Preface
      This	book	contains	the	solutions	of	all	the	exercises	of	my	book:	Principles	of
      Tensor	Calculus.	These	solutions	are	sufficiently	simplified	and	detailed	for	the
      benefit	of	readers	of	all	levels	particularly	those	at	introductory	levels.
      Taha	Sochi
      London,	September	2018
      	
      Table	of	Contents
      Preface
      Nomenclature
      Chapter	1	Preliminaries
      Chapter	2	Spaces,	Coordinate	Systems	and	Transformations
      Chapter	3	Tensors
      Chapter	4	Special	Tensors
      Chapter	5	Tensor	Differentiation
      Chapter	6	Differential	Operations
      Chapter	7	Tensors	in	Application
      Author	Notes
      Footnotes
      	
                     Nomenclature
                     In	the	following	list,	we	define	the	common	symbols,	notations	and
                     abbreviations	which	are	used	in	the	book	as	a	quick	reference	for	the	reader.
                     ∇                                nabla	differential	operator
                                   ;
                         	and	                        covariant	and	contravariant	differential	operators
                     ∇          ∇
                        ;
                     ∇f                               gradient	of	scalar	f
                     ∇⋅A                              divergence	of	tensor	A
                     ∇ × A                            curl	of	tensor	A
                        2
                         ,	    ,	                     Laplacian	operator
                     ∇ ∂ ∇
                             ii     ii
                     ∇v,	∂v                           velocity	gradient	tensor
                              i j
                     ,	(subscript)                    partial	derivative	with	respect	to	following	index(es)
                     ;	(subscript)                    covariant	derivative	with	respect	to	following	index(es)
                     hat	(e.g.	Â ,	Ê )                physical	representation	or	normalized	vector
                                      i    i
                                     i
                     bar	(e.g.	 ,	         )          transformed	quantity
                                   ũ Ã
                                           i
                     ○                                inner	or	outer	product	operator
                     ⊥                                perpendicular	to
                     1D,	2D,	3D,	nD                   one-,	two-,	three-,	n-dimensional
                     δ ⁄ δt                           absolute	derivative	operator	with	respect	to	t
                                                                                                                          th
                     ∂	and	∇                          partial	derivative	operator	with	respect	to	                           	variable
                                                                                                                         i
                       i          i
                                                                                                                               th
                     ∂                                covariant	derivative	operator	with	respect	to	                             	variable
                                                                                                                              i
                       ;i
                                                                                            st
                     [ij, k]                          Christoffel	symbol	of	                  	kind
                                                                                          1
                     A                                area
                     B,	B                             Finger	strain	tensor
                            ij
                         − 1        − 1
                            ,	                        Cauchy	strain	tensor
                     B        B
                                 ij
                     C                                curve
                        n
                                                      of	class	n
                     C
                     d,	d                             displacement	vector
                           i
                     det                              determinant	of	matrix
                  diag[⋯]                      diagonal	matrix	with	embraced	diagonal	elements
                  dr                           differential	of	position	vector
                  ds                           length	of	infinitesimal	element	of	curve
                  dσ                           area	of	infinitesimal	element	of	surface
                  dτ                           volume	of	infinitesimal	element	of	space
                                                th
                  e                               	vector	of	orthonormal	vector	set	(usually	Cartesian	basis
                                               i
                    i
                                               set)
                  e , e , e                    basis	vectors	of	spherical	coordinate	system
                    r   θ   φ
                  e , e , ⋯, e                 unit	dyads	of	spherical	coordinate	system
                    rr   rθ         φφ
                  e , e , e                    basis	vectors	of	cylindrical	coordinate	system
                    ρ   φ    z
                  e , e , ⋯, e                 unit	dyads	of	cylindrical	coordinate	system
                    ρρ   ρφ         zz
                  E,	E                         first	displacement	gradient	tensor
                        ij
                         i                      th
                     ,	                           	covariant	and	contravariant	basis	vectors
                  E E                          i
                    i
                                                th
                  ℰ                               	orthonormalized	covariant	basis	vector
                                               i
                     i
                  Eq./Eqs.                     Equation/Equations
                  g                            determinant	of	covariant	metric	tensor
                  g                            metric	tensor
                         ij   j
                     ,	   ,	                   covariant,	contravariant	and	mixed	metric	tensor	or	its
                  g g g
                    ij        i
                                               components
                  g , g , ⋯g                   coefficients	of	covariant	metric	tensor
                    11    12       nn
                    11    12       nn
                                               coefficients	of	contravariant	metric	tensor
                  g , g , ⋯g
                                                                     th
                  h                            scale	factor	for	       	coordinate
                                                                    i
                    i
                  iff                          if	and	only	if
                  J                            Jacobian	of	transformation	between	two	coordinate
                                               systems
                  J                            Jacobian	matrix	of	transformation	between	two	coordinate
                                               systems
                     − 1
                                               inverse	Jacobian	matrix	of	transformation
                  J
                  L                            length	of	curve
                  n,	n                         normal	vector	to	surface
                        i
                  P                            point
                  P(n, k)                      k-permutations	of	n	objects