1, A.J.Mcconnell, applications of tensor analysis, dover publications,Inc, NEW York 2, Garl E.pearson, handbook of applied mathematics, Van Nostrand Reinhold ; 2nd edition 3, I.s. sokolnikoff, tensor analysis, Walter de Gruyter ; 2Rev Ed edition 4. C.WREDE, introduction to vectors and tensor analysis, Dover Publications ; New Ed edition 5.WILHELM Flugge, tensor analysis and continuum mechanics, Springer ; 1 edition 6.Eutioquio C.young, vector and tensor analysis, CRC ; 2 edition 7. 黃克智, 薛明德, 陸明萬, 張量分析, 北京清華大學出版社

Tensor analysis 1 、 Vector in Euclidean 3-D 2 、 Tensors in Euclidean 3-D 3 、 general curvilinear coordinates in Euclidean 3-D 4 、 tensor calculus

1-1 Orthonormal base vector ： Let (e 1,e 2,e 3 ) be a right-handed set of three mutually perpendicular vector of unit magnitude 1 、 Vector in euclidean 3-D

e i (i = 1,2,3)may be used to define the kronecker delta δ ij and the permutation symbol e ijk by means of the equations and

[prove]

By setting r= i we recover the e – δ relation Thus δ ij = 1 for i=j ; e 123 =e 312 =e 231 =1;e 132 =e 321 =e 213 =-1; All other e ijk = 0. in turn,a pair of e ijk is related to a determinate of δ ij by (1-1-1)

Rotated set of orthonormal base vector is introduced. the corresponding new components of F may be computed in terms of the old ones by writing We have transformation rule Here 1-2 Cartesian component of vectors transformation rule

These direction cosines satisfy the useful relations (1-2-1) [prove]

1-3 General base vectors ： vector components with respect to a triad of base vector need not be resticted to the use of orthonomal vector.let ε 1, ε 2, ε 3 be any three noncoplanar vector that play the roles of general base vectors

and (metric tensor) (permutation tensor) From (1-1-1) the general vector identity can be established (*)(*)

Note that the volume v of the parallelopiped having the base vectors as edges equals ; consequently the permutation tensor and the permutation symbols are related by Denote by the determinate of the matrix having as its element then, by ( * ),, so that

1-4 General components of vectors ;transformation rules (convariant component) (contravariant component) The two kinds of components can be related with the help of the metric tensor. Substituting into yields [prove]

Use to denote the (i, j) th element of the inverse of the matrix [g] Where that the indices in the Kronecker delta have been placed in superscript and subscript position to conform to their placement on the left-hand side of the equation. When general vector components enter, the summation convention invariably applies to summation over a repeated index that appears once as a superscript and once as a subscript

for the transformed covariant components. We also find easily that The metric tensor can introduce an auxiliary set of base vectors by means of means of the definition Consider, finally, the question of base vector, a direct calculation gives

2-1 Dyads, dyadics, and second-order tensors The mathematical object denote by AB, where A and B are given vectors, is called a dyad. The meaning of AB is simply that the operation A sum of dyads, of the form Is called a dyadic 2 、 Tensors in Euclidean 3-D

Any dyadic can be expressed in terms of an arbitrary set of general base vectors ε i ;since It follows that T can always be written in the form (2-1-1) (contravariant components of the tensor).

We re-emphasize the basic meaning of T by noting that,for all vector V By introduce the contravariant base vectors є i we can define other kinds of components of the tensor T. thus, substituting Into (2-1-1), we get Where are nine quantities

are called the covariant component of the tensor. Similarly,we can define two, generally,kinds of mixed components That appear in the representation

Suppose new base vectors are introduced ； what are the new components of T ？ substitution Into (2-1-1) gives Is the desired transformation rule. Many different, but equivalent, relations are easily derived ； for example 2.2 Transformation rule

2.3 Cartesian components of second-order Tensors Cartesian components

2.4 Tensors operations Quotient laws

2.5 The metric Tensor Substituting

2.6 N th _ order Tensors A third-order tensor, or triadic, is the sum of triads,as follows ： It is easily established that any third-order tensor can written As well as in the alternative form N indices N base vectros

