2. The Gateaux Differential
We are presently concerned with Inner Product spaces
in our treatment of the Mechanics of Continua.
Consider a map,
๐ญ: V โ W
This maps from the domainV to W โ both of which are
Euclidean vector spaces. The concepts of limit and
continuity carries naturally from the real space to any
Euclidean vector space.
Dept of Systems Engineering, University of Lagos 2 oafak@unilag.edu.ng 12/27/2012
3. The Gateaux Differential
Let ๐0 โ V and ๐0 โ W, as usual we can say that the
limit
lim ๐ญ ๐ = ๐0
๐โ ๐0
if for any pre-assigned real number ๐ > 0, no matter
how small, we can always find a real number ๐ฟ > 0
such that ๐ญ ๐ โ ๐0 โค ๐ whenever ๐ โ ๐0 < ๐ฟ.
The function is said to be continuous at ๐0 if ๐ญ ๐0
exists and ๐ญ ๐0 = ๐0
Dept of Systems Engineering, University of Lagos 3 oafak@unilag.edu.ng 12/27/2012
4. The Gateaux Differential
Specifically, for ๐ผ โ โ let this map be:
๐ญ ๐ + ๐ผ๐ โ ๐ญ ๐ ๐
๐ท๐ญ ๐, ๐ โก lim = ๐ญ ๐ + ๐ผ๐
๐ผโ0 ๐ผ ๐๐ผ ๐ผ=0
We focus attention on the second variable โ while we allow
the dependency on ๐ to be as general as possible. We shall
show that while the above function can be any given function
of ๐ (linear or nonlinear), the above map is always linear in ๐
irrespective of what kind of Euclidean space we are mapping
from or into. It is called the Gateaux Differential.
Dept of Systems Engineering, University of Lagos 4 oafak@unilag.edu.ng 12/27/2012
5. The Gateaux Differential
Let us make the Gateaux differential a little more familiar in
real space in two steps: First, we move to the real space and
allow โ โ ๐๐ฅ and we obtain,
๐น ๐ฅ + ๐ผ๐๐ฅ โ ๐น ๐ฅ ๐
๐ท๐น(๐ฅ, ๐๐ฅ) = lim = ๐น ๐ฅ + ๐ผ๐๐ฅ
๐ผโ0 ๐ผ ๐๐ผ ๐ผ=0
And let ๐ผ๐๐ฅ โ ฮ๐ฅ, the middle term becomes,
๐น ๐ฅ + ฮ๐ฅ โ ๐น ๐ฅ ๐๐น
lim ๐๐ฅ = ๐๐ฅ
ฮ๐ฅโ0 ฮ๐ฅ ๐๐ฅ
from which it is obvious that the Gateaux derivative is a
generalization of the well-known differential from
elementary calculus. The Gateaux differential helps to
compute a local linear approximation of any function (linear
or nonlinear).
Dept of Systems Engineering, University of Lagos 5 oafak@unilag.edu.ng 12/27/2012
6. The Gateaux Differential
It is easily shown that the Gateaux differential is linear
in its second argument, ie, for ๐ผ โ R
๐ท๐ญ ๐, ๐ผ๐ = ๐ผ๐ท๐ญ ๐, ๐
Furthermore,
๐ท๐ญ ๐, ๐ + ๐ = ๐ท๐ญ ๐, ๐ + ๐ท๐ญ ๐, ๐
and that for ๐ผ, ๐ฝ โ R
๐ท๐ญ ๐, ๐ผ๐ + ๐ฝ๐ = ๐ผ๐ท๐ญ ๐, ๐ + ๐ฝ๐ท๐ญ ๐, ๐
Dept of Systems Engineering, University of Lagos 6 oafak@unilag.edu.ng 12/27/2012
7. The Frechรฉt Derivative
The Frechรฉt derivative or gradient of a differentiable
function is the linear operator, grad ๐ญ(๐) such that, for
๐๐ โ V,
๐๐ญ ๐
๐๐ โก grad ๐ญ ๐ ๐๐ โก ๐ท๐ญ ๐, ๐๐
๐๐
Obviously, grad ๐ญ(๐) is a tensor because it is a linear
transformation from one Euclidean vector space to
another. Its linearity derives from the second argument
of the RHS.
Dept of Systems Engineering, University of Lagos 7 oafak@unilag.edu.ng 12/27/2012
8. Differentiable Real-Valued Function
The Gradient of a Differentiable Real-Valued Function
For a real vector function, we have the map:
๐: V โ R
The Gateaux differential in this case takes two vector
arguments and maps to a real value:
๐ท๐: V ร V โ R
This is the classical definition of the inner product so that we
can define the differential as,
๐๐(๐)
โ ๐๐ โก ๐ท๐ ๐, ๐๐
๐๐
This quantity,
๐๐(๐)
๐๐
defined by the above equation, is clearly a vector. oafak@unilag.edu.ng 12/27/2012
Dept of Systems Engineering, University of Lagos 8
9. Real-Valued Function
It is the gradient of the scalar valued function of the
vector variable. We now proceed to find its components
in general coordinates. To do this, we choose a basis
๐ ๐ โ V. On such a basis, the function,
๐ = ๐ฃ๐ ๐ ๐
and,
๐ ๐ = ๐ ๐ฃ๐ ๐ ๐
we may also express the independent vector
๐๐ = ๐๐ฃ ๐ ๐ ๐
on this basis.
Dept of Systems Engineering, University of Lagos 9 oafak@unilag.edu.ng 12/27/2012
10. Real-Valued Function
The Gateaux differential in the direction of the basis vector ๐ ๐ is,
๐ ๐ + ๐ผ๐ ๐ โ ๐ ๐ ๐ ๐ฃ ๐ ๐ ๐ + ๐ผ๐ ๐ โ ๐ ๐ฃ ๐ ๐ ๐
๐ท๐ ๐, ๐ ๐ = lim = lim
๐ผโ0 ๐ผ ๐ผโ0 ๐ผ
๐
๐ ๐ฃ ๐ + ๐ผ๐ฟ ๐ ๐ ๐ โ ๐ ๐ฃ ๐ ๐ ๐
= lim
๐ผโ0 ๐ผ
As the function ๐ does not depend on the vector basis, we can
substitute the vector function by the real function of the components
๐ฃ 1 , ๐ฃ 2 , ๐ฃ 3 so that,
๐ ๐ฃ ๐ ๐ ๐ = ๐ ๐ฃ1, ๐ฃ 2, ๐ฃ 3
in which case, the above differential, similar to the real case discussed
earlier becomes,
๐๐ ๐ฃ 1 , ๐ฃ 2 , ๐ฃ 3
๐ท๐ ๐, ๐ ๐ =
๐๐ฃ ๐
Dept of Systems Engineering, University of Lagos 10 oafak@unilag.edu.ng 12/27/2012
11. ๐๐(๐)
Of course, it is now obvious that the gradient of
๐๐
the scalar valued vector function, expressed in its dual
basis must be,
๐๐(๐) ๐๐ ๐ฃ 1 , ๐ฃ 2 , ๐ฃ 3 ๐
= ๐
๐
๐๐ ๐๐ฃ
so that we can recover,
๐๐ ๐ ๐๐ ๐ฃ 1 , ๐ฃ 2 , ๐ฃ 3 ๐
๐ท๐ ๐, ๐ ๐ = โ ๐๐ = ๐
๐ โ ๐ ๐
๐๐ ๐๐ฃ
๐๐ ๐ฃ 1 , ๐ฃ 2 , ๐ฃ 3 ๐ ๐๐ ๐ฃ 1 , ๐ฃ 2 , ๐ฃ 3
= ๐
๐ฟ๐ =
๐๐ฃ ๐๐ฃ ๐
Hence we obtain the well-known result that
๐๐(๐) ๐๐ ๐ฃ 1 , ๐ฃ 2 , ๐ฃ 3 ๐
grad ๐ ๐ = = ๐
๐
๐๐ ๐๐ฃ
which defines the Frechรฉt derivative of a scalar valued
function with respect to its vector argument.
