Page 4 - Intro to Tensor Calculus

INTRODUCTION TO
TENSOR CALCULUS
AND
CONTINUUM MECHANICS

PART 1: INTRODUCTION TO TENSOR CALCULUS

§1.1 INDEX NOTATION .. .. ... .. .. ... .. .. 1
Exercise 1.1 . . . ... ... .. ... ... .. ... ... . 28
§1.2 TENSOR CONCEPTS AND TRANSFORMATIONS .. .. 35
Exercise 1.2 .. .. ... ... .. ... ... .. ... ... . 54
§1.3 SPECIAL TENSORS .. .. ... .. .. ... .. .. 65
Exercise 1.3 .. .. ... ... .. ... ... .. ... ... . 101
§1.4 DERIVATIVE OF A TENSOR ... .. .. ... .. .. 108
Exercise 1.4 .. .. ... ... .. ... ... .. ... ... . 123

§1.5 DIFFERENTIAL GEOMETRY AND RELATIVITY .. .. 129
Exercise 1.5 .. .. ... ... .. ... ... .. ... ... . 162

PART 2: INTRODUCTION TO CONTINUUM MECHANICS

§2.1 TENSOR NOTATION FOR VECTOR QUANTITIES .. .. 171
Exercise 2.1 .. .. ... ... .. ... ... .. ... ... . 182

§2.2 DYNAMICS . ... .. .. ... .. .. ... .. .. 187
Exercise 2.2 .. .. ... ... .. ... ... .. ... ... . 206
§2.3 BASIC EQUATIONS OF CONTINUUM MECHANICS .. . 211
Exercise 2.3 .. .. ... ... .. ... ... .. ... ... . 238
§2.4 CONTINUUM MECHANICS (SOLIDS) .. ... .. .. 243
Exercise 2.4 .. .. ... ... .. ... ... .. ... ... . 272
§2.5 CONTINUUM MECHANICS (FLUIDS) .. ... .. .. 282
Exercise 2.5 .. .. ... ... .. ... ... .. ... ... . 317
§2.6 ELECTRIC AND MAGNETIC FIELDS . . . ... .. .. 325
Exercise 2.6 .. .. ... ... .. ... ... .. ... ... . 347

BIBLIOGRAPHY ... .. .. ... .. .. ... .. .. 352
APPENDIX A UNITS OF MEASUREMENT ... .. .. 353
APPENDIX B CHRISTOFFEL SYMBOLS OF SECOND KIND 355
APPENDIX C VECTOR IDENTITIES . . . ... .. .. 362
INDEX . . . . . ... .. .. ... .. .. ... .. .. 363