differential geometry w notes from teacher_Part_44

# differential geometry w notes from teacher_Part_44 - 3.5...

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Unformatted text preview: 3.5. VECTOR ANALYSIS IN R3 87 • Therefore (∗dα)i = εi jk ∂ j αk , so that ∗dα = (∂2 α3 − ∂3 α2 )dx + (∂3 α1 − ∂1 α3 )dy + (∂1 α2 − ∂2 α1 )dz . • We see that ∗dα = curl α . • Two-Forms. For a 2-form β there holds (dβ)i jk = ∂i β jk + ∂ j βki + ∂k βi j , or dβ = (∂1 β23 + ∂2 β31 + ∂3 β12 )dx ∧ dy ∧ dz . • Hence, 1 ∗dβ = εi jk ∂i β jk = ∂1 β23 + ∂2 β31 + ∂3 β12 . 2 • Now let α be a 1-form α = α1 dx + α2 dy + α3 dz . Then ∗α = α1 dy ∧ dz − α2 dx ∧ dz + α3 dx ∧ dy , and d ∗ α = (∂1 α1 + ∂2 α2 + ∂3 α3 )dx ∧ dy ∧ dz , or ∗d ∗ α = ∂1 α1 + ∂2 α2 + ∂3 α3 . So, ∗d ∗ α = div α . diﬀgeom.tex; April 12, 2006; 17:59; p. 89 88 CHAPTER 3. DIFFERENTIAL FORMS diﬀgeom.tex; April 12, 2006; 17:59; p. 90 ...
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differential geometry w notes from teacher_Part_44 - 3.5...

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