Dept of Systems Engineering, University of Lagos 11 oafak@unilag.edu.ng 12/27/2012
12. Vector valued Function
๏ช The Gradient of a Differentiable Vector valued Function
The definition of the Frechรฉt clearly shows that the gradient
of a vector-valued function is itself a second order tensor. We
now find the components of this tensor in general
coordinates. To do this, we choose a basis ๐ ๐ โ V. On such
a basis, the function,
๐ญ ๐ = ๐น๐ ๐ ๐ ๐
The functional dependency on the basis vectors are ignorable
on account of the fact that the components themselves are
fixed with respect to the basis. We can therefore write,
๐ญ ๐ = ๐น ๐ ๐ฃ1 , ๐ฃ 2 , ๐ฃ 3 ๐ ๐
Dept of Systems Engineering, University of Lagos 12 oafak@unilag.edu.ng 12/27/2012
13. Vector Derivatives
๐ท๐ญ ๐, ๐๐ = ๐ท๐ญ ๐ฃ ๐ ๐ ๐ , ๐๐ฃ ๐ ๐ ๐
= ๐ท๐ญ ๐ฃ ๐ ๐ ๐ , ๐ ๐ ๐๐ฃ ๐
= ๐ท๐น ๐ ๐ฃ ๐ ๐ ๐ , ๐ ๐ ๐ ๐ ๐๐ฃ ๐
Again, upon noting that the functions ๐ท๐น ๐ ๐ฃ ๐ ๐ ๐ , ๐ ๐ ,
๐ = 1,2,3 are not functions of the vector basis, they can
be written as functions of the scalar components alone
so that we have, as before,
๐ท๐ญ ๐, ๐๐ = ๐ท๐น ๐ ๐ฃ ๐ ๐ ๐ , ๐ ๐ ๐ ๐ ๐๐ฃ ๐
= ๐ท๐น ๐ ๐, ๐ ๐ ๐ ๐ ๐๐ฃ ๐
Dept of Systems Engineering, University of Lagos 13 oafak@unilag.edu.ng 12/27/2012
14. Which we can compare to the earlier case of scalar valued
function and easily obtain,
๐๐น ๐ ๐ฃ 1 , ๐ฃ 2 , ๐ฃ 3 ๐
๐ท๐ญ ๐, ๐๐ = ๐ โ ๐ ๐ ๐ ๐ ๐๐ฃ ๐
๐๐ฃ ๐
๐๐น ๐ ๐ฃ 1 , ๐ฃ 2 , ๐ฃ 3
= ๐ ๐ โ ๐ ๐ ๐ ๐ ๐๐ฃ ๐
๐๐ฃ ๐
๐๐น ๐ ๐ฃ 1 , ๐ฃ 2 , ๐ฃ 3
= ๐ ๐ โ ๐ ๐ ๐๐
๐๐ฃ ๐
๐๐ญ ๐
= ๐๐ โก grad ๐ญ ๐ ๐๐
๐๐
Clearly, the tensor gradient of the vector-valued vector function
is,
๐๐ญ ๐ ๐๐น ๐ ๐ฃ 1 , ๐ฃ 2 , ๐ฃ 3
= grad ๐ญ ๐ = ๐ ๐ โ ๐ ๐
๐๐ ๐๐ฃ ๐
Where due attention should be paid to the covariance of the
quotient indices in contrast to the contravariance of the non
quotient indices.
Dept of Systems Engineering, University of Lagos 14 oafak@unilag.edu.ng 12/27/2012
15. The Trace & Divergence
The trace of this gradient is called the divergence of the
function:
๐๐น ๐ ๐ฃ 1 , ๐ฃ 2 , ๐ฃ 3
div ๐ญ ๐ = tr ๐ ๐ โ ๐ ๐
๐๐ฃ ๐
๐๐น ๐ ๐ฃ 1 , ๐ฃ 2 , ๐ฃ 3
= ๐ ๐ โ ๐ ๐
๐๐ฃ ๐
๐๐น ๐ ๐ฃ 1 , ๐ฃ 2 , ๐ฃ 3 ๐
= ๐
๐ฟ๐
๐๐ฃ
๐๐น ๐ ๐ฃ 1 , ๐ฃ 2 , ๐ฃ 3
=
๐๐ฃ ๐
Dept of Systems Engineering, University of Lagos 15 oafak@unilag.edu.ng 12/27/2012
16. Tensor-Valued Vector Function
Consider the mapping,
๐ป: V โ T
Which maps from an inner product space to a tensor space. The
Gateaux differential in this case now takes two vectors and
produces a tensor:
๐ท๐ป: V ร V โ T
With the usual notation, we may write,
๐ท๐ป ๐, ๐๐ = ๐ท๐ป ๐ฃ ๐ ๐ ๐ , ๐๐ฃ ๐ ๐ ๐ = ๐ท๐ป ๐ฃ ๐ ๐ ๐ , ๐๐ฃ ๐ ๐ ๐
๐๐ป ๐
= ๐๐ โก grad ๐ป ๐ ๐๐
๐๐
= ๐ท๐ ๐ผ๐ฝ ๐ฃ ๐ ๐ ๐ , ๐๐ฃ ๐ ๐ ๐ ๐ ๐ผ โ ๐ ๐ฝ
= ๐ท๐ ๐ผ๐ฝ ๐, ๐ ๐ ๐ ๐ผ โ ๐ ๐ฝ ๐๐ฃ ๐
Each of these nine functions look like the real differential of several
variables we considered earlier.
Dept of Systems Engineering, University of Lagos 16 oafak@unilag.edu.ng 12/27/2012
17. The independence of the basis vectors imply, as usual, that
๐๐ ๐ผ๐ฝ ๐ฃ 1 , ๐ฃ 2 , ๐ฃ 3
๐ท๐ ๐ผ๐ฝ ๐, ๐ ๐ =
๐๐ฃ ๐
so that,
๐ท๐ป ๐, ๐๐ = ๐ท๐ ๐ผ๐ฝ ๐, ๐ ๐ ๐ ๐ผ โ ๐ ๐ฝ ๐๐ฃ ๐
๐๐ ๐ผ๐ฝ ๐ฃ 1 , ๐ฃ 2 , ๐ฃ 3
= ๐ ๐ผ โ ๐ ๐ฝ ๐๐ฃ ๐
๐๐ฃ ๐
๐๐ ๐ผ๐ฝ ๐ฃ 1 , ๐ฃ 2 , ๐ฃ 3
= ๐ ๐ผ โ ๐ ๐ฝ ๐ ๐ โ ๐๐
๐๐ฃ ๐
๐๐ ๐ผ๐ฝ ๐ฃ 1 , ๐ฃ 2 , ๐ฃ 3 ๐๐ป ๐
= ๐ ๐ผ โ ๐ ๐ฝ โ ๐ ๐ ๐๐ = ๐๐
๐๐ฃ ๐ ๐๐
โก grad ๐ป ๐ ๐๐
Which defines the third-order tensor,
๐๐ป ๐ ๐๐ ๐ผ๐ฝ ๐ฃ 1 , ๐ฃ 2 , ๐ฃ 3
= grad ๐ป ๐ = ๐ ๐ผ โ ๐ ๐ฝ โ ๐ ๐
๐๐ ๐๐ฃ ๐
and with no further ado, we can see that a third-order tensor
transforms a vector into a second order tensor.
Dept of Systems Engineering, University of Lagos 17 oafak@unilag.edu.ng 12/27/2012
18. The Divergence of a Tensor Function
The divergence operation can be defined in several ways.
Most common is achieved by the contraction of the last two
basis so that,
๐๐ ๐ผ๐ฝ ๐ฃ 1 , ๐ฃ 2 , ๐ฃ 3
div ๐ป ๐ = ๐ ๐ผ โ ๐ ๐ฝ โ ๐ ๐
๐๐ฃ ๐
๐๐ ๐ผ๐ฝ ๐ฃ 1 , ๐ฃ 2 , ๐ฃ 3
= ๐ ๐ผ ๐ ๐ฝ โ ๐ ๐
๐๐ฃ ๐
๐๐ ๐ผ๐ฝ ๐ฃ 1 , ๐ฃ 2 , ๐ฃ 3 ๐ ๐๐ ๐ผ๐ ๐ฃ 1 , ๐ฃ 2 , ๐ฃ 3
= ๐
๐ ๐ผ ๐ฟ๐ฝ = ๐
๐ ๐ผ
๐๐ฃ ๐๐ฃ
It can easily be shown that this is the particular vector that
gives, for all constant vectors ๐,
div ๐ป ๐ โก div ๐ปT ๐
Dept of Systems Engineering, University of Lagos 18 oafak@unilag.edu.ng 12/27/2012
19. Real-Valued Tensor Functions
Two other kinds of functions are critically important in
our study. These are real valued functions of tensors
and tensor-valued functions of tensors. Examples in the
first case are the invariants of the tensor function that
we have already seen. We can express stress in terms of
strains and vice versa. These are tensor-valued functions
of tensors. The derivatives of such real and tensor
functions arise in our analysis of continua. In this section
these are shown to result from the appropriate Gateaux
differentials. The gradients or Frechรฉt derivatives will be
extracted once we can obtain the Gateaux differentials.
Dept of Systems Engineering, University of Lagos 19 oafak@unilag.edu.ng 12/27/2012
20. Frechรฉt Derivative
Consider the Map:
๐: T โ R
The Gateaux differential in this case takes two vector
arguments and maps to a real value:
๐ท๐: T ร T โ R
Which is, as usual, linear in the second argument. The Frechรฉt
derivative can be expressed as the first component of the
following scalar product:
๐๐(๐ป)
๐ท๐ ๐ป, ๐๐ป = : ๐๐ป
๐๐ป
which, we recall, is the trace of the contraction of one tensor
with the transpose of the other second-order tensors. This is
a scalar quantity.
