Page 99 - Intro to Tensor Calculus

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2
1
3
i
This is the vector λ i ,i =1, 2, 3. Finally, if we select λ ,i =1, 2, 3in the x direction, we set dx = dx =0
(2)
in the element of arc length squared and determine the unit vector
dx 3 1
3
2
1
λ =0, λ =0, λ = = .
ds h 3
i
This is the vector λ i ,i =1, 2, 3. Similarly, the unit vector µ can be selected as one of the above three
(3)
directions. Examining all nine possible combinations for selecting the unit vectors, we calculate the physical
components in an orthogonal coordinate system as:
A 11 A 12 A 13
A(11) = A(12) = A(13) =
h 1 h 1 h 1 h 2 h 1 h 3
A 21 A 22 A 23
A(21) = A(22) = A(23) =
h 1 h 2 h 2 h 2 h 2 h 3
A 31 A 32 A 33
A(31) = A(32) = A(33) =
h 3 h 1 h 3 h 2 h 3 h 3
These results can be written in the more compact form

A (i)(j)
A(ij)= no summation on i or j . (1.3.61)
h (i) h (j)
For mixed tensors we have

i1
i3
i2
i
A = g im A mj = g A 1j + g A 2j + g A 3j . (1.3.62)
j
From the fact g ij =0 for i 6= j, together with the physical components from equation (1.3.61), the equation
(1.3.62) reduces to
(i) (i)(i) 1
A = g A (i)(j) = · h (i) h (j) A(ij) no summation on i and i, j =1, 2or 3.
(j) h 2
(i)
This can also be written in the form

(i) h (i)
A(ij)= A no summation on i or j. (1.3.63)
(j)
h (j)
i
Hence, the physical components associated with the mixed tensor A in an orthogonal coordinate system
j
canbe expressedas
A(11) = A 1 1 h 1 1 h 1
1 A(12) = A A(13) = A
2 h 2 3 h 3
A(21) = A 2 h 2 2
1 A(22) = A h 2
h 1 2 A(23) = A 2 3
A(31) = A 3 h 3 A(32) = A 3 h 3 3 h 3
1 2 A(33) = A .
h 1 h 2 3
For second order contravariant tensors we may write

i3
ij
i2
i1
A g jm = A i = A g 1m + A g 2m + A g 3m .
m