Page 29 - Intro to Tensor Calculus

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The Green’s identity
ZZZ ZZ
2
2
φ∇ ψ − ψ∇ φ dτ = (φ∇ψ − ψ∇φ) · b n dσ
V S
~
~
is obtained by first letting F = φ∇ψ in the divergence theorem and then letting F = ψ∇φ in the divergence
theorem and then subtracting the results.
Determinants, Cofactors

For A =(a ij ),i, j =1,... ,n an n × n matrix, the determinant of A can be written as

a a a .
det A = |A| = e i 1 i 2 i 3 ...i n 1i 1 2i 2 3i 3 ...a ni n

This gives a summation of the n! permutations of products formed from the elements of the matrix A.The
result is a single number called the determinant of A.
EXAMPLE 1.1-24. In the case n =2 we have

a 11 a 12
|A| = = e nma 1n a 2m

a 21 a 22
= e 1m a 11 a 2m + e 2m a 12 a 2m
= e 12 a 11 a 22 + e 21 a 12 a 21

= a 11 a 22 − a 12 a 21

EXAMPLE 1.1-25. In the case n = 3 we can use either of the notations
   1 1 1 
a 11 a 12 a 13 a 1 a 2 a 3
A =   or A =  a 2 a 2 a 2 
a 23
a 22
a 21
1 2 3
a 3 a 3 a 3
a 31 a 32 a 33
1 2 3
and represent the determinant of A in any of the forms
det A = e ijk a 1i a 2j a 3k
det A = e ijk a i1 a j2 a k3
i j k
det A = e ijk a a a
1 2 3
1 2 3
det A = e ijk a a a .
i j k
These represent row and column expansions of the determinant.
i j k
An important identity results if we examine the quantity B rst = e ijk a a a . It is an easy exercise to
r s t
change the dummy summation indices and rearrange terms in this expression. For example,
k j i
i j k
i j k
i j k
B rst = e ijk a a a = e kji a a a = e kji a a a = −e ijk a a a = −B tsr ,
r s t
t s r
t s r
r s t
and by considering other permutations of the indices, one can establish that B rst is completely skew-
symmetric. In the exercises it is shown that any third order completely skew-symmetric system satisfies
B rst = B 123 e rst . But B 123 =det A and so we arrive at the identity
i j k
B rst = e ijk a a a = |A|e rst .
r s t