Successfully reported this slideshow.
We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. You can change your ad preferences anytime.

3. tensor calculus jan 2013

2,103 views

Published on

Published in: Education
  • Be the first to comment

3. tensor calculus jan 2013

  1. 1. Tensor CalculusDifferentials & Directional Derivatives
  2. 2. The Gateaux Differential We are presently concerned with Inner Product spaces in our treatment of the Mechanics of Continua. Consider a map, ๐‘ญ: 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. 3. The Gateaux Differential Let ๐’—0 โˆˆ V and ๐’˜0 โˆˆ W, as usual we can say that the limit lim ๐‘ญ ๐’— = ๐’˜0 ๐’—โ†’ ๐’—0 if for any pre-assigned real number ๐œ– > 0, no matter how small, we can always find a real number ๐›ฟ > 0 such that ๐‘ญ ๐’— โˆ’ ๐’˜0 โ‰ค ๐œ– whenever ๐’— โˆ’ ๐’—0 < ๐›ฟ. The function is said to be continuous at ๐’—0 if ๐‘ญ ๐’—0 exists and ๐‘ญ ๐’—0 = ๐’˜0Dept of Systems Engineering, University of Lagos 3 oafak@unilag.edu.ng 12/27/2012
  4. 4. The Gateaux Differential Specifically, for ๐›ผ โˆˆ โ„› let this map be: ๐‘ญ ๐’™ + ๐›ผ๐’‰ โˆ’ ๐‘ญ ๐’™ ๐‘‘ ๐ท๐‘ญ ๐’™, ๐’‰ โ‰ก lim = ๐‘ญ ๐’™ + ๐›ผ๐’‰ ๐›ผโ†’0 ๐›ผ ๐‘‘๐›ผ ๐›ผ=0 We focus attention on the second variable โ„Ž 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. 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„Ž โ†’ ๐‘‘๐‘ฅ and we obtain, ๐น ๐‘ฅ + ๐›ผ๐‘‘๐‘ฅ โˆ’ ๐น ๐‘ฅ ๐‘‘ ๐ท๐น(๐‘ฅ, ๐‘‘๐‘ฅ) = lim = ๐น ๐‘ฅ + ๐›ผ๐‘‘๐‘ฅ ๐›ผโ†’0 ๐›ผ ๐‘‘๐›ผ ๐›ผ=0 And let ๐›ผ๐‘‘๐‘ฅ โ†’ ฮ”๐‘ฅ, the middle term becomes, ๐น ๐‘ฅ + ฮ”๐‘ฅ โˆ’ ๐น ๐‘ฅ ๐‘‘๐น lim ๐‘‘๐‘ฅ = ๐‘‘๐‘ฅ ฮ”๐‘ฅโ†’0 ฮ”๐‘ฅ ๐‘‘๐‘ฅ 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. 6. The Gateaux Differential It is easily shown that the Gateaux differential is linear in its second argument, ie, for ๐›ผ โˆˆ R ๐ท๐‘ญ ๐’™, ๐›ผ๐’‰ = ๐›ผ๐ท๐‘ญ ๐’™, ๐’‰ Furthermore, ๐ท๐‘ญ ๐’™, ๐’ˆ + ๐’‰ = ๐ท๐‘ญ ๐’™, ๐’ˆ + ๐ท๐‘ญ ๐’™, ๐’‰ and that for ๐›ผ, ๐›ฝ โˆˆ R ๐ท๐‘ญ ๐’™, ๐›ผ๐’ˆ + ๐›ฝ๐’‰ = ๐›ผ๐ท๐‘ญ ๐’™, ๐’ˆ + ๐›ฝ๐ท๐‘ญ ๐’™, ๐’‰Dept of Systems Engineering, University of Lagos 6 oafak@unilag.edu.ng 12/27/2012
  7. 7. The Frechรฉt Derivative The Frechรฉt derivative or gradient of a differentiable function is the linear operator, grad ๐‘ญ(๐’—) such that, for ๐‘‘๐’— โˆˆ V, ๐‘‘๐‘ญ ๐’— ๐‘‘๐’— โ‰ก grad ๐‘ญ ๐’— ๐‘‘๐’— โ‰ก ๐ท๐‘ญ ๐’—, ๐‘‘๐’— ๐‘‘๐’— Obviously, grad ๐‘ญ(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. 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: ๐ท๐‘“: V ร— V โ†’ R This is the classical definition of the inner product so that we can define the differential as, ๐‘‘๐‘“(๐’—) โ‹… ๐‘‘๐’— โ‰ก ๐ท๐‘“ ๐’—, ๐‘‘๐’— ๐‘‘๐’— This quantity, ๐‘‘๐‘“(๐’—) ๐‘‘๐’— defined by the above equation, is clearly a vector. oafak@unilag.edu.ng 12/27/2012Dept of Systems Engineering, University of Lagos 8
  9. 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‘– โŠ‚ V. On such a basis, the function, ๐’— = ๐‘ฃ๐‘– ๐ ๐‘– and, ๐‘“ ๐’— = ๐‘“ ๐‘ฃ๐‘– ๐ ๐‘– we may also express the independent vector ๐‘‘๐’— = ๐‘‘๐‘ฃ ๐‘– ๐  ๐‘– on this basis.Dept of Systems Engineering, University of Lagos 9 oafak@unilag.edu.ng 12/27/2012
  10. 10. Real-Valued Function The Gateaux differential in the direction of the basis vector ๐  ๐‘– is, ๐‘“ ๐’— + ๐›ผ๐  ๐‘– โˆ’ ๐‘“ ๐’— ๐‘“ ๐‘ฃ ๐‘— ๐  ๐‘— + ๐›ผ๐  ๐‘– โˆ’ ๐‘“ ๐‘ฃ ๐‘– ๐  ๐‘– ๐ท๐‘“ ๐’—, ๐  ๐‘– = lim = lim ๐›ผโ†’0 ๐›ผ ๐›ผโ†’0 ๐›ผ ๐‘— ๐‘“ ๐‘ฃ ๐‘— + ๐›ผ๐›ฟ ๐‘– ๐  ๐‘— โˆ’ ๐‘“ ๐‘ฃ ๐‘– ๐  ๐‘– = lim ๐›ผโ†’0 ๐›ผ As the function ๐‘“ does not depend on the vector basis, we can substitute the vector function by the real function of the components ๐‘ฃ 1 , ๐‘ฃ 2 , ๐‘ฃ 3 so that, ๐‘“ ๐‘ฃ ๐‘– ๐  ๐‘– = ๐‘“ ๐‘ฃ1, ๐‘ฃ 2, ๐‘ฃ 3 in which case, the above differential, similar to the real case discussed earlier becomes, ๐œ•๐‘“ ๐‘ฃ 1 , ๐‘ฃ 2 , ๐‘ฃ 3 ๐ท๐‘“ ๐’—, ๐  ๐‘– = ๐œ•๐‘ฃ ๐‘–Dept of Systems Engineering, University of Lagos 10 oafak@unilag.edu.ng 12/27/2012
