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in direct correspondence to Eq. (1.7) we write
A = A 11 e 1 e 1 + A 12 e 1 e 2 + + A 33 e 3 e 3 .
(1.8)
Two vectors written sidebyside are to be multiplied dyadically, f
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The system flag, A or B, acts as a dotplaceholder and moves along with its associated index
in operations. Now that we are dealing with up to four basis triads, the components
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Covariant differentiation of covariant components Recalling, Eq. (5.63), we now consider
a similarlydefined basisdependent vector:
g k
P ik  ,
i
(5.90)
and analogously to E
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Curvilinear Analysis in a Euclidean Space
Rebecca M.Brannon, University of New Mexico, First draft: Fall 1998, Present draft: 6/17/04.
1. Introduction
This manuscript is a stude
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vantage of being applicable only for RCC systems. The same operation would be computed
differently in a nonRCC system the fundamental operation itself doesnt change; instead
th
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position vector would move when changing the associated coordinate, holding others con1
stant. We will call a basis regular if it consists of a righthanded orthonormal triad.
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Study Question 2.5 Recall that the covariant basis may be regarded as a transformation
of the righthanded orthonormal laboratory basis. Namely, g i = F e i .
(a) Prove that
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and lowering indices.
Study Question 2.9 Simplify the following expressions so that there are no metric coefficients:
a m g ml , g pk u p , f n g ni , r i g ij s j , g ij b k g
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2.1 Modified summation convention
Given that cfw_ g 1, g 2, g 3 is a basis, we know there exist unique coefficients cfw_ a 1, a 2, a 3
a can
be written a = a 1 g + a 2 g +
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components of A and the contravariant B mn components of B are known, then the inner
product between A and B is
A :B = ( A ji g i g j ): ( B mn g m g n ) = A ji B mn ( g i g j
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In terms of a general basis, the basis form for the alternating tensor is
k g i g j g = etc. ,
= ijk g i g j g k = ijk g i g j g k = ij
k
where
ijk = [ g i, g j, g k ]
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We showed in previous sections how the covariant components of a vector are related to
the contravariant components from the same system. For example, we showed that v i = g ij
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variant metric. Thus, we may solve for the derivative on the righthandside to obtain
1
J
 =  Jg ij
2
g ij
(2.30)
where we have again recalled that g o = J 2 .
Study Questio
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Study Question 4.3 Consider the same irregular base vectors in Study Question 4.1.
Namely,
g 1A = e 1 + 2e 2
g 2A = e 1 + e 2
and
and
g 2B
g 3A = e 3
g 1B = 2e 1 + e 2
and
g 2B
UNM SUPPLEMENTAL BOOK DRAFT
June 2004
Curvilinear Analysis in a Euclidean Space
Presented in a framework and notation customized for students and professionals
who are already familiar with Cartesian analysis in ordinary 3D physical engineering
space.
Reb
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m) so that
i B m g gn .
(3.8)
A B = A m
n i
Alternatively, we could have used the g jm to lower the m superscript on B (changing it to
a j) so that
(3.9)
A B = A ij B jn g
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second indices of v i W jk Z lmn gives ji v i W jk Z lmn , which simplifies to v i W ik Z lmn . Note that the
contraction operation has reduced the order of the result from a s
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ij
mn g iA g jA , the mixed tensors being ones whose components transform
according to T BB
= T AA
Bm Bn
Bj = T An g BA g jn or T iB = T mA g im g BA , etc.
according to T iB
m
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Our final index changing properties involve the mixed Kronecker delta itself. The mixed
Kronecker delta ij changes the index symbol without changing its level. For example,
v i
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thats needed is knowledge of tensor analysis in Cartesian coordinates. However, for more
complicated coordinate systems, a fully developed curvilinear theory is indispensable.
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Applying Eq. (5.41) with the contravariant basis of Eq. (5.26) gives
s
s e  s
ds +  e z ,
=  e r +  r
z
r
dx
which is the gradient formula typically found in math hand
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Study Question 3.3 Consider a tensor having components with respect to the orthonormal laboratory basis given by
[T ] =
2
1
0
1 2
2 2
0 2
.
(3.33)
ek ek
Prove that this tenso
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Study Question 2.8 Consider the same irregular base vectors in Study Question 2.2.
Namely, g 1 = e 1 + 2e 2 , g 2 = e 1 + e 2 , g 3 = e 3 .
Now consider three vectors, a , b ,
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Upon simplification, the matrix of dv dx with respect to the usual orthonormalized basis
cfw_ e r, e , e z is
dv

dx
=
v r
r
1 v r
 v
r
v r
z
v
r
v z
r
v
1 v
 +
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ces. Applying similar heuristic notational interpretation, the reader can verify that u C v
must be a scalar computed in RCC by u i C ij v j .
A first course in tensor analysi
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tered tensor analysis, you will begin to recognize its basic concepts in many other seemingly
unrelated fields of study. Your knowledge of tensors will therefore help you master
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th
increased, holding the other coordinates constant. Hence, the natural definition of the i covariant basis is the partial derivative of x with respect to i :
x
g i i .
(5.13
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This document is a teaching and learning tool. To assist with this goal, you will note that the
text is colorcoded as follows:
BLUE definition
RED important concept
Please dire
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might not be orthogonal and/or might not be of unit length and/or might not form a righthanded system. Again, keep in mind that we will be deriving new procedures for computing