Harley Flanders Differential Forms with Applications to the Physical Sciences. Dover, 1989 (1962) Contents FOREWORD

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1 Harley Fladers Dfferetal Forms wth Applcatos to the Physcal Sceces FORWORD Dover, 989 (962) Cotets PRFAC TO TH DOVR DITION PRFAC TO TH FIRST DITION.. xteror Dfferetal Forms.2. Comparso wth Tesors 2.. The Space of p-vectors 2.2. Determats 2.3. xteror Products 2.4. Lear Trasformatos 2.5. Ier Product Spaces 2.6. Ier Products of p-vectors 2.7. The Star Operator 2.8. Problems 3.. Dfferetal Forms 3.2. xteror Dervatve 3.3. Mappgs 3.4. Chage of Coordates 3.5. A xample from Mechacs I. Itroducto II. xteror algebra III. The xteror Dervatve 3.6. Coverse of the Pocare Lemma 3.7. A xample 3.8. Further Remarks 3.9. Problems 4.. Movg Frames 3 IV. Applcatos 4.2. Relato betwee Orthogoal ad Skew-symmetrc Matrces 4.3. The 6-dmesoal Frame Space 4.4. The Laplaca, Orthogoal Coordates 4.5. Surfaces 4.6. Maxwell's Feld quatos 4.7. Problems 5.. Itroducto V. Mafolds ad Itegrato

2 5.2. Mafolds 5.3. Taget Vectors 5.4. Dfferetal Forms 5.5. ucldea Smplces 5.6. Chas ad Boudares 5.7. Itegrato of Forms 5.8. Stokes' Theorem 5.9. Perods ad De Rham's Theorems 5.0. Surfaces; Some xamples 5.. Mappgs of Chas 5.2. Problems 6.. Volumes VI. Applcatos ucldea space 6.2. Wdg Numbers, Degree of a Mappg 6.3. The Hopf Ivarat 6.4. Lkg Numbers, the Gauss Itegral, Ampère's Law 7.. Potetal Theory 7.2. The Heat quato VII. Applcatos to Dfferetal quatos 7.3. The Frobeus Itegrato Theorem 7.4. Applcatos of the Frobeus Theorem 7.5. Systems of Ordary quatos 7.6. The Thrd Le Theorem VIII. Applcatos to Dfferetal Geometry 8.. Surfaces (Cotued) 8.2. Hypersurfaces 8.3. Remaa Geometry, Local Theory 8.4. Remaa Geometry, Harmoc Itegrals 8.5. Affe Coecto 8.6. Problems 9.. Le Groups 9.2. xamples of Le Groups 9.3. Matrx Groups 9.4. xamples of Matrx Groups 9.5. B-varat Forms 9.6. Problems 0.. Phase ad State Space IX. Applcatos to Group Theory X. Applcato to Physcs

3 0.2. Hamltoa Systems 0.3. Itegral-varats 0.4. Brackets 0.5. Cotact Trasformatos 0.6. Flud Mechacs 0.7. Problems BIBLIOGRAPHY GLOSSARY OF NOTATION INDX

4 R ucldea le. ucldea -space. GLOSSARY OF NOTATION A. Spaces The set of real umbers, also cosdered as U, V, Ope sets ( L, M, Vector spaces. p, or o a mafold). L The space of p-vectors o L. M, N, Mafolds. F p (U ) The collecto of all p-forms o U., the Cartesa product. If S ad T are arbtrary sets (collectos of objects), 2 ther cartesa product s the set S T cosstg of all ordered pars (s, t) where s belogs to S ad t to T. S Ths s the cartesa product S S of S wth tself. S T Smlarly 3 S = S S S, etc. Ths s the tersecto of the sets S ad T. For example, f S = {, 2, 5, 7} ad T = {2, 3, 7, 9}, the I The ut terval 0 t. B. Fuctos S T = {2, 7}. Mappg. A mappg s a smooth fucto φ from oe space M to aother N. We wrte Composte mappg. If φ : M N. φ : M N ad ψ : N P, the we may form the composte mappg by ψ o φ : M ( ψ o φ)( x) = ψ [ φ( x)] for x M. P. It s defed Lear fuctoal. A lear trasformato o a lear space L to the oe- dmesoal space R of real umbers. ( Jacoba. If u = u x, K, x ) ( =,,), the Jacoba of ths mappg s the determat u / x. j φ * φ The mappg o dfferetal forms duced by the mappg φ betwee spaces. The mappg o chas duced by the mappg φ betwee spaces. C. Specal symbols v

5 t A The traspose of the matrx A, obtaed from A by terchagg rows ad colums. A The determat of the lear trasformato (matrx) A. dm L The dmeso of the lear space L. H σ Here H = { h, K, hp }, a set of dces creasg order, p h < L, h ad σ = σ L σ p H h h. H' Ths s the complemetary set of dces. For example, f = 8 ad H = {2, 3, 5, 6}, the H' = {l, 4, 7, 8}. sgπ If π s a permutato o {l, 2,, }, the sgπ = f π s effected by a eve umber of terchages (of two umbers) ad sgπ =- f π s effected by a odd umber of terchages. * The star operator. ( α, β ) The er product. The boudary operator. L L meas (The crcumflex dcates omsso.) L + L. v