Dept of Systems Engineering, University of Lagos 20 oafak@unilag.edu.ng 12/27/2012
21. Guided by the previous result of the gradient of the scalar valued
function of a vector, it is not difficult to see that,
๐๐(๐ป) ๐๐(๐11 , ๐12 , โฆ , ๐ 33 ) ๐ผ
= ๐ โ ๐ ๐ฝ
๐๐ป ๐๐ ๐ผ๐ฝ
In the dual to the same basis, we can write,
๐๐ป = ๐๐ ๐๐ ๐ ๐ โ ๐ ๐
Clearly,
๐๐(๐ป) ๐๐(๐11 , ๐12 , โฆ , ๐ 33 ) ๐ผ
: ๐๐ป = ๐ โ ๐ ๐ฝ : ๐๐ ๐๐ ๐ ๐ โ ๐ ๐
๐๐ป ๐๐ ๐ผ๐ฝ
๐๐(๐11 , ๐12 , โฆ , ๐ 33 )
= ๐๐ ๐๐
๐๐ ๐๐
Again, note the covariance of the quotient indices.
Dept of Systems Engineering, University of Lagos 21 oafak@unilag.edu.ng 12/27/2012
22. Examples
Let ๐ be a symmetric and positive definite tensor and let
๐ผ1 ๐บ , ๐ผ2 ๐บ and ๐ผ3 ๐บ be the three principal invariants of
๐๐ฐ1 ๐
๐ show that (a) = ๐ the identity tensor, (b)
๐๐
๐๐ฐ2 ๐ ๐๐ผ3 ๐
= ๐ผ1 ๐ ๐ โ ๐ and (c) = ๐ผ3 ๐ ๐ โ1
๐๐ ๐๐
๐๐ฐ1 ๐
can be written in the invariant component form
๐๐
as,
๐๐ผ1 ๐ ๐๐ผ1 ๐
= ๐
๐ ๐ โ ๐ ๐
๐๐ ๐๐ ๐
Dept of Systems Engineering, University of Lagos 22 oafak@unilag.edu.ng 12/27/2012
23. (a) Continued
ฮฑ
Recall that ๐ผ1 ๐ = tr ๐ = ๐ฮฑ hence
๐๐ผ1 ๐ ๐๐ผ1 ๐
= ๐
๐ ๐ โ ๐ ๐
๐๐ ๐๐ ๐
ฮฑ
๐๐ฮฑ
= ๐
๐ ๐ โ ๐ ๐
๐๐ ๐
= ๐ฟ ๐๐ผ ๐ฟ ๐ ๐ผ ๐ ๐ โ ๐ ๐
= ๐ฟ ๐๐ ๐ ๐ โ ๐ ๐ = ๐
which is the identity tensor as expected.
Dept of Systems Engineering, University of Lagos 23 oafak@unilag.edu.ng 12/27/2012
26. Real Tensor Functions
Consider the function ๐(๐บ) = tr ๐บ ๐ where ๐ โ โ, and
๐บ is a tensor.
๐ ๐บ = tr ๐บ ๐ = ๐บ ๐ : ๐
To be specific, let ๐ = 3.
Dept of Systems Engineering, University of Lagos 26 oafak@unilag.edu.ng 12/27/2012
28. Real Tensor Functions
With the last equality coming from the definition of the
Inner product while noting that a circular permutation
does not alter the value of the trace. It is easy to
establish inductively that in the most general case, for
๐ > 0, we have,
๐โ1 T
๐ท๐ ๐บ, ๐๐บ = ๐ ๐บ : ๐๐บ
Clearly,
๐ T
tr ๐บ ๐ = ๐ ๐บ ๐โ1
๐๐บ
Dept of Systems Engineering, University of Lagos 28 oafak@unilag.edu.ng 12/27/2012
29. Real Tensor Functions
When ๐ = 1,
๐
๐ท๐ ๐บ, ๐๐บ = ๐ ๐บ + ๐ผ๐๐บ
๐๐ผ ๐ผ=0
๐
= tr ๐บ + ๐ผ๐๐บ
๐๐ผ ๐ผ=0
= tr ๐ ๐๐บ = ๐: ๐๐บ
Or that,
๐
tr ๐บ = ๐.
๐๐บ
Dept of Systems Engineering, University of Lagos 29 oafak@unilag.edu.ng 12/27/2012
30. Real Tensor Functions
Derivatives of the other two invariants of the tensor ๐บ
can be found as follows:
๐
๐ผ2 (๐บ)
๐๐บ
1 ๐
= tr 2 ๐บ โ tr ๐บ2
2 ๐๐บ
1
= 2tr ๐บ ๐ โ 2 ๐บT = tr ๐บ ๐ โ ๐บT
2
.
Dept of Systems Engineering, University of Lagos 30 oafak@unilag.edu.ng 12/27/2012
31. Real Tensor Functions
To Determine the derivative of the third invariant, we
begin with the trace of the Cayley-Hamilton for ๐บ:
tr ๐บ3 โ ๐ผ1 ๐บ2 + ๐ผ2 ๐บ โ ๐ผ3 ๐ฐ = tr(๐บ3 ) โ ๐ผ1 tr ๐บ2 + ๐ผ2 tr ๐บ
โ 3๐ผ3 = 0
Therefore,
3๐ผ3 = tr(๐บ3 ) โ ๐ผ1 tr ๐บ2 + ๐ผ2 tr ๐บ
1 2
๐ผ2 ๐บ = tr ๐บ โ tr ๐บ2
2
.
Dept of Systems Engineering, University of Lagos 31 oafak@unilag.edu.ng 12/27/2012
32. Differentiating the Invariants
We can therefore write,
3๐ผ3 ๐บ = tr(๐บ3 ) โ tr ๐บ tr ๐บ2
1 2
+ tr ๐บ โ tr ๐บ2 tr ๐บ
2
so that, in terms of traces only,
1 3
๐ผ3 ๐บ = tr ๐บ โ 3tr ๐บ tr ๐บ2 + 2tr(๐บ3 )
6
Dept of Systems Engineering, University of Lagos 32 oafak@unilag.edu.ng 12/27/2012
33. Real Tensor Functions
Clearly,
๐๐ผ3 (๐บ) 1
= 3tr 2 ๐บ ๐ โ 3tr ๐บ2 โ 3tr ๐บ 2๐บ ๐ + 2 ร 3 ๐บ2 ๐
๐๐บ 6
= ๐ผ2 ๐ โ tr ๐บ ๐บ ๐ + ๐บ2๐
Dept of Systems Engineering, University of Lagos 33 oafak@unilag.edu.ng 12/27/2012
34. Real Tensor Functions
.
Dept of Systems Engineering, University of Lagos 34 oafak@unilag.edu.ng 12/27/2012
35. The Euclidean Point Space
In Classical Theory, the world is a Euclidean Point Space
of dimension three. We shall define this concept now
and consequently give specific meanings to related
concepts such as
๏ง Frames of Reference,
๏ง Coordinate Systems and
๏ง Global Charts
Dept of Systems Engineering, University of Lagos 35 oafak@unilag.edu.ng 12/27/2012
36. The Euclidean Point Space
๏ง Scalars, Vectors and Tensors we will deal with in
general vary from point to point in the material. They
are therefore to be regarded as functions of position
in the physical space occupied.
๏ง Such functions, associated with positions in the
Euclidean point space, are called fields.
๏ง We will therefore be dealing with scalar, vector and
tensor fields.
Dept of Systems Engineering, University of Lagos 36 oafak@unilag.edu.ng 12/27/2012
37. The Euclidean Point Space
๏ช The Euclidean Point Space โฐ is a set of elements
called points. For each pair of points ๐, ๐ โ โฐ,
โ ๐ ๐, ๐ โ E with the following two properties:
1. ๐ ๐, ๐ = ๐ ๐, ๐ + ๐ ๐, ๐ โ๐, ๐, ๐ โ โฐ
2. ๐ ๐, ๐ = ๐ ๐, ๐ โ ๐ = ๐
Based on these two, we proceed to show that,
๐ ๐, ๐ = ๐
And that,
๐ ๐, ๐ = โ๐ ๐, ๐
Dept of Systems Engineering, University of Lagos 37 oafak@unilag.edu.ng 12/27/2012
38. The Euclidean Point Space
From property 1, let ๐ โ ๐ it is clear that,
๐ ๐, ๐ = ๐ ๐, ๐ + ๐ ๐, ๐
And it we further allow ๐ โ ๐, we find that,
๐ ๐, ๐ = ๐ ๐, ๐ + ๐ ๐, ๐ = 2๐ ๐, ๐
Which clearly shows that ๐ ๐, ๐ = ๐ the zero vector.