  11. 11. ๐‘‘๐‘“(๐’—) Of course, it is now obvious that the gradient of ๐‘‘๐’— the scalar valued vector function, expressed in its dual basis must be, ๐‘‘๐‘“(๐’—) ๐œ•๐‘“ ๐‘ฃ 1 , ๐‘ฃ 2 , ๐‘ฃ 3 ๐‘– = ๐‘– ๐  ๐‘‘๐’— ๐œ•๐‘ฃ so that we can recover, ๐‘‘๐‘“ ๐’— ๐œ•๐‘“ ๐‘ฃ 1 , ๐‘ฃ 2 , ๐‘ฃ 3 ๐‘— ๐ท๐‘“ ๐’—, ๐  ๐‘– = โ‹… ๐‘‘๐’— = ๐‘— ๐  โ‹… ๐ ๐‘– ๐‘‘๐’— ๐œ•๐‘ฃ ๐œ•๐‘“ ๐‘ฃ 1 , ๐‘ฃ 2 , ๐‘ฃ 3 ๐‘— ๐œ•๐‘“ ๐‘ฃ 1 , ๐‘ฃ 2 , ๐‘ฃ 3 = ๐‘— ๐›ฟ๐‘– = ๐œ•๐‘ฃ ๐œ•๐‘ฃ ๐‘– Hence we obtain the well-known result that ๐‘‘๐‘“(๐’—) ๐œ•๐‘“ ๐‘ฃ 1 , ๐‘ฃ 2 , ๐‘ฃ 3 ๐‘– grad ๐‘“ ๐’— = = ๐‘– ๐  ๐‘‘๐’— ๐œ•๐‘ฃ 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. 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‘– โŠ‚ V. On such a basis, the function, ๐‘ญ ๐’— = ๐น๐‘˜ 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, ๐‘ญ ๐’— = ๐น ๐‘˜ ๐‘ฃ1 , ๐‘ฃ 2 , ๐‘ฃ 3 ๐  ๐‘˜Dept of Systems Engineering, University of Lagos 12 oafak@unilag.edu.ng 12/27/2012
  13. 13. Vector Derivatives ๐ท๐‘ญ ๐’—, ๐‘‘๐’— = ๐ท๐‘ญ ๐‘ฃ ๐‘– ๐  ๐‘– , ๐‘‘๐‘ฃ ๐‘– ๐  ๐‘– = ๐ท๐‘ญ ๐‘ฃ ๐‘– ๐  ๐‘– , ๐  ๐‘– ๐‘‘๐‘ฃ ๐‘– = ๐ท๐น ๐‘˜ ๐‘ฃ ๐‘– ๐  ๐‘– , ๐  ๐‘– ๐  ๐‘˜ ๐‘‘๐‘ฃ ๐‘– Again, upon noting that the functions ๐ท๐น ๐‘˜ ๐‘ฃ 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, ๐ท๐‘ญ ๐’—, ๐‘‘๐’— = ๐ท๐น ๐‘˜ ๐‘ฃ ๐‘– ๐  ๐‘– , ๐  ๐‘– ๐  ๐‘˜ ๐‘‘๐‘ฃ ๐‘– = ๐ท๐น ๐‘˜ ๐’—, ๐  ๐‘– ๐  ๐‘˜ ๐‘‘๐‘ฃ ๐‘–Dept of Systems Engineering, University of Lagos 13 oafak@unilag.edu.ng 12/27/2012
  14. 14. Which we can compare to the earlier case of scalar valued function and easily obtain, ๐œ•๐น ๐‘˜ ๐‘ฃ 1 , ๐‘ฃ 2 , ๐‘ฃ 3 ๐‘— ๐ท๐‘ญ ๐’—, ๐‘‘๐’— = ๐  โ‹… ๐  ๐‘– ๐  ๐‘˜ ๐‘‘๐‘ฃ ๐‘– ๐œ•๐‘ฃ ๐‘— ๐œ•๐น ๐‘˜ ๐‘ฃ 1 , ๐‘ฃ 2 , ๐‘ฃ 3 = ๐  ๐‘˜ โŠ— ๐  ๐‘— ๐  ๐‘– ๐‘‘๐‘ฃ ๐‘– ๐œ•๐‘ฃ ๐‘— ๐œ•๐น ๐‘˜ ๐‘ฃ 1 , ๐‘ฃ 2 , ๐‘ฃ 3 = ๐  ๐‘˜ โŠ— ๐  ๐‘— ๐‘‘๐’— ๐œ•๐‘ฃ ๐‘— ๐‘‘๐‘ญ ๐’— = ๐‘‘๐’— โ‰ก grad ๐‘ญ ๐’— ๐‘‘๐’— ๐‘‘๐’— Clearly, the tensor gradient of the vector-valued vector function is, ๐‘‘๐‘ญ ๐’— ๐œ•๐น ๐‘˜ ๐‘ฃ 1 , ๐‘ฃ 2 , ๐‘ฃ 3 = grad ๐‘ญ ๐’— = ๐  ๐‘˜ โŠ— ๐ ๐‘— ๐‘‘๐’— ๐œ•๐‘ฃ 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. 15. The Trace & Divergence The trace of this gradient is called the divergence of the function: ๐œ•๐น ๐‘˜ ๐‘ฃ 1 , ๐‘ฃ 2 , ๐‘ฃ 3 div ๐‘ญ ๐’— = tr ๐  ๐‘˜ โŠ— ๐ ๐‘— ๐œ•๐‘ฃ ๐‘— ๐œ•๐น ๐‘˜ ๐‘ฃ 1 , ๐‘ฃ 2 , ๐‘ฃ 3 = ๐  ๐‘˜ โ‹… ๐ ๐‘— ๐œ•๐‘ฃ ๐‘— ๐œ•๐น ๐‘˜ ๐‘ฃ 1 , ๐‘ฃ 2 , ๐‘ฃ 3 ๐‘— = ๐‘— ๐›ฟ๐‘˜ ๐œ•๐‘ฃ ๐œ•๐น ๐‘— ๐‘ฃ 1 , ๐‘ฃ 2 , ๐‘ฃ 3 = ๐œ•๐‘ฃ ๐‘—Dept of Systems Engineering, University of Lagos 15 oafak@unilag.edu.ng 12/27/2012
  16. 16. Tensor-Valued Vector Function Consider the mapping, ๐‘ป: 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: ๐ท๐‘ป: V ร— V โ†’ T With the usual notation, we may write, ๐ท๐‘ป ๐’—, ๐‘‘๐’— = ๐ท๐‘ป ๐‘ฃ ๐‘– ๐  ๐‘– , ๐‘‘๐‘ฃ ๐‘– ๐  ๐‘– = ๐ท๐‘ป ๐‘ฃ ๐‘– ๐  ๐‘– , ๐‘‘๐‘ฃ ๐‘– ๐  ๐‘– ๐‘‘๐‘ป ๐’— = ๐‘‘๐’— โ‰ก grad ๐‘ป ๐’— ๐‘‘๐’— ๐‘‘๐’— = ๐ท๐‘‡ ๐›ผ๐›ฝ ๐‘ฃ ๐‘– ๐  ๐‘– , ๐‘‘๐‘ฃ ๐‘– ๐  ๐‘– ๐  ๐›ผ โŠ— ๐  ๐›ฝ = ๐ท๐‘‡ ๐›ผ๐›ฝ ๐’—, ๐  ๐‘– ๐  ๐›ผ โŠ— ๐  ๐›ฝ ๐‘‘๐‘ฃ 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. 17. The independence of the basis vectors imply, as usual, that ๐œ•๐‘‡ ๐›ผ๐›ฝ ๐‘ฃ 1 , ๐‘ฃ 2 , ๐‘ฃ 3 ๐ท๐‘‡ ๐›ผ๐›ฝ ๐’—, ๐  ๐‘– = ๐œ•๐‘ฃ ๐‘– so that, ๐ท๐‘ป ๐’—, ๐‘‘๐’— = ๐ท๐‘‡ ๐›ผ๐›ฝ ๐’—, ๐  ๐‘– ๐  ๐›ผ โŠ— ๐  ๐›ฝ ๐‘‘๐‘ฃ ๐‘– ๐œ•๐‘‡ ๐›ผ๐›ฝ ๐‘ฃ 1 , ๐‘ฃ 2 , ๐‘ฃ 3 = ๐  ๐›ผ โŠ— ๐  ๐›ฝ ๐‘‘๐‘ฃ ๐‘– ๐œ•๐‘ฃ ๐‘– ๐œ•๐‘‡ ๐›ผ๐›ฝ ๐‘ฃ 1 , ๐‘ฃ 2 , ๐‘ฃ 3 = ๐  ๐›ผ โŠ— ๐  ๐›ฝ ๐  ๐‘– โ‹… ๐‘‘๐’— ๐œ•๐‘ฃ ๐‘– ๐œ•๐‘‡ ๐›ผ๐›ฝ ๐‘ฃ 1 , ๐‘ฃ 2 , ๐‘ฃ 3 ๐‘‘๐‘ป ๐’— = ๐  ๐›ผ โŠ— ๐  ๐›ฝ โŠ— ๐  ๐‘– ๐‘‘๐’— = ๐‘‘๐’— ๐œ•๐‘ฃ ๐‘– ๐‘‘๐’— โ‰ก grad ๐‘ป ๐’— ๐‘‘๐’— Which defines the third-order tensor, ๐‘‘๐‘ป ๐’— ๐œ•๐‘‡ ๐›ผ๐›ฝ ๐‘ฃ 1 , ๐‘ฃ 2 , ๐‘ฃ 3 = grad ๐‘ป ๐’— = ๐  ๐›ผ โŠ— ๐  ๐›ฝ โŠ— ๐ ๐‘– ๐‘‘๐’— ๐œ•๐‘ฃ 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. 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, ๐œ•๐‘‡ ๐›ผ๐›ฝ ๐‘ฃ 1 , ๐‘ฃ 2 , ๐‘ฃ 3 div ๐‘ป ๐’— = ๐  ๐›ผ โŠ— ๐  ๐›ฝ โ‹… ๐ ๐‘– ๐œ•๐‘ฃ ๐‘– ๐œ•๐‘‡ ๐›ผ๐›ฝ ๐‘ฃ 1 , ๐‘ฃ 2 , ๐‘ฃ 3 = ๐  ๐›ผ ๐  ๐›ฝ โ‹… ๐ ๐‘– ๐œ•๐‘ฃ ๐‘– ๐œ•๐‘‡ ๐›ผ๐›ฝ ๐‘ฃ 1 , ๐‘ฃ 2 , ๐‘ฃ 3 ๐‘– ๐œ•๐‘‡ ๐›ผ๐‘– ๐‘ฃ 1 , ๐‘ฃ 2 , ๐‘ฃ 3 = ๐‘– ๐  ๐›ผ ๐›ฟ๐›ฝ = ๐‘– ๐ ๐›ผ ๐œ•๐‘ฃ ๐œ•๐‘ฃ It can easily be shown that this is the particular vector that gives, for all constant vectors ๐’‚, div ๐‘ป ๐’‚ โ‰ก div ๐‘ปT ๐’‚Dept of Systems Engineering, University of Lagos 18 oafak@unilag.edu.ng 12/27/2012
