Tensor Analysis and Curvilinear Coordinates Phil Lucht Rimrock Digital Technology, Salt Lake City, Utah 84103 last update: May 19, 2016 Maple code is available upon request. Comments and errata are welcome. The material in this document is copyrighted by the author. The graphics look ratty in Windows Adobe PDF viewers when not scaled up, but look just fine in this excellent freeware viewer: http://www.tracker-software.com/pdf-xchange-products-comparison-chart . The table of contents has live links. Most PDF viewers provide these links as bookmarks on the left. Overview and Summary.........................................................................................................................7 1. The Transformation F: invertibility, coordinate lines, and level surfaces..................................12 Example 1: Polar coordinates (N=2)..................................................................................................13 Example 2: Spherical coordinates (N=3)...........................................................................................14 Cartesian Space and Quasi-Cartesian Space.......................................................................................15 Pictures A,B,C and D..........................................................................................................................16 Coordinate Lines.................................................................................................................................16 Example 1: Polar coordinates, coordinate lines .................................................................................17 Example 2: Spherical coordinates, coordinate lines ..........................................................................18 Level Surfaces.....................................................................................................................................18 2. Linear Local Transformations associated with F : scalars and two kinds of vectors .................20 2.1 Linear Local Transformations.......................................................................................................20 2.2 Scalars...........................................................................................................................................21 2.3 Contravariant vectors ....................................................................................................................22 2.4 Covariant vectors ..........................................................................................................................22 2.5 Bar notation...................................................................................................................................23 2.6 Origin of the names contravariant and covariant ..........................................................................23 2.7 Other vector types? .......................................................................................................................24 2.8 Linear transformations ..................................................................................................................25 2.9 Vectors that are contravariant by definition..................................................................................26 2.10 Vector Fields...............................................................................................................................26 2.11 Names and symbols ....................................................................................................................27 2.12 Definition of the words "scalar", "vector" and "tensor"..............................................................28 3. Tangent Base Vectors en and Inverse Tangent Base Vectors u'n ..................................................30 3.1 Differential Displacements ...........................................................................................................30 3.2 Definition of the en ; the en are the columns of S.........................................................................31 3.3 en as a contravariant vector ...........................................................................................................33 3.4 A semantic question: unit vectors ................................................................................................33 Example 1: Polar coordinates, tangent base vectors ..........................................................................34 Example 2: Spherical Coordinates, tangent base vectors...................................................................35 3.5 The inverse tangent base vectors u'n and inverse coordinate lines................................................36 Example 1: Polar coordinates: inverse tangent base vectors and inverse coordinate lines...............37 1 4. Notions of length, distance and scalar product in Cartesian Space..............................................39 5. The Metric Tensor ............................................................................................................................41 5.1 The Picture D Context ..................................................................................................................41 5.2 Definition of the metric tensor......................................................................................................41 5.3 Inverse of the metric tensor...........................................................................................................44 5.4 A metric tensor is symmetric ........................................................................................................44 5.5 det(g) and gnn of a Cartesian-generated metric tensor are non-negative.......................................45 5.6 Definition of two kinds of rank-2 tensors .....................................................................................45 5.7 Proof that the metric tensor and its inverse are both rank-2 tensors .............................................46 5.8 Metric tensor converts vector types ..............................................................................................49 5.9 Vectors in Cartesian space ............................................................................................................49 5.10 The covariant dot product A • B and norm |A| .........................................................................50 5.11 Metric tensor and tangent base vectors: scale factors and orthogonal coordinates...................52 5.12 The Jacobian J.............................................................................................................................55 5.13 Some relations between g, R and S in Pictures B and C (Cartesian x-space).............................58 Example 1: Polar coordinates: metric tensor and Jacobian...............................................................60 Example 2: Spherical coordinates: metric tensor and Jacobian ........................................................61 5.14 Special Relativity and its Metric Tensor: vectors and spinors...................................................62 5.15 General Relativity and its Metric Tensor....................................................................................65 5.16 Continuum Mechanics and its Metric Tensors............................................................................66 6. Reciprocal Base Vectors En and Inverse Reciprocal Base Vectors U'n........................................73 6.1 Definition of the En .......................................................................................................................73 6.2 The en and En Dot Products and Reciprocity (Duality) ................................................................74 6.3 Covariant partners for en and En ..................................................................................................78 6.4 Summary of the basic facts about en and En .................................................................................80 6.5 Repeat the above for the inverse transformation: definition of the U'n........................................80 6.6 Expanding vectors on different sets of basis vectors ....................................................................81 6.7 Another way to write the En..........................................................................................................84 6.8 Comparison of ¯en and En ..............................................................................................................85 6.9 Handedness of coordinate systems: the en , the sign of det(S), and Parity ..................................86 7. Translation to the Standard Notation .............................................................................................89 7.1 Outer Products ..............................................................................................................................89 7.2 Mixed Tensors and Notation Issues..............................................................................................90 7.3 The up/down bell goes off ............................................................................................................91 7.4 Some Preliminary Translations; raising and lowering tensor indices with g...............................92 7.5 Dealing with the matrices R and S ; various Rules and Theorems ...............................................96 7.6 Orthogonality Rules, Inversion Rules, Cancellation