Similarly, from the above, we find that,
๐ ๐, ๐ = ๐ = ๐ ๐, ๐ + ๐ ๐, ๐
So that ๐ ๐, ๐ = โ๐ ๐, ๐
Dept of Systems Engineering, University of Lagos 38 oafak@unilag.edu.ng 12/27/2012
39. Dept of Systems Engineering, University of Lagos 39 oafak@unilag.edu.ng 12/27/2012
40. Coordinate System
Dept of Systems Engineering, University of Lagos 40 oafak@unilag.edu.ng 12/27/2012
41. Coordinate System
Dept of Systems Engineering, University of Lagos 41 oafak@unilag.edu.ng 12/27/2012
42. Position Vectors
Note that โฐ is NOT a vector space. In our discussion, the
vector space E to which ๐ belongs, is associated as we
have shown. It is customary to oscillate between these
two spaces. When we are talking about the vectors,
they are in E while the points are in โฐ.
We adopt the convention that ๐ ๐ โก ๐ ๐, ๐ referring
to the vector ๐. If therefore we choose an arbitrarily
fixed point ๐ โ โฐ, we are associating ๐ ๐ , ๐ ๐ and
๐ ๐ respectively with ๐ ๐, ๐ , ๐ ๐, ๐ and ๐ ๐, ๐ .
These are vectors based on the points ๐, ๐ and ๐ with
reference to the origin chosen. To emphasize the
association with both points of โฐ as well as members of
E they are called Position Vectors.
Dept of Systems Engineering, University of Lagos 42 oafak@unilag.edu.ng 12/27/2012
43. Length in the Point Space
Recall that by property 1,
๐ ๐ โก ๐ ๐, ๐ = ๐ ๐, ๐ + ๐ ๐, ๐
Furthermore, we have deduced that ๐ ๐, ๐ = โ๐(๐, ๐)
We may therefore write that,
๐ ๐ = ๐ ๐ โ ๐(๐)
Which, when there is no ambiguity concerning the
chosen origin, we can write as,
๐ ๐ = ๐โ ๐
And the distance between the two is,
๐ ๐ ๐ = ๐ ๐โ ๐ = ๐โ ๐
Dept of Systems Engineering, University of Lagos 43 oafak@unilag.edu.ng 12/27/2012
44. Metric Properties
๏ช The norm of a vector is the square root of the inner
product of the vector with itself. If the coordinates of
๐ and ๐ on a set of independent vectors are ๐ฅ ๐ and ๐ฆ ๐ ,
then the distance we seek is,
๐ ๐โ ๐ = ๐โ ๐ = ๐ ๐๐ (๐ฅ ๐ โ ๐ฆ ๐ )(๐ฅ ๐ โ ๐ฆ ๐ )
The more familiar Pythagorean form occurring only
when ๐ ๐๐ = 1.
Dept of Systems Engineering, University of Lagos 44 oafak@unilag.edu.ng 12/27/2012
45. Coordinate Transformations
๏ช Consider a Cartesian coordinate system
๐ฅ, ๐ฆ, ๐ง or ๐ฆ1 , ๐ฆ 2 and ๐ฆ 3 with an orthogonal basis. Let us
now have the possibility of transforming to another
coordinate system of an arbitrary nature: ๐ฅ 1 , ๐ฅ 2 , ๐ฅ 3 .
We can represent the transformation and its inverse
in the equations:
๏ช ๐ฆ ๐ = ๐ฆ ๐ ๐ฅ 1 , ๐ฅ 2 , ๐ฅ 3 , ๐ฅ ๐ = ๐ฅ ๐ (๐ฆ1 , ๐ฆ 2 , ๐ฆ 3 )
๏ช And if the Jacobian of transformation,
Dept of Systems Engineering, University of Lagos 45 oafak@unilag.edu.ng 12/27/2012
46. Jacobian Determinant
๐๐ฆ1 ๐๐ฆ1 ๐๐ฆ1
๐๐ฅ 1 ๐๐ฅ 2 ๐๐ฅ 3
๐๐ฆ ๐ ๐ ๐ฆ1 , ๐ฆ 2 , ๐ฆ 3 ๐๐ฆ 2 ๐๐ฆ 2 ๐๐ฆ 2
๐
โก 1, ๐ฅ2, ๐ฅ3 โก
๐๐ฅ ๐ ๐ฅ ๐๐ฅ 1 ๐๐ฅ 2 ๐๐ฅ 3
๐๐ฆ 3 ๐๐ฆ 3 ๐๐ฆ 3
๐๐ฅ 1 ๐๐ฅ 2 ๐๐ฅ 3
does not vanish, then the inverse transformation will exist.
So that,
๐ฅ ๐ = ๐ฅ ๐ (๐ฆ1 , ๐ฆ 2 , ๐ฆ 3 )
Dept of Systems Engineering, University of Lagos 46 oafak@unilag.edu.ng 12/27/2012
47. Given a position vector
๐ซ = ๐ซ(๐ฅ 1 , ๐ฅ 2 , ๐ฅ 3 )
๏ช In the new coordinate system, we can form a set of three
vectors,
๐๐ซ
๐ ๐ = ๐
, ๐ = 1,2,3
๐๐ฅ
Which represent vectors along the tangents to the
coordinate lines. (This is easily established for the Cartesian
system and it is true in all systems. ๐ซ = ๐ซ ๐ฅ 1 , ๐ฅ 2 , ๐ฅ 3 = ๐ฆ1 ๐ข +
๐ฆ2 ๐ฃ + ๐ฆ3 ๐ค
So that
๐๐ซ ๐๐ซ ๐๐ซ
๐ข = 1 , ๐ฃ = 2 and ๐ค = 3
๐๐ฆ ๐๐ฆ ๐๐ฆ
The fact that this is true in the general case is easily seen
when we consider that along those lines, only the variable we
are differentiating with respect to varies. Consider the
general case where,
๐ซ = ๐ซ ๐ฅ1 , ๐ฅ 2 , ๐ฅ 3
The total differential of ๐ซ is simply,
Dept of Systems Engineering, University of Lagos 47 oafak@unilag.edu.ng 12/27/2012
48. Natural Dual Bases
๐๐ซ 1+
๐๐ซ 2+
๐๐ซ
๐๐ซ = 1 ๐๐ฅ ๐๐ฅ ๐๐ฅ 3
๐๐ฅ ๐๐ฅ 2 ๐๐ฅ 3
โก ๐๐ฅ 1 ๐ 1 + ๐๐ฅ 2 ๐ 2 + ๐๐ฅ 3 ๐ 3
With ๐ ๐ ๐ฅ 1 , ๐ฅ 2 , ๐ฅ 3 , ๐ = 1,2,3 now depending in general
on ๐ฅ 1 , ๐ฅ 2 and ๐ฅ 3 now forming a basis on which we can
describe other vectors in the coordinate system. We
have no guarantees that this vectors are unit in length
nor that they are orthogonal to one another. In the
Cartesian case, ๐ ๐ , ๐ = 1,2,3 are constants, normalized
and orthogonal. They are our familiar
๐๐ซ ๐๐ซ ๐๐ซ
๐ข = 1 , ๐ฃ = 2 and ๐ค = 3 .
๐๐ฆ ๐๐ฆ ๐๐ฆ
Dept of Systems Engineering, University of Lagos 48 oafak@unilag.edu.ng 12/27/2012
49. We now proceed to show that the set of basis vectors,
๐ ๐ โก ๐ป๐ฅ ๐
is reciprocal to ๐ ๐ . The total differential ๐๐ซ can be written in terms
of partial differentials as,
๐๐ซ
๐๐ซ = ๐๐ฆ ๐
๐๐ฆ ๐
Which when expressed in component form yields,
๐๐ฅ 1 1 ๐๐ฅ 1 2 ๐๐ฅ 1 3
๐๐ฅ 1 = 1 ๐๐ฆ + 2 ๐๐ฆ + 3 ๐๐ฆ
๐๐ฆ ๐๐ฆ ๐๐ฆ
๐๐ฅ 2 1 ๐๐ฅ 2 2 ๐๐ฅ 2 3
๐๐ฅ 2 = 1 ๐๐ฆ + 2 ๐๐ฆ + 3 ๐๐ฆ
๐๐ฆ ๐๐ฆ ๐๐ฆ
๐๐ฅ 3 1 ๐๐ฅ 3 2 ๐๐ฅ 3 3
๐๐ฅ 3 = 1 ๐๐ฆ + 2 ๐๐ฆ + 3 ๐๐ฆ
๐๐ฆ ๐๐ฆ ๐๐ฆ
Or, more compactly that,
๐ =
๐๐ฅ ๐
๐๐ฅ ๐๐ฆ ๐ = ๐ป๐ฅ ๐ โ ๐๐ซ
๐๐ฆ ๐
Dept of Systems Engineering, University of Lagos 49 oafak@unilag.edu.ng 12/27/2012
50. In Cartesian coordinates, ๐ฆ ๐ , ๐ = 1, . . , 3 for any scalar field
๐ (๐ฆ1 , ๐ฆ 2 , ๐ฆ 3 ),
๐๐ ๐ =
๐๐ ๐๐ ๐๐
๐๐ฆ ๐๐ฅ + ๐๐ฆ + ๐๐ง = grad ๐ โ ๐๐ซ
๐๐ฆ ๐ ๐๐ฅ ๐๐ฆ ๐๐ง
We are treating each curvilinear coordinate as a scalar field,
๐ฅ ๐ = ๐ฅ ๐ (๐ฆ1 , ๐ฆ 2 , ๐ฆ 3 ).