  19. 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. 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: ๐ท๐‘“: 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: ๐‘‘๐‘“(๐‘ป) ๐ท๐‘“ ๐‘ป, ๐‘‘๐‘ป = : ๐‘‘๐‘ป ๐‘‘๐‘ป 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. 21. Guided by the previous result of the gradient of the scalar valued function of a vector, it is not difficult to see that, ๐‘‘๐‘“(๐‘ป) ๐œ•๐‘“(๐‘‡11 , ๐‘‡12 , โ€ฆ , ๐‘‡ 33 ) ๐›ผ = ๐  โŠ— ๐ ๐›ฝ ๐‘‘๐‘ป ๐œ•๐‘‡ ๐›ผ๐›ฝ In the dual to the same basis, we can write, ๐‘‘๐‘ป = ๐‘‘๐‘‡ ๐‘–๐‘— ๐  ๐‘– โŠ— ๐  ๐‘— Clearly, ๐‘‘๐‘“(๐‘ป) ๐œ•๐‘“(๐‘‡11 , ๐‘‡12 , โ€ฆ , ๐‘‡ 33 ) ๐›ผ : ๐‘‘๐‘ป = ๐  โŠ— ๐  ๐›ฝ : ๐‘‘๐‘‡ ๐‘–๐‘— ๐  ๐‘– โŠ— ๐  ๐‘— ๐‘‘๐‘ป ๐œ•๐‘‡ ๐›ผ๐›ฝ ๐œ•๐‘“(๐‘‡11 , ๐‘‡12 , โ€ฆ , ๐‘‡ 33 ) = ๐‘‘๐‘‡ ๐‘–๐‘— ๐œ•๐‘‡ ๐‘–๐‘— Again, note the covariance of the quotient indices.Dept of Systems Engineering, University of Lagos 21 oafak@unilag.edu.ng 12/27/2012
  22. 22. Examples Let ๐’ be a symmetric and positive definite tensor and let ๐ผ1 ๐‘บ , ๐ผ2 ๐‘บ and ๐ผ3 ๐‘บ be the three principal invariants of ๐œ•๐‘ฐ1 ๐’ ๐’ show that (a) = ๐Ÿ the identity tensor, (b) ๐œ•๐’ ๐œ•๐‘ฐ2 ๐’ ๐œ•๐ผ3 ๐’ = ๐ผ1 ๐’ ๐Ÿ โˆ’ ๐’ and (c) = ๐ผ3 ๐’ ๐’ โˆ’1 ๐œ•๐’ ๐œ•๐’ ๐œ•๐‘ฐ1 ๐’ can be written in the invariant component form ๐œ•๐’ as, ๐œ•๐ผ1 ๐’ ๐œ•๐ผ1 ๐’ = ๐‘— ๐ ๐‘– โŠ— ๐ ๐‘— ๐œ•๐’ ๐œ•๐‘† ๐‘–Dept of Systems Engineering, University of Lagos 22 oafak@unilag.edu.ng 12/27/2012
  23. 23. (a) Continued ฮฑ Recall that ๐ผ1 ๐’ = tr ๐’ = ๐‘†ฮฑ hence ๐œ•๐ผ1 ๐’ ๐œ•๐ผ1 ๐’ = ๐‘— ๐ ๐‘– โŠ— ๐ ๐‘— ๐œ•๐’ ๐œ•๐‘† ๐‘– ฮฑ ๐œ•๐‘†ฮฑ = ๐‘— ๐ ๐‘– โŠ— ๐ ๐‘— ๐œ•๐‘† ๐‘– = ๐›ฟ ๐‘–๐›ผ ๐›ฟ ๐‘— ๐›ผ ๐  ๐‘– โŠ— ๐  ๐‘— = ๐›ฟ ๐‘—๐‘– ๐  ๐‘– โŠ— ๐  ๐‘— = ๐Ÿ which is the identity tensor as expected.Dept of Systems Engineering, University of Lagos 23 oafak@unilag.edu.ng 12/27/2012
  24. 24. ๐œ•๐ผ2 ๐’ (b) = ๐ผ1 ๐’ ๐Ÿ โˆ’ ๐’ ๐œ•๐’ ๐œ•๐ผ2 ๐’ in a similar way can be written in the invariant component form ๐œ•๐’ as, ๐œ•๐ผ2 ๐’ 1 ๐œ•๐ผ1 ๐’ ฮฑ ๐›ฝ ๐›ฝ = ๐‘†ฮฑ ๐‘† ๐›ฝ โˆ’ ๐‘† ฮฑ ๐‘†ฮฑ ๐  ๐‘– โŠ— ๐  ๐‘— ๐›ฝ ๐œ•๐’ 2 ๐œ•๐‘† ๐‘— ๐‘– 1 where we have utilized the fact that ๐ผ2 ๐’ = 2 tr 2 ๐’ โˆ’ tr(๐’ 2 ) . Consequently, ๐œ•๐ผ2 ๐’ 1 ๐œ• ฮฑ ๐›ฝ ๐›ฝ = ๐‘†ฮฑ ๐‘† ๐›ฝ โˆ’ ๐‘† ฮฑ ๐‘†ฮฑ ๐  ๐‘– โŠ— ๐  ๐‘— ๐›ฝ ๐œ•๐’ 2 ๐œ•๐‘† ๐‘— ๐‘– 1 ๐‘– ๐›ผ ๐›ฝ ๐‘– ๐›ฝ ฮฑ ๐›ฝ ๐›ฝ = ๐›ฟ ๐›ผ ๐›ฟ ๐‘— ๐‘† ๐›ฝ + ๐›ฟ ๐›ฝ ๐›ฟ ๐‘— ๐‘†ฮฑ โˆ’ ๐›ฟ ๐›ฝ ๐›ฟ ๐‘— ๐›ผ ๐‘†ฮฑ โˆ’ ๐›ฟ ๐‘–๐›ผ ๐›ฟ ๐‘— ๐‘† ฮฑ ๐  ๐‘– โŠ— ๐  ๐‘— ๐‘– ๐›ฝ 2 1 ๐‘– ๐›ฝ ๐‘— ๐‘— = ๐›ฟ ๐‘— ๐‘† ๐›ฝ + ๐›ฟ ๐‘—๐‘– ๐‘†ฮฑ โˆ’ ๐‘† ๐‘– โˆ’ ๐‘† ๐‘– ๐  ๐‘– โŠ— ๐  ๐‘— ฮฑ 2 ๐‘— = ๐›ฟ ๐‘—๐‘– ๐‘†ฮฑ โˆ’ ๐‘† ๐‘– ๐  ๐‘– โŠ— ๐  ๐‘— = ๐ผ1 ๐’ ๐Ÿ โˆ’ ๐’ ฮฑDept of Systems Engineering, University of Lagos 24 oafak@unilag.edu.ng 12/27/2012
  25. 25. ๐œ•๐‘ฐ3 ๐’ ๐œ•det ๐’๏€ช = = ๐’๐œ ๐œ•๐’ ๐œ•๐’๏€ช the cofactor of ๐’. Clearly ๐’ ๐œ = det ๐’ ๐’โˆ’1 = ๐‘ฐ3 ๐’ ๐’โˆ’1 as required. Details of this for the contravariant components of a tensor is presented below. Let 1 ๐‘–๐‘—๐‘˜ ๐‘Ÿ๐‘ ๐‘ก ๐’ = ๐‘†= ๐‘’ ๐‘’ ๐‘† ๐‘–๐‘Ÿ ๐‘† ๐‘—๐‘  ๐‘† ๐‘˜๐‘ก 3!Differentiating wrt ๐‘† ๐›ผ๐›ฝ , we obtain, ๐œ•๐‘† 1 ๐‘–๐‘—๐‘˜ ๐‘Ÿ๐‘ ๐‘ก ๐œ•๐‘† ๐‘–๐‘Ÿ ๐œ•๐‘† ๐‘—๐‘  ๐œ•๐‘† ๐‘˜๐‘ก ๐  โŠ— ๐  ๐›ฝ= ๐‘’ ๐‘’ ๐‘† ๐‘† + ๐‘† ๐‘–๐‘Ÿ ๐‘† + ๐‘† ๐‘–๐‘Ÿ ๐‘† ๐‘—๐‘  ๐  โŠ— ๐ ๐›ฝ ๐œ•๐‘† ๐›ผ๐›ฝ ๐›ผ 3! ๐œ•๐‘† ๐›ผ๐›ฝ ๐‘—๐‘  ๐‘˜๐‘ก ๐œ•๐‘† ๐›ผ๐›ฝ ๐‘˜๐‘ก ๐œ•๐‘† ๐›ผ๐›ฝ ๐›ผ 1 ๐‘–๐‘—๐‘˜ ๐‘Ÿ๐‘ ๐‘ก ๐›ผ ๐›ฝ ๐›ฝ ๐›ฝ = ๐‘’ ๐‘’ ๐›ฟ ๐‘– ๐›ฟ ๐‘Ÿ ๐‘† ๐‘—๐‘  ๐‘† ๐‘˜๐‘ก + ๐‘† ๐‘–๐‘Ÿ ๐›ฟ ๐‘— ๐›ผ ๐›ฟ ๐‘  ๐‘† ๐‘˜๐‘ก + ๐‘† ๐‘–๐‘Ÿ ๐‘† ๐‘—๐‘  ๐›ฟ ๐‘˜๐›ผ ๐›ฟ ๐‘ก ๐  ๐›ผ โŠ— ๐  ๐›ฝ 3! 1 ๐›ผ๐‘—๐‘˜ ๐›ฝ๐‘ ๐‘ก = ๐‘’ ๐‘’ ๐‘† ๐‘—๐‘  ๐‘† ๐‘˜๐‘ก + ๐‘† ๐‘—๐‘  ๐‘† ๐‘˜๐‘ก + ๐‘† ๐‘—๐‘  ๐‘† ๐‘˜๐‘ก ๐  ๐›ผ โŠ— ๐  ๐›ฝ 3! 1 ๐›ผ๐‘—๐‘˜ ๐›ฝ๐‘ ๐‘ก = ๐‘’ ๐‘’ ๐‘† ๐‘—๐‘  ๐‘† ๐‘˜๐‘ก ๐  ๐›ผ โŠ— ๐  ๐›ฝ โ‰ก ๐‘† c ๐›ผ๐›ฝ ๐  ๐›ผ โŠ— ๐  ๐›ฝ 2!Which is the cofactor of ๐‘† ๐›ผ๐›ฝDept of Systems Engineering, University of Lagos 25 oafak@unilag.edu.ng 12/27/2012
