differential geometry w notes from teacher_Part_44

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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online