๐ = ๐ป๐ฅ ๐ โ
๐๐ซ
๐๐ฅ ๐๐ฅ ๐
๐๐ฅ ๐
= ๐ ๐ โ ๐ ๐ ๐๐ฅ ๐
๐
= ๐ฟ ๐ ๐๐ฅ ๐ .
The last equality arises from the fact that this is the only way
one component of the coordinate differential can equal
another.
๐
โ ๐ ๐ โ ๐ ๐ = ๐ฟ ๐
Which recovers for us the reciprocity relationship and shows
that ๐ ๐ and ๐ ๐ are reciprocal systems.
Dept of Systems Engineering, University of Lagos 50 oafak@unilag.edu.ng 12/27/2012
54. For any two tensor fields ๐ฎ and ๐ฏ, Show that,
๐ฎ ร : grad ๐ฏ = ๐ฎ โ curl ๐ฏ
๐ฎ ร : grad ๐ฏ = ๐ ๐๐๐ ๐ข ๐ ๐ ๐ โ ๐ ๐ : ๐ฃ ๐ผ , ๐ฝ ๐ ๐ผ โ ๐ ๐ฝ
= ๐ ๐๐๐ ๐ข ๐ ๐ฃ ๐ผ , ๐ฝ ๐ ๐ โ ๐ ๐ผ ๐ ๐ โ ๐ ๐ฝ
๐ฝ
= ๐ ๐๐๐ ๐ข ๐ ๐ฃ ๐ผ , ๐ฝ ๐ฟ ๐ ๐ผ ๐ฟ ๐ = ๐ ๐๐๐ ๐ข ๐ ๐ฃ ๐ , ๐
= ๐ฎ โ curl ๐ฏ
Dept of Systems Engineering, University of Lagos 54 oafak@unilag.edu.ng 12/27/2012
55. For a vector field ๐, show that ๐ ๐ซ๐๐ ๐ ร is a third
ranked tensor. Hence or otherwise show that
๐๐ข๐ฏ ๐ ร = โ๐๐ฎ๐ซ๐ฅ ๐.
The secondโorder tensor ๐ ร is defined as ๐ ๐๐๐ ๐ข ๐ ๐ ๐ โ
๐ ๐ . Taking the covariant derivative with an independent
base, we have
grad ๐ ร = ๐ ๐๐๐ ๐ข ๐ , ๐ ๐ ๐ โ ๐ ๐ โ ๐ ๐
This gives a third order tensor as we have seen.
Contracting on the last two bases,
div ๐ ร = ๐ ๐๐๐ ๐ข ๐ , ๐ ๐ ๐ โ ๐ ๐ โ ๐ ๐
= ๐ ๐๐๐ ๐ข ๐ , ๐ ๐ ๐ ๐ฟ ๐๐
= ๐ ๐๐๐ ๐ข ๐ , ๐ ๐ ๐
= โcurl ๐
Dept of Systems Engineering, University of Lagos 55 oafak@unilag.edu.ng 12/27/2012
56. Show that ๐๐ฎ๐ซ๐ฅ ๐๐ = grad ๐ ร
Note that ๐๐ = ๐๐ ๐ผ๐ฝ ๐ ๐ผ โ ๐ ๐ฝ , and that curl ๐ป =
๐ ๐๐๐ ๐ ๐ผ๐ , ๐ ๐ ๐ โ ๐ ๐ผ so that,
curl ๐๐ = ๐ ๐๐๐ ๐๐ ๐ผ๐ , ๐ ๐ ๐ โ ๐ ๐ผ
= ๐ ๐๐๐ ๐, ๐ ๐ ๐ผ๐ ๐ ๐ โ ๐ ๐ผ = ๐ ๐๐๐ ๐, ๐ ๐ ๐ โ ๐ ๐
= grad ๐ ร
Show that curl ๐ ร = div ๐ ๐ โ grad ๐
๐ ร = ๐ ๐ผ๐ฝ๐ ๐ฃ ๐ฝ ๐ ๐ผ โ ๐ ๐
curl ๐ป = ๐ ๐๐๐ ๐ ๐ผ๐ , ๐ ๐ ๐ โ ๐ ๐ผ
so that
curl ๐ ร = ๐ ๐๐๐ ๐ ๐ผ๐ฝ๐ ๐ฃ ๐ฝ , ๐ ๐ ๐ โ ๐ ๐ผ
= ๐ ๐๐ผ ๐ ๐๐ฝ โ ๐ ๐๐ฝ ๐ ๐๐ผ ๐ฃ ๐ฝ , ๐ ๐ ๐ โ ๐ ๐ผ
= ๐ฃ ๐, ๐ ๐ ๐ผ โ ๐ ๐ผ โ ๐ฃ ๐, ๐ ๐ ๐ โ ๐ ๐
= div ๐ ๐ โ grad ๐
Dept of Systems Engineering, University of Lagos 56 oafak@unilag.edu.ng 12/27/2012
57. Show that ๐๐ข๐ฏ ๐ ร ๐ = ๐ โ ๐๐ฎ๐ซ๐ฅ ๐ โ ๐ โ ๐๐ฎ๐ซ๐ฅ ๐
div ๐ ร ๐ = ๐ ๐๐๐ ๐ข ๐ ๐ฃ ๐ , ๐
Noting that the tensor ๐ ๐๐๐ behaves as a constant under
a covariant differentiation, we can write,
div ๐ ร ๐ = ๐ ๐๐๐ ๐ข ๐ ๐ฃ ๐ , ๐
= ๐ ๐๐๐ ๐ข ๐ , ๐ ๐ฃ ๐ + ๐ ๐๐๐ ๐ข ๐ ๐ฃ ๐ , ๐
= ๐ โ curl ๐ โ ๐ โ curl ๐
Given a scalar point function ฯ and a vector field ๐ฏ,
show that ๐๐ฎ๐ซ๐ฅ ๐๐ฏ = ๐ curl ๐ฏ + grad๐ ร ๐ฏ.
curl ๐๐ฏ = ๐ ๐๐๐ ๐๐ฃ ๐ , ๐ ๐ ๐
= ๐ ๐๐๐ ๐, ๐ ๐ฃ ๐ + ๐๐ฃ ๐ , ๐ ๐ ๐
= ๐ ๐๐๐ ๐, ๐ ๐ฃ ๐ ๐ ๐ + ๐ ๐๐๐ ๐๐ฃ ๐ , ๐ ๐ ๐
= grad๐ ร ๐ฏ + ๐ curl ๐ฏ
Dept of Systems Engineering, University of Lagos 57 oafak@unilag.edu.ng 12/27/2012
58. ๐
Given that ๐ ๐ก = ๐(๐ก) , Show that ๐ ๐ก = : ๐
๐ ๐ก
๐2 โก ๐: ๐
Now,
๐ 2 = 2๐
๐๐ ๐๐ ๐๐ ๐๐
๐ = : ๐ + ๐: = 2๐:
๐๐ก ๐๐ก ๐๐ก ๐๐ก ๐๐ก
as inner product is commutative. We can therefore
write that
๐๐ ๐ ๐๐ ๐
= : = : ๐
๐๐ก ๐ ๐๐ก ๐ ๐ก
as required.
Dept of Systems Engineering, University of Lagos 58 oafak@unilag.edu.ng 12/27/2012
59. Given a tensor field ๐ป, obtain the vector ๐ โก ๐ป ๐ ๐ฏ and
show that its divergence is ๐ป: ๐๐ฏ + ๐ฏ โ ๐๐ข๐ฏ ๐ป
The divergence of ๐ is the scalar sum , ๐๐๐ ๐ฃ ๐ , ๐ .