  26. 26. Real Tensor Functions Consider the function ๐‘“(๐‘บ) = tr ๐‘บ ๐‘˜ where ๐‘˜ โˆˆ โ„›, and ๐‘บ is a tensor. ๐‘“ ๐‘บ = tr ๐‘บ ๐‘˜ = ๐‘บ ๐‘˜ : ๐Ÿ To be specific, let ๐‘˜ = 3.Dept of Systems Engineering, University of Lagos 26 oafak@unilag.edu.ng 12/27/2012
  27. 27. Real Tensor Functions The Gateaux differential in this case, ๐‘‘ ๐‘‘ 3 ๐ท๐‘“ ๐‘บ, ๐‘‘๐‘บ = ๐‘“ ๐‘บ + ๐›ผ๐‘‘๐‘บ = tr ๐‘บ + ๐›ผ๐‘‘๐‘บ ๐‘‘๐›ผ ๐›ผ=0 ๐‘‘๐›ผ ๐›ผ=0 ๐‘‘ = tr ๐‘บ + ๐›ผ๐‘‘๐‘บ ๐‘บ + ๐›ผ๐‘‘๐‘บ ๐‘บ + ๐›ผ๐‘‘๐‘บ ๐‘‘๐›ผ ๐›ผ=0 ๐‘‘ = tr ๐‘บ + ๐›ผ๐‘‘๐‘บ ๐‘บ + ๐›ผ๐‘‘๐‘บ ๐‘บ + ๐›ผ๐‘‘๐‘บ ๐‘‘๐›ผ ๐›ผ=0 = tr ๐‘‘๐‘บ ๐‘บ + ๐›ผ๐‘‘๐‘บ ๐‘บ + ๐›ผ๐‘‘๐‘บ + ๐‘บ + ๐›ผ๐‘‘๐‘บ ๐‘‘๐‘บ ๐‘บ + ๐›ผ๐‘‘๐‘บ + ๐‘บ + ๐›ผ๐‘‘๐‘บ ๐‘บ + ๐›ผ๐‘‘๐‘บ ๐‘‘๐‘บ ๐›ผ=0 = tr ๐‘‘๐‘บ ๐‘บ ๐‘บ + ๐‘บ ๐‘‘๐‘บ ๐‘บ + ๐‘บ ๐‘บ ๐’…๐‘บ = 3 ๐‘บ2 T : ๐‘‘๐‘บDept of Systems Engineering, University of Lagos 27 oafak@unilag.edu.ng 12/27/2012
  28. 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, ๐‘˜โˆ’1 T ๐ท๐‘“ ๐‘บ, ๐‘‘๐‘บ = ๐‘˜ ๐‘บ : ๐‘‘๐‘บ Clearly, ๐‘‘ T tr ๐‘บ ๐‘˜ = ๐‘˜ ๐‘บ ๐‘˜โˆ’1 ๐‘‘๐‘บDept of Systems Engineering, University of Lagos 28 oafak@unilag.edu.ng 12/27/2012
  29. 29. Real Tensor Functions When ๐‘˜ = 1, ๐‘‘ ๐ท๐‘“ ๐‘บ, ๐‘‘๐‘บ = ๐‘“ ๐‘บ + ๐›ผ๐‘‘๐‘บ ๐‘‘๐›ผ ๐›ผ=0 ๐‘‘ = tr ๐‘บ + ๐›ผ๐‘‘๐‘บ ๐‘‘๐›ผ ๐›ผ=0 = tr ๐Ÿ ๐‘‘๐‘บ = ๐Ÿ: ๐‘‘๐‘บ Or that, ๐‘‘ tr ๐‘บ = ๐Ÿ. ๐‘‘๐‘บDept of Systems Engineering, University of Lagos 29 oafak@unilag.edu.ng 12/27/2012
  30. 30. Real Tensor Functions Derivatives of the other two invariants of the tensor ๐‘บ can be found as follows: ๐‘‘ ๐ผ2 (๐‘บ) ๐‘‘๐‘บ 1 ๐‘‘ = tr 2 ๐‘บ โˆ’ tr ๐‘บ2 2 ๐‘‘๐‘บ 1 = 2tr ๐‘บ ๐Ÿ โˆ’ 2 ๐‘บT = tr ๐‘บ ๐Ÿ โˆ’ ๐‘บT 2 .Dept of Systems Engineering, University of Lagos 30 oafak@unilag.edu.ng 12/27/2012
  31. 31. Real Tensor Functions To Determine the derivative of the third invariant, we begin with the trace of the Cayley-Hamilton for ๐‘บ: tr ๐‘บ3 โˆ’ ๐ผ1 ๐‘บ2 + ๐ผ2 ๐‘บ โˆ’ ๐ผ3 ๐‘ฐ = tr(๐‘บ3 ) โˆ’ ๐ผ1 tr ๐‘บ2 + ๐ผ2 tr ๐‘บ โˆ’ 3๐ผ3 = 0 Therefore, 3๐ผ3 = tr(๐‘บ3 ) โˆ’ ๐ผ1 tr ๐‘บ2 + ๐ผ2 tr ๐‘บ 1 2 ๐ผ2 ๐‘บ = tr ๐‘บ โˆ’ tr ๐‘บ2 2 .Dept of Systems Engineering, University of Lagos 31 oafak@unilag.edu.ng 12/27/2012
  32. 32. Differentiating the Invariants We can therefore write, 3๐ผ3 ๐‘บ = tr(๐‘บ3 ) โˆ’ tr ๐‘บ tr ๐‘บ2 1 2 + tr ๐‘บ โˆ’ tr ๐‘บ2 tr ๐‘บ 2 so that, in terms of traces only, 1 3 ๐ผ3 ๐‘บ = tr ๐‘บ โˆ’ 3tr ๐‘บ tr ๐‘บ2 + 2tr(๐‘บ3 ) 6Dept of Systems Engineering, University of Lagos 32 oafak@unilag.edu.ng 12/27/2012
  33. 33. Real Tensor Functions Clearly, ๐‘‘๐ผ3 (๐‘บ) 1 = 3tr 2 ๐‘บ ๐Ÿ โˆ’ 3tr ๐‘บ2 โˆ’ 3tr ๐‘บ 2๐‘บ ๐‘‡ + 2 ร— 3 ๐‘บ2 ๐‘‡ ๐‘‘๐‘บ 6 = ๐ผ2 ๐Ÿ โˆ’ tr ๐‘บ ๐‘บ ๐‘‡ + ๐‘บ2๐‘‡Dept of Systems Engineering, University of Lagos 33 oafak@unilag.edu.ng 12/27/2012
  34. 34. Real Tensor Functions .Dept of Systems Engineering, University of Lagos 34 oafak@unilag.edu.ng 12/27/2012
  35. 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 ChartsDept of Systems Engineering, University of Lagos 35 oafak@unilag.edu.ng 12/27/2012
  36. 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. 37. The Euclidean Point Space ๏€ช The Euclidean Point Space โ„ฐ is a set of elements called points. For each pair of points ๐’™, ๐’š โˆˆ โ„ฐ, โˆƒ ๐’– ๐’™, ๐’š โˆˆ E with the following two properties: 1. ๐’– ๐’™, ๐’š = ๐’– ๐’™, ๐’› + ๐’– ๐’›, ๐’š โˆ€๐’™, ๐’š, ๐’› โˆˆ โ„ฐ 2. ๐’– ๐’™, ๐’š = ๐’– ๐’™, ๐’› โ‡” ๐’š = ๐’› Based on these two, we proceed to show that, ๐’– ๐’™, ๐’™ = ๐ŸŽ And that, ๐’– ๐’™, ๐’› = โˆ’๐’– ๐’›, ๐’™Dept of Systems Engineering, University of Lagos 37 oafak@unilag.edu.ng 12/27/2012