2.7 The permutation tensor Choose a particular set of base vectors ， and define the third –order tensor Since it follow that E has the same from as (2-4-1) with respect to all sets of base vector, so is indeed a tensor (2-4-1)

3 、 general curvilinear coordinates in Euclidean 3-D Suppose that general coordinates are( ); this means that the position vectors x of a point is a known function of, and, then the choice that is usually made for the base vectors is For consistency with the right-handedness of the ε i, the coordinates must be numbered in such a way that 3-1 coordinate system and general in Euclidean 3-D

With in terms of the Cartesian coordinates and the Cartesian base vectors, where, we have And so As an example, consider the cylindrical coordinate

We have already seen that is a tensor ; it will now be shown why it is called the metric tensor The definition, together with,give Note that So that an element of arc length satisfies 3-2 metric tensor and jacobian

The jacobian of the transformation relating Cartesian coordinates and curvilinear coordinates is determinant of the array and the element of volume having the vectors As edges is Note that is the same as

3-3 Transformation rule for change of coordinates Suppose a new set general coordinates is introduced, with the understanding that the relations between and are known, at least in principle. The rule for changing to new tensor components is

4-1 gradient of a scalar If f is a scalar function, then But grad ; hence An alternative way to conclude that is a vector is to note that is a scalar for all recall that is a vector,and invoke the appropriate quotient law. 4 、 tensor calculus is the convariant component of

4-2 Derivative of a vector ; christoffel` symbol; covariant derivative Consider the partial derivative of a vector F. with F =, we have write the contravriant component of the derivative with respect to of the base vector. Note that christoffel system of the second kind

We can now write Introduce the notation This means that called the convariant derivative of --- is definded as the contravariant component of the vector comparing (4-2-1) and (4- 2-2) then gives us the formula Although is not necessarily a tensor, is one, for (4-2-1) (4-2-2)

The covariant derivative of writing as, is defined as the convariant component of ; hence A direct calculation of is more instructive; with F=,we have Now, whence And therefore consequently And while this be the same as (4-2-3) it shows the explicit addition to needed to provide the covariant derivative of (4-2-3)

Other notations are common for convariant derivatiives; they are, in approximate order of popularity Although is not a third-order tensor, the superscript can nevertheless be lowered by means of the operation and the resultant quantity, denoted by, is the Christoffel symbol of the first kind. The following relations are easily verified

(4-2-4) [Prove] ：

4-3 covariant derivatives of N th –order Tensors Let us work out the formula for the covariant derivatives of. write By definition This leads directly to the formula

4-4 divergence of a vector A useful formula for will be developed for general coordinate systems. We have But, by determinant theory Hence And therefore

4-5 Riemann-Christoffel Tensor Since the order of differentiation in repeated partial differentiation of Cartesian tensor is irrelevant, it follows that the indices in repeated covariant differentiation of general tensors in Euclidean 3-D may also be interchanged at will. Thus, identities like and Eq (4-5-1) is easily verified directly, since (4-5-1) However, the assertion of (4-5-1) in Euclidean 3-D leads to some nontrivial information. By direct calculation it can be shown that

With help of (4-2-4) it can be shown that, the Riemann-christoffel tensor, is given by But since the left-hand side of vanishes for all vectors, it follows that Although (4-5-2) represents 81equations, most of them are either identities or redundant, since. Only six distinct nontrivial conditions are specified by (4-5-2), and they may be written as (4-5-2) (4-5-3)

[Note] [prove]

Since is antisymmetrical in i and j as well as in p and k, no information is lost if (4-5-2) is multiplied by. Consequently, a set of six equations equivalent to (4-5-3) is given neatly by Where is the symmetrical, second-order tensor The tensor is related simply to the Ricci tensor So that (4-5-3) is also equivalent to the assertion

The familiar divergence theorem relating integrals over a volume V and its boundary surface S can obvious be written in tensor notation as Where N i is the unit outward normal vector to S. Similar stokes ， theorem for integrals over a surface S and its boundary line C is just Where t k is the unit tangent vector to C, and the usual handedness rules apply for direction of N i and t i 4-6 Integral Relations