Expanding the product covariant derivative we obtain,
div ๐ปT ๐ฏ = ๐๐๐ ๐ฃ ๐ , ๐ = ๐๐๐ , ๐ ๐ฃ ๐ + ๐๐๐ ๐ฃ ๐ , ๐
= div ๐ป โ ๐ฏ + tr ๐ปT grad๐ฏ
= div ๐ป โ ๐ฏ + ๐ป: grad ๐ฏ
Recall that scalar product of two vectors is
commutative so that
div ๐ปT ๐ฏ = ๐ป: grad๐ฏ + ๐ฏ โ div ๐ป
Dept of Systems Engineering, University of Lagos 59 oafak@unilag.edu.ng 12/27/2012
60. For a second-order tensor ๐ป define
curl ๐ป โก ๐ ๐๐๐ ๐ป ๐ผ๐ , ๐ ๐ ๐ โ ๐ ๐ถ show that for any constant
vector ๐, curl ๐ป ๐ = curl ๐ป ๐ป ๐
Express vector ๐ in the invariant form with
contravariant components as ๐ = ๐ ๐ฝ ๐ ๐ฝ . It follows that
curl ๐ป ๐ = ๐ ๐๐๐ ๐ ๐ผ๐ , ๐ ๐ ๐ โ ๐ ๐ผ ๐
= ๐ ๐๐๐ ๐ ๐ผ๐ , ๐ ๐ ๐ฝ ๐ ๐ โ ๐ ๐ผ ๐ ๐ฝ
= ๐ ๐๐๐ ๐ ๐ผ๐ , ๐ ๐ ๐ฝ ๐ ๐ ๐ฟ ๐ฝ๐ผ
= ๐ ๐๐๐ ๐ ๐ผ๐ , ๐ ๐ ๐ ๐ ๐ผ
= ๐ ๐๐๐ ๐ ๐ผ๐ ๐ ๐ผ , ๐ ๐ ๐
The last equality resulting from the fact that vector ๐ is
a constant vector. Clearly,
curl ๐ป ๐ = curl ๐ป ๐ ๐
Dept of Systems Engineering, University of Lagos 60 oafak@unilag.edu.ng 12/27/2012
61. For any two vectors ๐ฎ and ๐ฏ, show that curl ๐ฎ โ ๐ฏ =
grad ๐ฎ ๐ฏ ร ๐ป + curl ๐ฏ โ ๐ where ๐ฏ ร is the skew tensor
๐ ๐๐๐ ๐ฃ ๐ ๐ ๐ โ ๐ ๐ .
Recall that the curl of a tensor ๐ป is defined by curl ๐ป โก
๐ ๐๐๐ ๐ ๐ผ๐ , ๐ ๐ ๐ โ ๐ ๐ผ . Clearly therefore,
curl ๐ โ ๐ = ๐ ๐๐๐ ๐ข ๐ผ ๐ฃ ๐ , ๐ ๐ ๐ โ ๐ ๐ผ
= ๐ ๐๐๐ ๐ข ๐ผ , ๐ ๐ฃ ๐ + ๐ข ๐ผ ๐ฃ ๐ , ๐ ๐ ๐ โ ๐ ๐ผ
= ๐ ๐๐๐ ๐ข ๐ผ , ๐ ๐ฃ ๐ ๐ ๐ โ ๐ ๐ผ + ๐ ๐๐๐ ๐ข ๐ผ ๐ฃ ๐ , ๐ ๐ ๐ โ ๐ ๐ผ
= ๐ ๐๐๐ ๐ฃ ๐ ๐ ๐ โ ๐ข ๐ผ , ๐ ๐ ๐ผ + ๐ ๐๐๐ ๐ฃ ๐ , ๐ ๐ ๐ โ ๐ข ๐ผ ๐ ๐ผ
= ๐ ๐๐๐ ๐ฃ ๐ ๐ ๐ โ ๐ ๐ ๐ข ๐ผ , ๐ฝ ๐ ๐ฝ โ ๐ ๐ผ + ๐ ๐๐๐ ๐ฃ ๐ , ๐ ๐ ๐
โ ๐ข ๐ผ ๐ ๐ผ = โ ๐ ร grad ๐ ๐ป + curl ๐ โ ๐
= grad ๐ ๐ ร ๐ป + curl ๐ โ ๐
upon noting that the vector cross is a skew tensor.
Dept of Systems Engineering, University of Lagos 61 oafak@unilag.edu.ng 12/27/2012
63. Given a scalar point function ๐ and a second-order tensor field ๐ป,
show that ๐๐ฎ๐ซ๐ฅ ๐๐ป = ๐ ๐๐ฎ๐ซ๐ฅ ๐ป + grad๐ ร ๐ป ๐ป where
grad๐ ร is the skew tensor ๐ ๐๐๐ ๐, ๐ ๐ ๐ โ ๐ ๐
curl ๐๐ป โก ๐ ๐๐๐ ๐๐ ๐ผ๐ , ๐ ๐ ๐ โ ๐ ๐ผ
= ๐ ๐๐๐ ๐, ๐ ๐ ๐ผ๐ + ๐๐ ๐ผ๐ , ๐ ๐ ๐ โ ๐ ๐ผ
= ๐ ๐๐๐ ๐, ๐ ๐ ๐ผ๐ ๐ ๐ โ ๐ ๐ผ + ๐๐ ๐๐๐ ๐ ๐ผ๐ , ๐ ๐ ๐ โ ๐ ๐ผ
= ๐ ๐๐๐ ๐, ๐ ๐ ๐ โ ๐ ๐ ๐ ๐ผ๐ฝ ๐ ๐ฝ โ ๐ ๐ผ + ๐๐ ๐๐๐ ๐ ๐ผ๐ , ๐ ๐ ๐ โ ๐ ๐ผ
= ๐ curl ๐ป + grad๐ ร ๐ป ๐
Dept of Systems Engineering, University of Lagos 63 oafak@unilag.edu.ng 12/27/2012
64. For a second-order tensor field ๐ป, show that
div curl ๐ป = curl div ๐ป ๐
Define the second order tensor ๐ as
curl ๐ป โก ๐ ๐๐๐ ๐ ๐ผ๐ , ๐ ๐ ๐ โ ๐ ๐ผ = ๐.๐ผ ๐ ๐ โ ๐ ๐ผ
๐
The gradient of ๐บ is
๐.๐ผ , ๐ฝ ๐ ๐ โ ๐ ๐ผ โ ๐ ๐ฝ = ๐ ๐๐๐ ๐ ๐ผ๐ , ๐๐ฝ ๐ ๐ โ ๐ ๐ผ โ ๐ ๐ฝ
๐
Clearly,
div curl ๐ป = ๐ ๐๐๐ ๐ ๐ผ๐ , ๐๐ฝ ๐ ๐ โ ๐ ๐ผ โ ๐ ๐ฝ
= ๐ ๐๐๐ ๐ ๐ผ๐ , ๐๐ฝ ๐ ๐ ๐ ๐ผ๐ฝ
= ๐ ๐๐๐ ๐ ๐ฝ ๐ , ๐๐ฝ ๐ ๐ = curl div ๐ปT
Dept of Systems Engineering, University of Lagos 64 oafak@unilag.edu.ng 12/27/2012
65. If the volume ๐ฝ is enclosed by the surface ๐บ, the position vector
๐ = ๐ ๐ ๐ ๐ and ๐ is the external unit normal to each surface
๐
element, show that ๐บ ๐ ๐ โ ๐ โ ๐๐ ๐บ equals the volume
๐
contained in ๐ฝ.
๐ โ ๐ = ๐ฅ ๐ ๐ฅ ๐ ๐ ๐ โ ๐ ๐ = ๐ฅ ๐ ๐ฅ ๐ ๐ ๐๐
By the Divergence Theorem,
๐ป ๐ โ ๐ โ ๐๐๐ = ๐ปโ ๐ป ๐โ ๐ ๐๐
๐ ๐
= ๐ ๐ ๐ ๐ ๐ฅ ๐ ๐ฅ ๐ ๐ ๐๐ ๐ ๐ โ ๐ ๐ ๐๐
๐
= ๐ ๐ ๐ ๐๐ ๐ฅ ๐ , ๐ ๐ฅ ๐ + ๐ฅ ๐ ๐ฅ ๐ , ๐ ๐ ๐ โ ๐ ๐ ๐๐
๐
๐
= ๐ ๐๐ ๐ ๐๐ ๐ฟ ๐๐ ๐ฅ ๐ + ๐ฅ ๐ ๐ฟ ๐ , ๐ ๐๐ = 2๐ ๐๐ ๐ ๐๐ ๐ฅ ๐ , ๐ ๐๐
๐ ๐
= 2๐ฟ ๐๐ ๐ฟ ๐๐ ๐๐ = 6 ๐๐
๐ ๐
Dept of Systems Engineering, University of Lagos 65 oafak@unilag.edu.ng 12/27/2012
66. For any Euclidean coordinate system, show that div ๐ฎ ร ๐ฏ =
๐ฏ curl ๐ฎ โ ๐ฎ curl ๐ฏ
Given the contravariant vector ๐ข ๐ and ๐ฃ ๐ with their associated
vectors ๐ข ๐ and ๐ฃ ๐ , the contravariant component of the above cross
product is ๐ ๐๐๐ ๐ข ๐ ๐ฃ ๐ .The required divergence is simply the
contraction of the covariant ๐ฅ ๐ derivative of this quantity:
๐ ๐๐๐ ๐ข ๐ ๐ฃ ๐ = ๐ ๐๐๐ ๐ข ๐,๐ ๐ฃ ๐ + ๐ ๐๐๐ ๐ข ๐ ๐ฃ ๐,๐
,๐
where we have treated the tensor ๐ ๐๐๐ as a constant under the
covariant derivative.