  38. 38. The Euclidean Point Space From property 1, let ๐’š 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’› = โˆ’๐’– ๐’›, ๐’™Dept of Systems Engineering, University of Lagos 38 oafak@unilag.edu.ng 12/27/2012
  39. 39. Dept of Systems Engineering, University of Lagos 39 oafak@unilag.edu.ng 12/27/2012
  40. 40. Coordinate SystemDept of Systems Engineering, University of Lagos 40 oafak@unilag.edu.ng 12/27/2012
  41. 41. Coordinate SystemDept of Systems Engineering, University of Lagos 41 oafak@unilag.edu.ng 12/27/2012
  42. 42. Position Vectors Note that โ„ฐ 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 โ„ฐ. We adopt the convention that ๐’™ ๐’š โ‰ก ๐’– ๐’™, ๐’š referring to the vector ๐’–. If therefore we choose an arbitrarily fixed point ๐ŸŽ โˆˆ โ„ฐ, we are associating ๐’™ ๐ŸŽ , ๐’š ๐ŸŽ and ๐’› ๐ŸŽ respectively with ๐’– ๐’™, ๐ŸŽ , ๐’– ๐’š, ๐ŸŽ and ๐’– ๐’›, ๐ŸŽ . These are vectors based on the points ๐’™, ๐’š and ๐’› with reference to the origin chosen. To emphasize the association with both points of โ„ฐ 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. 43. Length in the Point Space Recall that by property 1, ๐’™ ๐’š โ‰ก ๐’– ๐’™, ๐’š = ๐’– ๐’™, ๐ŸŽ + ๐’– ๐ŸŽ, ๐’š Furthermore, we have deduced that ๐’– ๐ŸŽ, ๐’š = โˆ’๐’–(๐’š, ๐ŸŽ) We may therefore write that, ๐’™ ๐’š = ๐’™ ๐ŸŽ โˆ’ ๐’š(๐ŸŽ) Which, when there is no ambiguity concerning the chosen origin, we can write as, ๐’™ ๐’š = ๐’™โˆ’ ๐’š And the distance between the two is, ๐’… ๐’™ ๐’š = ๐’… ๐’™โˆ’ ๐’š = ๐’™โˆ’ ๐’šDept of Systems Engineering, University of Lagos 43 oafak@unilag.edu.ng 12/27/2012
  44. 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’š on a set of independent vectors are ๐‘ฅ ๐‘– and ๐‘ฆ ๐‘– , then the distance we seek is, ๐’… ๐’™โˆ’ ๐’š = ๐’™โˆ’ ๐’š = ๐‘” ๐‘–๐‘— (๐‘ฅ ๐‘– โˆ’ ๐‘ฆ ๐‘– )(๐‘ฅ ๐‘— โˆ’ ๐‘ฆ ๐‘— ) The more familiar Pythagorean form occurring only when ๐‘” ๐‘–๐‘— = 1.Dept of Systems Engineering, University of Lagos 44 oafak@unilag.edu.ng 12/27/2012
  45. 45. Coordinate Transformations ๏€ช Consider a Cartesian coordinate system ๐‘ฅ, ๐‘ฆ, ๐‘ง or ๐‘ฆ1 , ๐‘ฆ 2 and ๐‘ฆ 3 with an orthogonal basis. Let us now have the possibility of transforming to another coordinate system of an arbitrary nature: ๐‘ฅ 1 , ๐‘ฅ 2 , ๐‘ฅ 3 . We can represent the transformation and its inverse in the equations: ๏€ช ๐‘ฆ ๐‘– = ๐‘ฆ ๐‘– ๐‘ฅ 1 , ๐‘ฅ 2 , ๐‘ฅ 3 , ๐‘ฅ ๐‘– = ๐‘ฅ ๐‘– (๐‘ฆ1 , ๐‘ฆ 2 , ๐‘ฆ 3 ) ๏€ช And if the Jacobian of transformation,Dept of Systems Engineering, University of Lagos 45 oafak@unilag.edu.ng 12/27/2012
  46. 46. Jacobian Determinant ๐œ•๐‘ฆ1 ๐œ•๐‘ฆ1 ๐œ•๐‘ฆ1 ๐œ•๐‘ฅ 1 ๐œ•๐‘ฅ 2 ๐œ•๐‘ฅ 3 ๐œ•๐‘ฆ ๐‘– ๐œ• ๐‘ฆ1 , ๐‘ฆ 2 , ๐‘ฆ 3 ๐œ•๐‘ฆ 2 ๐œ•๐‘ฆ 2 ๐œ•๐‘ฆ 2 ๐‘— โ‰ก 1, ๐‘ฅ2, ๐‘ฅ3 โ‰ก ๐œ•๐‘ฅ ๐œ• ๐‘ฅ ๐œ•๐‘ฅ 1 ๐œ•๐‘ฅ 2 ๐œ•๐‘ฅ 3 ๐œ•๐‘ฆ 3 ๐œ•๐‘ฆ 3 ๐œ•๐‘ฆ 3 ๐œ•๐‘ฅ 1 ๐œ•๐‘ฅ 2 ๐œ•๐‘ฅ 3 does not vanish, then the inverse transformation will exist. So that, ๐‘ฅ ๐‘– = ๐‘ฅ ๐‘– (๐‘ฆ1 , ๐‘ฆ 2 , ๐‘ฆ 3 )Dept of Systems Engineering, University of Lagos 46 oafak@unilag.edu.ng 12/27/2012
  47. 47. Given a position vector ๐ซ = ๐ซ(๐‘ฅ 1 , ๐‘ฅ 2 , ๐‘ฅ 3 ) ๏€ช In the new coordinate system, we can form a set of three vectors, ๐œ•๐ซ ๐ ๐‘– = ๐‘– , ๐‘– = 1,2,3 ๐œ•๐‘ฅ 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. ๐ซ = ๐ซ ๐‘ฅ 1 , ๐‘ฅ 2 , ๐‘ฅ 3 = ๐‘ฆ1 ๐ข + ๐‘ฆ2 ๐ฃ + ๐‘ฆ3 ๐ค So that ๐œ•๐ซ ๐œ•๐ซ ๐œ•๐ซ ๐ข = 1 , ๐ฃ = 2 and ๐ค = 3 ๐œ•๐‘ฆ ๐œ•๐‘ฆ ๐œ•๐‘ฆ 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, ๐ซ = ๐ซ ๐‘ฅ1 , ๐‘ฅ 2 , ๐‘ฅ 3 The total differential of ๐ซ is simply,Dept of Systems Engineering, University of Lagos 47 oafak@unilag.edu.ng 12/27/2012
