SolitaryRoad.com

Website owner:  James Miller


[ Home ] [ Up ] [ Info ] [ Mail ]

TENSORS, CONTRAVARIANT AND COVARIANT COMPONENTS OF A VECTOR IN A CURVINILINEAR COORDINATE SYSTEM, NOTATIONS AND ABBREVIATIONS



Contravariant and covariant components of a vector in a curvinilinear coordinate system. Suppose we are given a system of equations


            u1 = u1(x, y, z)

            u2 = u2(x, y, z)

            u3 = u3(x, y, z)

 

and, the inverse transformation


            x = x(u1, u2, u3)

            y = y(u1, u2, u3)

            z = z(u1, u2, u3)


defining a curvilinear coordinate system in space in some region R.


Contravariant components. Consider a coordinate system with its origin located at a point P in region R and with unitary base vectors


             ole.gif


where r is the position vector of P given by


            r = x(u1, u2, u3) i + y(u1, u2, u3) j + z(u1, u2, u3) k .


Then any vector A can be expressed with respect to this coordinate system in terms of its base vectors as


             ole1.gif


where C1, C2, C3 are called the contravariant components of A.



Covariant components. Consider a coordinate system with its origin located at a point P in region R and with unitary base vectors



 

ole2.gif


Then any vector A can be expressed with respect to this coordinate system in terms of its base vectors as


             ole3.gif


where c1, c2, c3 are called the covariant components of A.




Question. Let A be a given vector defined with respect to two general curvilinear coordinate systems ole4.gif and ole5.gif . 1. What is the relationship between the contravariant components of A in the two systems? 2. What is the relationship between the covariant components of A in the two systems?


Answer. We answer the two questions:

 

1. Contravariant components. Let vector A’s contravariant components with respect to system ole6.gif be ole7.gif and its contravariant components with respect to system ole8.gif be ole9.gif Then



             ole10.gif


or more briefly


ole11.gif


Proof.


Due to symmetry, by interchanging the roles of the ole12.gif and ole13.gif , we see that


  ole14.gif



These results lead us to adopt the following definition: If three quantities ole15.gif of a coordinate system ole16.gif are related to three other quantities ole17.gif of another coordinate system ole18.gif by the transformation equations 1) or 2), then the quantities are called components of a contravariant vector or a contravariant tensor of the first rank.  



2. Covariant components. Let vector A’s covariant components with respect to system ole19.gif be ole20.gif and its covariant components with respect to system ole21.gif be ole22.gif Then



             ole23.gif



or more briefly



ole24.gif



Proof.


Due to symmetry, by interchanging the roles of the ole25.gif and ole26.gif , we see that



ole27.gif


These results lead us to adopt the following definition: If three quantities ole28.gif of a coordinate system ole29.gif are related to three other quantities ole30.gif of another coordinate system ole31.gif by the transformation equations 3) or 4), then the quantities are called components of a covariant vector or a covariant tensor of the first rank.



In generalizing these concepts to higher dimensions and in generalizing the concept of vector we are lead to tensor analysis.





Notations and abbreviations of tensor analysis



Notations for variables. In tensor analysis we use both superscripts and subscripts in denoting variables. We may refer to a set of variables x1, x2, ..., xn or to the set of variables x1, x2, ..., xn .



Notations for coordinate transformations. Let ole32.gif and ole33.gif be the coordinates of a point with respect to two different frames of reference and suppose a transformation of coordinates is defined between the two frames of reference by the system



             ole34.gif



This system can be denoted by the shorthand notation


ole35.gif


There will then be an inverse transformation given by


  ole36.gif



The summation convention. The summation convention is the convention of letting the repetition of an index (subscript or superscript) in a given term denote a summation with respect to that index over its range. For example, the sum


            a1x1 + a2x2 + ... + anx6


is denoted by the shorthand notation aixi. Instead of using the index i in this example we could have used another letter, say q, and the sum could be written aqxq. A repeated index, such as i in aixi, is called a dummy index of umbral index.


Free index. An index that occurs only once in a given term is called a free index. A free index stands for any of the numbers 1, 2, ... , n. An example is k in the equations 5) and 6) above.




Contravariant tensor of the first rank (or first order). If n quantities ole37.gif in a coordinate system ole38.gif are related to n other quantities ole39.gif in another coordinate system ole40.gif by the transformation equations


                         ole41.gif


or, by the conventions and abbreviations we have adopted,


                         ole42.gif


they are called components of a covariant tensor of the first rank (or first order). [they may also be called components of a contravariant vector]



Covariant tensor of the first rank (or first order). If n quantities ole43.gif in a coordinate system ole44.gif are related to n other quantities ole45.gif in another coordinate system ole46.gif by the transformation equations


                         ole47.gif


or, by our conventions,


                         ole48.gif


they are called components of a covariant tensor of the first rank (or first order). [they may also be called components of a covariant vector]



Subscript and superscript conventions. Superscripts are used to indicate contravariant components. Subscripts are used to indicate covariant components. Either subscripts or superscripts can be used for coordinates.



Note. Instead of speaking of a tensor whose components are Ap or Ap we often refer simply to the tensor Ap or Ap.



Contravariant tensor of the second rank. If n2 quantities ole49.gif in a coordinate system ole50.gif are related to n2 other quantities ole51.gif in another coordinate system ole52.gif by the transformation equations


                         ole53.gif


or, by our conventions,


                         ole54.gif


they are called components of a contravariant tensor of the second rank (or of rank two).


Note. The tensors ole55.gif and ole56.gif are two dimensional arrays containing n rows and n columns. The integers p, r and q, s are row and column indices pointing to elements within the arrays. We can speak of a tensor ole57.gif (tensor array) or we can speak of an element within the tensor, say ole58.gif or ole59.gif . Thus we use the same notation for both the tensor and elements within the tensor, a possible cause for confusion. These arrays represent a generalization of the concept of a vector leading into tensor analysis.




Covariant tensor of the second rank. If n2 quantities ole60.gif in a coordinate system ole61.gif are related to n2 other quantities ole62.gif in another coordinate system ole63.gif by the transformation equations


                         ole64.gif


or, by our conventions,


                         ole65.gif


they are called components of a covariant tensor of the second rank.




Mixed tensor of the second rank. If n2 quantities ole66.gif in a coordinate system ole67.gif are related to n2 other quantities ole68.gif in another coordinate system ole69.gif by the transformation equations


                         ole70.gif


or, by our conventions,


                         ole71.gif


they are called components of a mixed tensor of the second rank.




Tensors of rank greater than two. Tensors of rank greater than two are easily defined. For example, the tensor ole72.gif , a mixed tensor of rank 5, contravariant of order 3 and covariant of order 2, is defined by the following relation


             ole73.gif



Note the pattern of the indices in the successive fractions


             ole74.gif


Each index of ole75.gif is paired with its counterpart in ole76.gif i.e. p is paired with q, r is paired with s, m is paired with t, etc. The order of the fractions is the same as the order of the indices in ole77.gif i.e. p, r, m, i, j. The covariant fractions are the same as the contravariant fractions, but inverted.




References.


Spiegel. Vector Analysis. Chap. 7, 8.


[ Home ] [ Up ] [ Info ] [ Mail ]