Cyclically rearranging the RHS we obtain,
๐ ๐๐๐ ๐ข ๐ ๐ฃ ๐ = ๐ฃ ๐ ๐ ๐๐๐ ๐ข ๐,๐ + ๐ข ๐ ๐ ๐๐๐ ๐ฃ ๐,๐ = ๐ฃ ๐ ๐ ๐๐๐ ๐ข ๐,๐ + ๐ข ๐ ๐ ๐๐๐ ๐ฃ ๐,๐
,๐
where we have used the anti-symmetric property of the tensor
๐ ๐๐๐ . The last expression shows clearly that
div ๐ฎ ร ๐ฏ = ๐ฏ curl ๐ฎ โ ๐ฎ curl ๐ฏ
as required.
Dept of Systems Engineering, University of Lagos 66 oafak@unilag.edu.ng 12/27/2012
67. Liouville Formula
๐๐ ๐
For a scalar variable ๐ผ, if the tensor ๐ = ๐(๐ผ) and ๐ โก ๐๐ผ, Show that ๐๐ผ det ๐ =
det ๐ tr(๐ ๐ โ๐ )
Proof:
Let ๐ โก ๐ ๐ โ1 so that, ๐ = ๐๐. In component form, we have ๐๐ ๐ = ๐ด ๐ ๐ ๐๐ ๐ .
Therefore,
๐ ๐
det ๐ = ๐ ๐๐๐ ๐๐1 ๐๐2 ๐ 3 = ๐ ๐๐๐ ๐๐1 ๐๐2 ๐ 3 + ๐๐1 ๐๐2 ๐ 3 + ๐๐1 ๐๐2 ๐ 3
๐ ๐ ๐ ๐
๐๐ผ ๐๐ผ
= ๐ ๐๐๐ ๐ด1 ๐๐ ๐ ๐๐2 ๐ 3 + ๐๐1 ๐ด2๐ ๐๐ ๐ ๐ 3 + ๐๐1 ๐๐2 ๐ด3๐ ๐ ๐๐
๐ ๐ ๐
= ๐ ๐๐๐ ๐ด1 ๐๐1 + ๐ด1 ๐๐2 + ๐ด1 ๐๐3
1 2 3 ๐๐2 ๐ 3
๐
+ ๐๐1 ๐ด1 ๐๐1 + ๐ด2 ๐๐2 + ๐ด2 ๐๐3
2
2 3 ๐ 3 + ๐๐1 ๐๐2
๐ ๐ด1 ๐ 1 + ๐ด3 ๐ 2 + ๐ด3 ๐ 3
3
๐ 2 ๐ 3 ๐
๏ช All the boxed terms in the above equation vanish on account of the contraction
of a symmetric tensor with an antisymmetric one.
Dept of Systems Engineering, University of Lagos 67 oafak@unilag.edu.ng 12/27/2012
68. Liouville Theorem Contโd
(For example, the first boxed term yields, ๐ ๐๐๐ ๐ด1 ๐๐2 ๐๐2 ๐ 3
2 ๐
Which is symmetric as well as antisymmetric in ๐ and ๐. It
therefore vanishes. The same is true for all other such terms.)
๐
det ๐ = ๐ ๐๐๐ ๐ด1 ๐๐1 ๐๐2 ๐ 3 + ๐๐1 ๐ด2 ๐๐2 ๐ 3 + ๐๐1 ๐๐2 ๐ด3 ๐ 3
1 ๐ 2 ๐ 3 ๐
๐๐ผ
= ๐ด ๐ ๐ ๐๐๐ ๐๐1 ๐๐2 ๐ 3
๐ ๐
= tr ๐ ๐ โ1 det ๐
as required.
Dept of Systems Engineering, University of Lagos 68 oafak@unilag.edu.ng 12/27/2012
70. When ๐ป is symmetric, show that tr(curl ๐ป) vanishes.
curl ๐ป = ๐ ๐๐๐ ๐ ๐ฝ๐ , ๐ ๐ ๐ โ ๐ ๐ฝ
tr curl ๐ป = ๐ ๐๐๐ ๐ ๐ฝ๐ , ๐ ๐ ๐ โ ๐ ๐ฝ
๐ฝ
= ๐ ๐๐๐ ๐ ๐ฝ๐ , ๐ ๐ฟ ๐ = ๐ ๐๐๐ ๐๐๐ , ๐
which obviously vanishes on account of the symmetry
and antisymmetry in ๐ and ๐. In this case,
curl curl ๐ป
= ๐ป 2 tr ๐ป โ div div ๐ป ๐ฐ โ grad grad tr๐ป
+ 2 grad div ๐ป โ ๐ป 2 ๐ป
T
as grad div ๐ป = grad div ๐ป if the order of
differentiation is immaterial and T is symmetric.
Dept of Systems Engineering, University of Lagos 70 oafak@unilag.edu.ng 12/27/2012
71. For a scalar function ฮฆ and a vector ๐ฃ ๐ show that the
divergence of the vector ๐ฃ ๐ ฮฆ is equal to, ๐ฏ โ gradฮฆ +
ฮฆ div ๐ฏ
๐ฃ ๐ ฮฆ ,๐ = ฮฆ๐ฃ ๐ ,๐ + ๐ฃ ๐ ฮฆ,i
Hence the result.
Dept of Systems Engineering, University of Lagos 71 oafak@unilag.edu.ng 12/27/2012
72. Show that ๐๐๐๐ ๐ฎ ร ๐ฏ = ๐ฏ โ ๐๐ฎ + ๐ฎ โ div ๐ฏ โ ๐ฏ โ div ๐ฎ โ
๐ฎโ ๐ ๐ฏ
Taking the associated (covariant) vector of the expression for
the cross product in the last example, it is straightforward to
see that the LHS in indicial notation is,
๐ ๐๐๐ ๐ ๐๐๐ ๐ข ๐ ๐ฃ ๐
,๐
Expanding in the usual way, noting the relation between the
alternating tensors and the Kronecker deltas,
๐๐๐
๐ ๐๐๐ ๐ ๐๐๐ ๐ข ๐ ๐ฃ ๐ = ๐ฟ ๐๐๐ ๐ข ๐ ,๐ ๐ฃ ๐ โ ๐ข ๐ ๐ฃ ๐ ,๐
,๐
๐๐ ๐ฟ ๐๐ ๐ฟ๐ ๐
= ๐ฟ ๐๐ ๐ข ๐ ,๐ ๐ฃ ๐ โ ๐ข ๐ ๐ฃ ๐ ,๐ = ๐ข ๐ ,๐ ๐ฃ ๐ โ ๐ข ๐ ๐ฃ ๐ ,๐
๐ฟ ๐๐ ๐ฟ ๐๐
= ๐ฟ ๐๐ ๐ฟ ๐๐ โ ๐ฟ ๐๐ ๐ฟ ๐ ๐ ๐ข ๐ ,๐ ๐ฃ ๐ โ ๐ข ๐ ๐ฃ ๐ ,๐
= ๐ฟ ๐๐ ๐ฟ ๐๐ ๐ข ๐ ,๐ ๐ฃ ๐ โ ๐ฟ ๐๐ ๐ฟ ๐๐ ๐ข ๐ ๐ฃ ๐ ,๐ + ๐ฟ ๐๐ ๐ฟ ๐ ๐ ๐ข ๐ ,๐ ๐ฃ ๐
โ ๐ฟ ๐๐ ๐ฟ ๐ ๐ ๐ข ๐ ๐ฃ ๐ ,๐
= ๐ข ๐ ,๐ ๐ฃ ๐ โ ๐ข ๐ ,๐ ๐ฃ ๐ + ๐ข ๐ ๐ฃ ๐ ,๐ โ ๐ข ๐ ๐ฃ ๐ ,๐
Which is the result we seek in indicial notation.