  48. 48. Natural Dual Bases ๐œ•๐ซ 1+ ๐œ•๐ซ 2+ ๐œ•๐ซ ๐‘‘๐ซ = 1 ๐‘‘๐‘ฅ ๐‘‘๐‘ฅ ๐‘‘๐‘ฅ 3 ๐œ•๐‘ฅ ๐œ•๐‘ฅ 2 ๐œ•๐‘ฅ 3 โ‰ก ๐‘‘๐‘ฅ 1 ๐ 1 + ๐‘‘๐‘ฅ 2 ๐  2 + ๐‘‘๐‘ฅ 3 ๐  3 With ๐  ๐‘– ๐‘ฅ 1 , ๐‘ฅ 2 , ๐‘ฅ 3 , ๐‘– = 1,2,3 now depending in general on ๐‘ฅ 1 , ๐‘ฅ 2 and ๐‘ฅ 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 ๐œ•๐ซ ๐œ•๐ซ ๐œ•๐ซ ๐ข = 1 , ๐ฃ = 2 and ๐ค = 3 . ๐œ•๐‘ฆ ๐œ•๐‘ฆ ๐œ•๐‘ฆDept of Systems Engineering, University of Lagos 48 oafak@unilag.edu.ng 12/27/2012
  49. 49. We now proceed to show that the set of basis vectors, ๐  ๐‘– โ‰ก ๐›ป๐‘ฅ ๐‘– is reciprocal to ๐  ๐‘– . The total differential ๐‘‘๐ซ can be written in terms of partial differentials as, ๐œ•๐ซ ๐‘‘๐ซ = ๐‘‘๐‘ฆ ๐‘– ๐œ•๐‘ฆ ๐‘– Which when expressed in component form yields, ๐œ•๐‘ฅ 1 1 ๐œ•๐‘ฅ 1 2 ๐œ•๐‘ฅ 1 3 ๐‘‘๐‘ฅ 1 = 1 ๐‘‘๐‘ฆ + 2 ๐‘‘๐‘ฆ + 3 ๐‘‘๐‘ฆ ๐œ•๐‘ฆ ๐œ•๐‘ฆ ๐œ•๐‘ฆ ๐œ•๐‘ฅ 2 1 ๐œ•๐‘ฅ 2 2 ๐œ•๐‘ฅ 2 3 ๐‘‘๐‘ฅ 2 = 1 ๐‘‘๐‘ฆ + 2 ๐‘‘๐‘ฆ + 3 ๐‘‘๐‘ฆ ๐œ•๐‘ฆ ๐œ•๐‘ฆ ๐œ•๐‘ฆ ๐œ•๐‘ฅ 3 1 ๐œ•๐‘ฅ 3 2 ๐œ•๐‘ฅ 3 3 ๐‘‘๐‘ฅ 3 = 1 ๐‘‘๐‘ฆ + 2 ๐‘‘๐‘ฆ + 3 ๐‘‘๐‘ฆ ๐œ•๐‘ฆ ๐œ•๐‘ฆ ๐œ•๐‘ฆ Or, more compactly that, ๐‘— = ๐œ•๐‘ฅ ๐‘— ๐‘‘๐‘ฅ ๐‘‘๐‘ฆ ๐‘– = ๐›ป๐‘ฅ ๐‘— โ‹… ๐‘‘๐ซ ๐œ•๐‘ฆ ๐‘–Dept of Systems Engineering, University of Lagos 49 oafak@unilag.edu.ng 12/27/2012
  50. 50. In Cartesian coordinates, ๐‘ฆ ๐‘– , ๐‘– = 1, . . , 3 for any scalar field ๐œ™ (๐‘ฆ1 , ๐‘ฆ 2 , ๐‘ฆ 3 ), ๐œ•๐œ™ ๐‘– = ๐œ•๐œ™ ๐œ•๐œ™ ๐œ•๐œ™ ๐‘‘๐‘ฆ ๐‘‘๐‘ฅ + ๐‘‘๐‘ฆ + ๐‘‘๐‘ง = grad ๐œ™ โ‹… ๐‘‘๐ซ ๐œ•๐‘ฆ ๐‘– ๐œ•๐‘ฅ ๐œ•๐‘ฆ ๐œ•๐‘ง We are treating each curvilinear coordinate as a scalar field, ๐‘ฅ ๐‘— = ๐‘ฅ ๐‘— (๐‘ฆ1 , ๐‘ฆ 2 , ๐‘ฆ 3 ). ๐‘— = ๐›ป๐‘ฅ ๐‘— โ‹… ๐œ•๐ซ ๐‘‘๐‘ฅ ๐‘‘๐‘ฅ ๐‘š ๐œ•๐‘ฅ ๐‘š = ๐  ๐‘— โ‹… ๐  ๐‘š ๐‘‘๐‘ฅ ๐‘š ๐‘— = ๐›ฟ ๐‘š ๐‘‘๐‘ฅ ๐‘š . 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‘š = ๐›ฟ ๐‘š Which recovers for us the reciprocity relationship and shows that ๐  ๐‘— and ๐  ๐‘š are reciprocal systems.Dept of Systems Engineering, University of Lagos 50 oafak@unilag.edu.ng 12/27/2012
  51. 51. The position vector in Cartesian coordinates is ๐’“ = ๐’™ ๐’Š ๐’† ๐’Š. Show that (a) ๐๐ข๐ฏ ๐’“ = ๐Ÿ‘, (b) ๐๐ข๐ฏ ๐’“ โŠ— ๐’“ = ๐Ÿ’๐’“, (c) ๐๐ข๐ฏ ๐’“ = ๐Ÿ‘, and (d) ๐ ๐ซ๐š๐ ๐’“ = ๐Ÿ and (e) ๐œ๐ฎ๐ซ๐ฅ ๐’“ โŠ— ๐’“ = โˆ’๐’“ ร— grad ๐’“ = ๐‘ฅ ๐‘– , ๐’‹ ๐’† ๐‘– โŠ— ๐’† ๐’‹ = ๐›ฟ ๐‘–๐‘— ๐’† ๐‘– โŠ— ๐’† ๐’‹ = ๐Ÿ div ๐’“ = ๐‘ฅ ๐‘– , ๐’‹ ๐’† ๐‘– โ‹… ๐’† ๐’‹ = ๐›ฟ ๐‘–๐‘— ๐›ฟ ๐‘–๐‘— = ๐›ฟ ๐‘—๐‘— = 3. ๐’“ โŠ— ๐’“ = ๐‘ฅ ๐‘– ๐’† ๐‘– โŠ— ๐‘ฅ๐‘— ๐’†๐‘— = ๐‘ฅ ๐‘– ๐‘ฅ๐‘— ๐’† ๐‘– โŠ— ๐’† ๐’‹ grad ๐’“ โŠ— ๐’“ = ๐‘ฅ ๐‘– ๐‘ฅ ๐‘— , ๐‘˜ ๐’† ๐‘– โŠ— ๐’† ๐’‹ โŠ— ๐’† ๐’Œ div ๐’“ โŠ— ๐’“ = ๐‘ฅ ๐‘– , ๐‘˜ ๐‘ฅ ๐‘— + ๐‘ฅ ๐‘– ๐‘ฅ ๐‘— , ๐‘˜ ๐’† ๐‘– โŠ— ๐’† ๐’‹ โ‹… ๐’† ๐’Œ = ๐›ฟ ๐‘–๐‘˜ ๐’™ ๐’‹ + ๐’™ ๐’Š ๐›ฟ ๐‘—๐‘˜ ๐›ฟ ๐‘—๐‘˜ ๐’† ๐‘– = ๐›ฟ ๐‘–๐‘˜ ๐’™ ๐’Œ + ๐’™ ๐’Š ๐›ฟ ๐‘—๐‘— ๐’† ๐‘– = 4๐‘ฅ ๐‘– ๐’† ๐‘– = 4๐’“ curl ๐’“ โŠ— ๐’“ = ๐œ– ๐›ผ๐›ฝ๐›พ ๐‘ฅ ๐‘– ๐‘ฅ ๐›พ , ๐›ฝ ๐’† ๐›ผ โŠ— ๐’† ๐’Š = ๐œ– ๐›ผ๐›ฝ๐›พ ๐‘ฅ ๐‘– , ๐›ฝ ๐‘ฅ ๐›พ + ๐‘ฅ ๐‘– ๐‘ฅ ๐›พ , ๐›ฝ ๐’† ๐›ผ โŠ— ๐’† ๐’Š = ๐œ– ๐›ผ๐›ฝ๐›พ ๐›ฟ ๐‘–๐›ฝ ๐‘ฅ ๐›พ + ๐‘ฅ ๐‘– ๐›ฟ ๐›พ๐›ฝ ๐’† ๐›ผ โŠ— ๐’† ๐’Š = ๐œ– ๐›ผ๐‘–๐›พ ๐‘ฅ ๐›พ ๐’† ๐›ผ โŠ— ๐’† ๐’Š + ๐œ– ๐›ผ๐›ฝ๐›ฝ ๐‘ฅ ๐‘– ๐’† ๐›ผ โŠ— ๐’† ๐’Š = โˆ’๐œ– ๐›ผ๐›พ๐‘– ๐‘ฅ ๐›พ ๐’† ๐›ผ โŠ— ๐’† ๐’Š = โˆ’๐’“ ร—Dept of Systems Engineering, University of Lagos 51 oafak@unilag.edu.ng 12/27/2012
  52. 52. For a scalar field ๐“ and a tensor field ๐“ show that ๐ ๐ซ๐š๐ ๐“๐“ = ๐“๐ ๐ซ๐š๐ ๐“ + ๐“ โŠ— ๐ ๐ซ๐š๐๐“. Also show that ๐๐ข๐ฏ ๐“๐“ = ๐“ ๐๐ข๐ฏ ๐“ + ๐“๐ ๐ซ๐š๐๐“. grad ๐œ™๐“ = ๐œ™๐‘‡ ๐‘–๐‘— , ๐‘˜ ๐  ๐‘– โŠ— ๐  ๐‘— โŠ— ๐  ๐‘˜ = ๐œ™, ๐‘˜ ๐‘‡ ๐‘–๐‘— + ๐œ™๐‘‡ ๐‘–๐‘— , ๐‘˜ ๐  ๐‘– โŠ— ๐  ๐‘— โŠ— ๐  ๐‘˜ = ๐“ โŠ— grad๐œ™ + ๐œ™grad ๐“ Furthermore, we can contract the last two bases and obtain, div ๐œ™๐“ = ๐œ™, ๐‘˜ ๐‘‡ ๐‘–๐‘— + ๐œ™๐‘‡ ๐‘–๐‘— , ๐‘˜ ๐  ๐‘– โŠ— ๐  ๐‘— โ‹… ๐  ๐‘˜ = ๐œ™, ๐‘˜ ๐‘‡ ๐‘–๐‘— + ๐œ™๐‘‡ ๐‘–๐‘— , ๐‘˜ ๐  ๐‘– ๐›ฟ ๐‘— ๐‘˜ = ๐‘‡ ๐‘–๐‘˜ ๐œ™, ๐‘˜ ๐  ๐‘– + ๐œ™๐‘‡ ๐‘–๐‘˜ , ๐‘˜ ๐  ๐‘– = ๐“grad๐œ™ + ๐œ™ div ๐“Dept of Systems Engineering, University of Lagos 52 oafak@unilag.edu.ng 12/27/2012
