansion can be written as functions of the distance to the origin, <math>r</math>—most frequently, as a [[Laurent series]] in powers of <math>r</math>. For example, to describe the electromagnetic potential, <math>V</math>, from a source in a small region near the origin, the coefficients may be written as: |
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<math display="block">V(r,\theta,\varphi) = \sum_{\ell=0}^\infty\, \sum_{m=- |
<math display="block">V(r,\theta,\varphi) = \sum_{\ell=0}^\infty\, \sum_{m=-l}^\ell C^m_\ell(r)\, Y^m_\ell(\theta,\varphi)= \sum_{j=1}^\infty\, \sum_{\ell=0}^\infty\, \sum_{m=-l}^\ell \frac{D^m_{\ell,j}}{r^j}\, Y^m_\ell(\theta,\varphi) .</math> |
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==Applications== |
==Applications== |
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===Expansion in Cartesian coordinates=== |
===Expansion in Cartesian coordinates=== |
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Assume {{math|1=''v''('''r''') |
Assume {{math|1=''v''('''r''')=''v''(-'''r''')}} for convenience. The [[Taylor series|Taylor expansion]] of {{math|1=''v''('''r''' − '''R''')}} around the origin {{math|1='''r''' = '''0'''}} can be written as |
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<math display="block">v(\mathbf{r}- \mathbf{R}) = v(\mathbf{R}) - \sum_{\alpha=x,y,z} r_\alpha v_\alpha(\mathbf{R}) +\frac{1}{2} \sum_{\alpha=x,y,z}\sum_{\beta=x,y,z} r_\alpha r_\beta v_{\alpha\beta}(\mathbf{R}) |
<math display="block">v(\mathbf{r}- \mathbf{R}) = v(\mathbf{R}) - \sum_{\alpha=x,y,z} r_\alpha v_\alpha(\mathbf{R}) +\frac{1}{2} \sum_{\alpha=x,y,z}\sum_{\beta=x,y,z} r_\alpha r_\beta v_{\alpha\beta}(\mathbf{R}) |
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- \cdots + \cdots</math> |
- \cdots + \cdots</math> |
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and the expansion can be rewritten in terms of the components of a traceless Cartesian second rank [[tensor]]: |
and the expansion can be rewritten in terms of the components of a traceless Cartesian second rank [[tensor]]: |
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<math display="block">\sum_{\alpha=x,y,z}\sum_{\beta=x,y,z} r_\alpha r_\beta v_{\alpha\beta}(\mathbf{R}) |
<math display="block">\sum_{\alpha=x,y,z}\sum_{\beta=x,y,z} r_\alpha r_\beta v_{\alpha\beta}(\mathbf{R}) |
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= \frac{1}{3} \sum_{\alpha=x,y,z}\sum_{\beta=x,y,z} |
= \frac{1}{3} \sum_{\alpha=x,y,z}\sum_{\beta=x,y,z} (3r_\alpha r_\beta - \delta_{\alpha\beta} r^2) v_{\alpha\beta}(\mathbf{R}) ,</math> |
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where {{math|''δ''<sub>''αβ''</sub>}} is the [[Kronecker delta]] and {{math|''r''<sup>2</sup> ≡ {{abs|'''r'''}}<sup>2</sup>}}. Removing the trace is common, because it takes the rotationally invariant {{math|''r''<sup>2</sup>}} out of the second rank tensor. |
where {{math|''δ''<sub>''αβ''</sub>}} is the [[Kronecker delta]] and {{math|''r''<sup>2</sup> ≡ {{abs|'''r'''}}<sup>2</sup>}}. Removing the trace is common, because it takes the rotationally invariant {{math|''r''<sup>2</sup>}} out of the second rank tensor. |
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Copy and paste: – — ° ′ ″ ≈ ≠ ≤ ≥ ± − × ÷ ← → · § Cite your sources: <ref></ref>
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