Dept of Systems Engineering, University of Lagos 72 oafak@unilag.edu.ng 12/27/2012
73. Differentiation of Fields
Given a vector point function ๐ ๐ฅ 1 , ๐ฅ 2 , ๐ฅ 3 and its covariant
components, ๐ข ๐ผ ๐ฅ 1 , ๐ฅ 2 , ๐ฅ 3 , ๐ผ = 1,2,3, with ๐ ๐ผ as the reciprocal
basis vectors, Let
๐ ๐๐ข ๐ผ ๐ผ ๐๐ ๐ผ
๐ = ๐ข ๐ผ ๐ ๐ผ , then ๐๐ = ๐ข ๐ผ ๐ ๐ผ ๐๐ฅ ๐ = ๐ + ๐ข ๐ผ ๐๐ฅ ๐
๐๐ฅ ๐ ๐๐ฅ ๐ ๐๐ฅ ๐
Clearly,
๐๐ ๐๐ข ๐ผ ๐ผ ๐๐ ๐ผ
๐
= ๐
๐ + ๐ข
๐ ๐ผ
๐๐ฅ ๐๐ฅ ๐๐ฅ
And the projection of this quantity on the ๐ ๐ direction is,
๐๐ ๐๐ข ๐ผ ๐ผ ๐๐ ๐ผ ๐๐ข ๐ผ ๐ผ ๐๐ ๐ผ
โ ๐ = ๐ + ๐ข โ ๐ ๐ = ๐ โ ๐ ๐ + โ ๐ ๐ข
๐๐ฅ ๐ ๐ ๐๐ฅ ๐ ๐๐ฅ ๐ ๐ผ ๐๐ฅ ๐ ๐๐ฅ ๐ ๐ ๐ผ
๐๐ข ๐ผ ๐ผ ๐๐ ๐ผ
= ๐ฟ +
๐ ๐ ๐
โ ๐ ๐ ๐ข ๐ผ
๐๐ฅ ๐๐ฅ
Dept of Systems Engineering, University of Lagos 73 oafak@unilag.edu.ng 12/27/2012
74. Christoffel Symbols
๐
๏ช Now, ๐ ๐ โ ๐ ๐ = ๐ฟ ๐ so that
๐๐ ๐ ๐โ
๐๐ ๐
๐
โ ๐ ๐ + ๐ ๐
= 0.
๐๐ฅ ๐๐ฅ
๐๐ ๐ ๐
๐๐ ๐ ๐
โ ๐ ๐ = โ๐ โ โกโ .
๐๐ฅ ๐ ๐๐ฅ ๐ ๐๐
This important quantity, necessary to quantify the
derivative of a tensor in general coordinates, is called
the Christoffel Symbol of the second kind.
Dept of Systems Engineering, University of Lagos 74 oafak@unilag.edu.ng 12/27/2012
75. Derivatives in General Coordinates
Using this, we can now write that,
๐๐ ๐๐ข ๐ผ ๐ผ ๐๐ ๐ผ ๐๐ข ๐ ๐ผ
โ ๐ ๐ = ๐ฟ๐ + โ ๐ ๐ ๐ข ๐ผ = โ ๐ข
๐๐ฅ ๐ ๐๐ฅ ๐ ๐๐ฅ ๐ ๐๐ฅ ๐ ๐๐ ๐ผ
๏ช The quantity on the RHS is the component of the derivative of vector ๐
along the ๐ ๐ direction using covariant components. It is the covariant
derivative of ๐. Using contravariant components, we could write,
๐๐ข ๐ ๐๐ ๐ ๐ ๐๐ข ๐ ๐ผ
๐๐ = ๐ ๐ + ๐ข ๐๐ฅ ๐ = ๐ ๐ + ๐ ๐ข ๐ ๐๐ฅ ๐
๐๐ฅ ๐ ๐๐ฅ ๐ ๐๐ฅ ๐ ๐๐ ๐ผ
๏ช So that,
๐๐ ๐๐ข ๐ผ ๐๐ ๐ผ ๐ผ
= ๐ ๐ผ+ ๐ข
๐๐ฅ ๐ ๐๐ฅ ๐ ๐๐ฅ ๐
๏ช The components of this in the direction of ๐ ๐ can be obtained by taking a
dot product as before:
๐๐ ๐ =
๐๐ข ๐ผ ๐๐ ๐ผ ๐ผ ๐ =
๐๐ข ๐ผ ๐ ๐๐ ๐ผ ๐ ๐ข๐ผ =
๐๐ข ๐ ๐
โ ๐ ๐ ๐ผ+ ๐ข โ ๐ ๐ฟ๐ผ + โ ๐ + ๐ข๐ผ
๐๐ฅ ๐ ๐๐ฅ ๐ ๐๐ฅ ๐ ๐๐ฅ ๐ ๐๐ฅ ๐ ๐๐ฅ ๐ ๐ผ๐
Dept of Systems Engineering, University of Lagos 75 oafak@unilag.edu.ng 12/27/2012
76. Covariant Derivatives
The two results above are represented symbolically as,
๐๐ข ๐ ๐ผ ๐ ๐๐ข ๐ผ ๐
๐ข ๐,๐ = โ ๐ข ๐ผ and ๐ข,๐ = + ๐ข๐ผ
๐๐ฅ ๐ ๐๐ ๐๐ฅ ๐ ๐ผ๐
๏ช Which are the components of the covariant derivatives in
terms of the covariant and contravariant components
respectively.
๏ช It now becomes useful to establish the fact that our
definition of the Christoffel symbols here conforms to the
definition you find in the books using the transformation
rules to define the tensor quantities.
Dept of Systems Engineering, University of Lagos 76 oafak@unilag.edu.ng 12/27/2012
78. The First Kind
๏ช Which is the quantity defined as the Christoffel
symbols of the first kind in the textbooks. It is
therefore possible for us to write,
๐๐ ๐ ๐๐ ๐ 1 ๐๐ ๐๐ ๐๐ ๐๐ ๐๐ ๐๐
๐๐, ๐ โก โ ๐ ๐ = โ ๐ ๐ = + โ
๐๐ฅ ๐ ๐๐ฅ ๐ 2 ๐๐ฅ ๐ ๐๐ฅ ๐ ๐๐ฅ ๐
It should be emphasized that the Christoffel symbols,
even though the play a critical role in several tensor
relationships, are themselves NOT tensor quantities.
(Prove this). However, notice their symmetry in the
๐ and ๐. The extension of this definition to the
Christoffel symbols of the second kind is immediate:
Dept of Systems Engineering, University of Lagos 78 oafak@unilag.edu.ng 12/27/2012
79. The Second Kind
๏ช Contract the above equation with the conjugate metric
tensor, we have,
๐๐ผ ๐๐ผ
๐๐ ๐ ๐๐ผ
๐๐ ๐ ๐๐ ๐ ๐
๐ ๐๐, ๐ผ โก ๐ โ ๐ ๐ผ= ๐ โ ๐ ๐ผ= โ ๐ ๐=
๐๐ฅ ๐ ๐๐ฅ ๐ ๐๐ฅ ๐ ๐๐
๐๐ ๐
=โ ๐
โ ๐ ๐
๐๐ฅ
Which connects the common definition of the second
Christoffel symbol with the one defined in the above
derivation. The relationship,
๐ ๐๐ผ ๐๐, ๐ผ = ๐
๐๐
apart from defining the relationship between the Christoffel
symbols of the first kind and that second kind, also
highlights, once more, the index-raising property of the
conjugate metric tensor.
Dept of Systems Engineering, University of Lagos 79 oafak@unilag.edu.ng 12/27/2012
80. Two Christoffel Symbols Related
We contract the above equation with ๐ ๐๐ฝ and obtain,
๐๐ผ ๐๐, ๐ผ = ๐ ๐
๐ ๐๐ฝ ๐ ๐๐ฝ ๐๐
๐ผ ๐
๐ฟ ๐ฝ ๐๐, ๐ผ = ๐๐, ๐ฝ = ๐ ๐๐ฝ
๐๐
so that,
๐ผ
๐ ๐๐ผ ๐๐ = ๐๐, ๐
Showing that the metric tensor can be used to lower
the contravariant index of the Christoffel symbol of the
second kind to obtain the Christoffel symbol of the first
kind.
Dept of Systems Engineering, University of Lagos 80 oafak@unilag.edu.ng 12/27/2012
81. Higher Order Tensors
We are now in a position to express the derivatives of
higher order tensor fields in terms of the Christoffel
symbols.
For a second-order tensor ๐, we can express the
components in dyadic form along the product basis as
follows:
๐ = ๐๐๐ ๐ ๐ โ ๐ ๐ = ๐ ๐๐ ๐ ๐ โจ๐ ๐ = ๐.๐ ๐ ๐ โ ๐ ๐ = ๐๐.๐ ๐ ๐ โจ๐ ๐
๐
This is perfectly analogous to our expanding vectors in
terms of basis and reciprocal bases. Derivatives of the
tensor may therefore be expressible in any of these
product bases. As an example, take the product
covariant bases.
Dept of Systems Engineering, University of Lagos 81 oafak@unilag.edu.ng 12/27/2012