  53. 53. ๏€ช For two arbitrary vectors, ๐’– and ๐’—, show that ๐ ๐ซ๐š๐ ๐’– โŠ— ๐’— = ๐’– ร— ๐ ๐ซ๐š๐๐’— โˆ’ ๐’— ร— ๐ ๐ซ๐š๐๐’– ๏€ช grad ๐’– โŠ— ๐’— = ๐œ– ๐‘–๐‘—๐‘˜ ๐‘ข ๐‘— ๐‘ฃ ๐‘˜ , ๐‘™ ๐  ๐‘– โŠ— ๐  ๐‘™ = ๐œ– ๐‘–๐‘—๐‘˜ ๐‘ข ๐‘— , ๐‘™ ๐‘ฃ ๐‘˜ + ๐œ– ๐‘–๐‘—๐‘˜ ๐‘ข ๐‘— ๐‘ฃ ๐‘˜ , ๐‘™ ๐  ๐‘– โŠ— ๐  ๐‘™ = ๐‘ข ๐‘— , ๐‘™ ๐œ– ๐‘–๐‘—๐‘˜ ๐‘ฃ ๐‘˜ + ๐‘ฃ ๐‘˜ , ๐‘™ ๐œ– ๐‘–๐‘—๐‘˜ ๐‘ข ๐‘— ๐  ๐‘– โŠ— ๐  ๐‘™ = โˆ’ ๐’— ร— grad๐’– + ๐’– ร— grad๐’—Dept of Systems Engineering, University of Lagos 53 oafak@unilag.edu.ng 12/27/2012
  54. 54. For any two tensor fields ๐ฎ and ๐ฏ, Show that, ๐ฎ ร— : grad ๐ฏ = ๐ฎ โ‹… curl ๐ฏ ๐ฎ ร— : grad ๐ฏ = ๐œ– ๐‘–๐‘—๐‘˜ ๐‘ข ๐‘— ๐  ๐‘– โŠ— ๐  ๐‘˜ : ๐‘ฃ ๐›ผ , ๐›ฝ ๐  ๐›ผ โŠ— ๐  ๐›ฝ = ๐œ– ๐‘–๐‘—๐‘˜ ๐‘ข ๐‘— ๐‘ฃ ๐›ผ , ๐›ฝ ๐  ๐‘– โ‹… ๐  ๐›ผ ๐  ๐‘˜ โ‹… ๐  ๐›ฝ ๐›ฝ = ๐œ– ๐‘–๐‘—๐‘˜ ๐‘ข ๐‘— ๐‘ฃ ๐›ผ , ๐›ฝ ๐›ฟ ๐‘– ๐›ผ ๐›ฟ ๐‘˜ = ๐œ– ๐‘–๐‘—๐‘˜ ๐‘ข ๐‘— ๐‘ฃ ๐‘– , ๐‘˜ = ๐ฎ โ‹… curl ๐ฏDept of Systems Engineering, University of Lagos 54 oafak@unilag.edu.ng 12/27/2012
  55. 55. For a vector field ๐’–, show that ๐ ๐ซ๐š๐ ๐’– ร— is a third ranked tensor. Hence or otherwise show that ๐๐ข๐ฏ ๐’– ร— = โˆ’๐œ๐ฎ๐ซ๐ฅ ๐’–. The secondโ€“order tensor ๐’– ร— is defined as ๐œ– ๐‘–๐‘—๐‘˜ ๐‘ข ๐‘— ๐  ๐‘– โŠ— ๐  ๐‘˜ . Taking the covariant derivative with an independent base, we have grad ๐’– ร— = ๐œ– ๐‘–๐‘—๐‘˜ ๐‘ข ๐‘— , ๐‘™ ๐  ๐‘– โŠ— ๐  ๐‘˜ โŠ— ๐  ๐‘™ This gives a third order tensor as we have seen. Contracting on the last two bases, div ๐’– ร— = ๐œ– ๐‘–๐‘—๐‘˜ ๐‘ข ๐‘— , ๐‘™ ๐  ๐‘– โŠ— ๐  ๐‘˜ โ‹… ๐  ๐‘™ = ๐œ– ๐‘–๐‘—๐‘˜ ๐‘ข ๐‘— , ๐‘™ ๐  ๐‘– ๐›ฟ ๐‘™๐‘˜ = ๐œ– ๐‘–๐‘—๐‘˜ ๐‘ข ๐‘— , ๐‘˜ ๐  ๐‘– = โˆ’curl ๐’–Dept of Systems Engineering, University of Lagos 55 oafak@unilag.edu.ng 12/27/2012
  56. 56. Show that ๐œ๐ฎ๐ซ๐ฅ ๐œ™๐Ÿ = grad ๐œ™ ร— Note that ๐œ™๐Ÿ = ๐œ™๐‘” ๐›ผ๐›ฝ ๐  ๐›ผ โŠ— ๐  ๐›ฝ , and that curl ๐‘ป = ๐œ– ๐‘–๐‘—๐‘˜ ๐‘‡ ๐›ผ๐‘˜ , ๐‘— ๐  ๐‘– โŠ— ๐  ๐›ผ so that, curl ๐œ™๐Ÿ = ๐œ– ๐‘–๐‘—๐‘˜ ๐œ™๐‘” ๐›ผ๐‘˜ , ๐’‹ ๐  ๐‘– โŠ— ๐  ๐›ผ = ๐œ– ๐‘–๐‘—๐‘˜ ๐œ™, ๐’‹ ๐‘” ๐›ผ๐‘˜ ๐  ๐‘– โŠ— ๐  ๐›ผ = ๐œ– ๐‘–๐‘—๐‘˜ ๐œ™, ๐’‹ ๐  ๐‘– โŠ— ๐  ๐‘˜ = grad ๐œ™ ร— Show that curl ๐’— ร— = div ๐’— ๐Ÿ โˆ’ grad ๐’— ๐’— ร— = ๐œ– ๐›ผ๐›ฝ๐‘˜ ๐‘ฃ ๐›ฝ ๐  ๐›ผ โŠ— ๐  ๐‘˜ curl ๐‘ป = ๐œ– ๐‘–๐‘—๐‘˜ ๐‘‡ ๐›ผ๐‘˜ , ๐‘— ๐  ๐‘– โŠ— ๐  ๐›ผ so that curl ๐’— ร— = ๐œ– ๐‘–๐‘—๐‘˜ ๐œ– ๐›ผ๐›ฝ๐‘˜ ๐‘ฃ ๐›ฝ , ๐‘— ๐  ๐‘– โŠ— ๐  ๐›ผ = ๐‘” ๐‘–๐›ผ ๐‘” ๐‘—๐›ฝ โˆ’ ๐‘” ๐‘–๐›ฝ ๐‘” ๐‘—๐›ผ ๐‘ฃ ๐›ฝ , ๐‘— ๐  ๐‘– โŠ— ๐  ๐›ผ = ๐‘ฃ ๐‘—, ๐‘— ๐  ๐›ผ โŠ— ๐  ๐›ผ โˆ’ ๐‘ฃ ๐‘–, ๐‘— ๐  ๐‘– โŠ— ๐  ๐‘— = div ๐’— ๐Ÿ โˆ’ grad ๐’—Dept of Systems Engineering, University of Lagos 56 oafak@unilag.edu.ng 12/27/2012
  57. 57. Show that ๐๐ข๐ฏ ๐’– ร— ๐’— = ๐’— โ‹… ๐œ๐ฎ๐ซ๐ฅ ๐’– โˆ’ ๐’– โ‹… ๐œ๐ฎ๐ซ๐ฅ ๐’— div ๐’– ร— ๐’— = ๐œ– ๐‘–๐‘—๐‘˜ ๐‘ข ๐‘— ๐‘ฃ ๐‘˜ , ๐‘– Noting that the tensor ๐œ– ๐‘–๐‘—๐‘˜ behaves as a constant under a covariant differentiation, we can write, div ๐’– ร— ๐’— = ๐œ– ๐‘–๐‘—๐‘˜ ๐‘ข ๐‘— ๐‘ฃ ๐‘˜ , ๐‘– = ๐œ– ๐‘–๐‘—๐‘˜ ๐‘ข ๐‘— , ๐‘– ๐‘ฃ ๐‘˜ + ๐œ– ๐‘–๐‘—๐‘˜ ๐‘ข ๐‘— ๐‘ฃ ๐‘˜ , ๐‘– = ๐’— โ‹… curl ๐’– โˆ’ ๐’– โ‹… curl ๐’— Given a scalar point function ฯ• and a vector field ๐ฏ, show that ๐œ๐ฎ๐ซ๐ฅ ๐œ™๐ฏ = ๐œ™ curl ๐ฏ + grad๐œ™ ร— ๐ฏ. curl ๐œ™๐ฏ = ๐œ– ๐‘–๐‘—๐‘˜ ๐œ™๐‘ฃ ๐‘˜ , ๐‘— ๐  ๐‘– = ๐œ– ๐‘–๐‘—๐‘˜ ๐œ™, ๐‘— ๐‘ฃ ๐‘˜ + ๐œ™๐‘ฃ ๐‘˜ , ๐‘— ๐  ๐‘– = ๐œ– ๐‘–๐‘—๐‘˜ ๐œ™, ๐‘— ๐‘ฃ ๐‘˜ ๐  ๐‘– + ๐œ– ๐‘–๐‘—๐‘˜ ๐œ™๐‘ฃ ๐‘˜ , ๐‘— ๐  ๐‘– = grad๐œ™ ร— ๐ฏ + ๐œ™ curl ๐ฏDept of Systems Engineering, University of Lagos 57 oafak@unilag.edu.ng 12/27/2012
  58. 58. ๐€ Given that ๐œ‘ ๐‘ก = ๐€(๐‘ก) , Show that ๐œ‘ ๐‘ก = : ๐€ ๐€ ๐‘ก ๐œ‘2 โ‰ก ๐€: ๐€ Now, ๐‘‘ 2 = 2๐œ‘ ๐‘‘๐œ‘ ๐‘‘๐€ ๐‘‘๐€ ๐‘‘๐€ ๐œ‘ = : ๐€ + ๐€: = 2๐€: ๐‘‘๐‘ก ๐‘‘๐‘ก ๐‘‘๐‘ก ๐‘‘๐‘ก ๐‘‘๐‘ก as inner product is commutative. We can therefore write that ๐‘‘๐œ‘ ๐€ ๐‘‘๐€ ๐€ = : = : ๐€ ๐‘‘๐‘ก ๐œ‘ ๐‘‘๐‘ก ๐€ ๐‘ก as required.Dept of Systems Engineering, University of Lagos 58 oafak@unilag.edu.ng 12/27/2012
  59. 59. Given a tensor field ๐‘ป, obtain the vector ๐’˜ โ‰ก ๐‘ป ๐“ ๐ฏ and show that its divergence is ๐‘ป: ๐›๐ฏ + ๐ฏ โ‹… ๐๐ข๐ฏ ๐‘ป The divergence of ๐’˜ is the scalar sum , ๐‘‡๐‘—๐‘– ๐‘ฃ ๐‘— , ๐‘– . Expanding the product covariant derivative we obtain, div ๐‘ปT ๐ฏ = ๐‘‡๐‘—๐‘– ๐‘ฃ ๐‘— , ๐‘– = ๐‘‡๐‘—๐‘– , ๐‘– ๐‘ฃ ๐‘— + ๐‘‡๐‘—๐‘– ๐‘ฃ ๐‘— , ๐‘– = div ๐‘ป โ‹… ๐ฏ + tr ๐‘ปT grad๐ฏ = div ๐‘ป โ‹… ๐ฏ + ๐‘ป: grad ๐ฏ Recall that scalar product of two vectors is commutative so that div ๐‘ปT ๐ฏ = ๐‘ป: grad๐ฏ + ๐ฏ โ‹… div ๐‘ปDept of Systems Engineering, University of Lagos 59 oafak@unilag.edu.ng 12/27/2012
  60. 60. For a second-order tensor ๐‘ป define curl ๐‘ป โ‰ก ๐œ– ๐‘–๐‘—๐‘˜ ๐‘ป ๐›ผ๐‘˜ , ๐‘— ๐  ๐’Š โŠ— ๐  ๐œถ show that for any constant vector ๐’‚, curl ๐‘ป ๐’‚ = curl ๐‘ป ๐‘ป ๐’‚ Express vector ๐’‚ in the invariant form with contravariant components as ๐’‚ = ๐‘Ž ๐›ฝ ๐  ๐›ฝ . It follows that curl ๐‘ป ๐’‚ = ๐œ– ๐‘–๐‘—๐‘˜ ๐‘‡ ๐›ผ๐‘˜ , ๐‘— ๐  ๐‘– โŠ— ๐  ๐›ผ ๐’‚ = ๐œ– ๐‘–๐‘—๐‘˜ ๐‘‡ ๐›ผ๐‘˜ , ๐‘— ๐‘Ž ๐›ฝ ๐  ๐‘– โŠ— ๐  ๐›ผ ๐  ๐›ฝ = ๐œ– ๐‘–๐‘—๐‘˜ ๐‘‡ ๐›ผ๐‘˜ , ๐‘— ๐‘Ž ๐›ฝ ๐  ๐‘– ๐›ฟ ๐›ฝ๐›ผ = ๐œ– ๐‘–๐‘—๐‘˜ ๐‘‡ ๐›ผ๐‘˜ , ๐‘— ๐  ๐‘– ๐‘Ž ๐›ผ = ๐œ– ๐‘–๐‘—๐‘˜ ๐‘‡ ๐›ผ๐‘˜ ๐‘Ž ๐›ผ , ๐‘— ๐  ๐‘– The last equality resulting from the fact that vector ๐’‚ is a constant vector. Clearly, curl ๐‘ป ๐’‚ = curl ๐‘ป ๐‘‡ ๐’‚Dept of Systems Engineering, University of Lagos 60 oafak@unilag.edu.ng 12/27/2012
  61. 61. For any two vectors ๐ฎ and ๐ฏ, show that curl ๐ฎ โŠ— ๐ฏ = grad ๐ฎ ๐ฏ ร— ๐‘ป + curl ๐ฏ โŠ— ๐’– where ๐ฏ ร— is the skew tensor ๐œ– ๐‘–๐‘˜๐‘— ๐‘ฃ ๐‘˜ ๐  ๐’Š โŠ— ๐  ๐’‹ . Recall that the curl of a tensor ๐‘ป is defined by curl ๐‘ป โ‰ก ๐œ– ๐‘–๐‘—๐‘˜ ๐‘‡ ๐›ผ๐‘˜ , ๐‘— ๐  ๐‘– โŠ— ๐  ๐›ผ . Clearly therefore, curl ๐’– โŠ— ๐’— = ๐œ– ๐‘–๐‘—๐‘˜ ๐‘ข ๐›ผ ๐‘ฃ ๐‘˜ , ๐‘— ๐  ๐‘– โŠ— ๐  ๐›ผ = ๐œ– ๐‘–๐‘—๐‘˜ ๐‘ข ๐›ผ , ๐‘— ๐‘ฃ ๐‘˜ + ๐‘ข ๐›ผ ๐‘ฃ ๐‘˜ , ๐‘— ๐  ๐‘– โŠ— ๐  ๐›ผ = ๐œ– ๐‘–๐‘—๐‘˜ ๐‘ข ๐›ผ , ๐‘— ๐‘ฃ ๐‘˜ ๐  ๐‘– โŠ— ๐  ๐›ผ + ๐œ– ๐‘–๐‘—๐‘˜ ๐‘ข ๐›ผ ๐‘ฃ ๐‘˜ , ๐‘— ๐  ๐‘– โŠ— ๐  ๐›ผ = ๐œ– ๐‘–๐‘—๐‘˜ ๐‘ฃ ๐‘˜ ๐  ๐‘– โŠ— ๐‘ข ๐›ผ , ๐‘— ๐  ๐›ผ + ๐œ– ๐‘–๐‘—๐‘˜ ๐‘ฃ ๐‘˜ , ๐‘— ๐  ๐‘– โŠ— ๐‘ข ๐›ผ ๐  ๐›ผ = ๐œ– ๐‘–๐‘—๐‘˜ ๐‘ฃ ๐‘˜ ๐  ๐‘– โŠ— ๐  ๐‘— ๐‘ข ๐›ผ , ๐›ฝ ๐  ๐›ฝ โŠ— ๐  ๐›ผ + ๐œ– ๐‘–๐‘—๐‘˜ ๐‘ฃ ๐‘˜ , ๐‘— ๐  ๐‘– โŠ— ๐‘ข ๐›ผ ๐  ๐›ผ = โˆ’ ๐’— ร— grad ๐’– ๐‘ป + curl ๐’— โŠ— ๐’– = grad ๐’– ๐’— ร— ๐‘ป + curl ๐’— โŠ— 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
  62. 62. Show that curl ๐’– ร— ๐’— = div ๐’– โŠ— ๐’— โˆ’ ๐’— โŠ— ๐’– The vector ๐’˜ โ‰ก ๐’– ร— ๐’— = ๐‘ค ๐’Œ ๐  ๐’Œ = ๐œ– ๐‘˜๐›ผ๐›ฝ ๐‘ข ๐›ผ ๐‘ฃ ๐›ฝ ๐  ๐’Œ and curl ๐’˜ = ๐œ– ๐‘–๐‘—๐‘˜ ๐‘ค ๐‘˜ , ๐‘— ๐  ๐‘– . Therefore, curl ๐’– ร— ๐’— = ๐œ– ๐‘–๐‘—๐‘˜ ๐‘ค ๐‘˜ , ๐‘— ๐  ๐‘– = ๐œ– ๐‘–๐‘—๐‘˜ ๐œ– ๐‘˜๐›ผ๐›ฝ ๐‘ข ๐›ผ ๐‘ฃ ๐›ฝ , ๐‘— ๐  ๐‘– ๐‘— ๐‘– ๐‘— = ๐›ฟ ๐‘–๐›ผ ๐›ฟ ๐›ฝ โˆ’ ๐›ฟ ๐›ฝ ๐›ฟ ๐›ผ ๐‘ข๐›ผ๐‘ฃ ๐›ฝ ,๐‘— ๐ ๐‘– ๐‘— ๐‘– ๐‘— = ๐›ฟ ๐‘–๐›ผ ๐›ฟ ๐›ฝ โˆ’ ๐›ฟ ๐›ฝ ๐›ฟ ๐›ผ ๐‘ข ๐›ผ, ๐‘— ๐‘ฃ ๐›ฝ + ๐‘ข ๐›ผ ๐‘ฃ ๐›ฝ, ๐‘— ๐  ๐‘– = ๐‘ข ๐‘–, ๐‘— ๐‘ฃ ๐‘— + ๐‘ข ๐‘– ๐‘ฃ ๐‘—, ๐‘— โˆ’ ๐‘ข ๐‘—, ๐‘— ๐‘ฃ ๐‘– + ๐‘ข ๐‘— ๐‘ฃ ๐‘–, ๐‘— ๐ ๐‘– = ๐‘ข๐‘– ๐‘ฃ ๐‘— ,๐‘— โˆ’ ๐‘ข ๐‘— ๐‘ฃ๐‘– ,๐‘— ๐ ๐‘– = div ๐’– โŠ— ๐’— โˆ’ ๐’— โŠ— ๐’– since div ๐’– โŠ— ๐’— = ๐‘ข๐‘– ๐‘ฃ ๐‘— , ๐›ผ ๐ ๐‘– โŠ— ๐  ๐‘— โ‹… ๐  ๐›ผ = ๐‘ข ๐‘– ๐‘ฃ ๐‘— , ๐‘— ๐  ๐‘–.Dept of Systems Engineering, University of Lagos 62 oafak@unilag.edu.ng 12/27/2012
  63. 63. Given a scalar point function ๐œ™ and a second-order tensor field ๐‘ป,show that ๐œ๐ฎ๐ซ๐ฅ ๐œ™๐‘ป = ๐œ™ ๐œ๐ฎ๐ซ๐ฅ ๐‘ป + grad๐œ™ ร— ๐‘ป ๐‘ป where grad๐œ™ ร— is the skew tensor ๐œ– ๐‘–๐‘—๐‘˜ ๐œ™, ๐‘— ๐  ๐’Š โŠ— ๐  ๐’Œcurl ๐œ™๐‘ป โ‰ก ๐œ– ๐‘–๐‘—๐‘˜ ๐œ™๐‘‡ ๐›ผ๐‘˜ , ๐‘— ๐  ๐‘– โŠ— ๐  ๐›ผ = ๐œ– ๐‘–๐‘—๐‘˜ ๐œ™, ๐‘— ๐‘‡ ๐›ผ๐‘˜ + ๐œ™๐‘‡ ๐›ผ๐‘˜ , ๐‘— ๐  ๐‘– โŠ— ๐  ๐›ผ = ๐œ– ๐‘–๐‘—๐‘˜ ๐œ™, ๐‘— ๐‘‡ ๐›ผ๐‘˜ ๐  ๐‘– โŠ— ๐  ๐›ผ + ๐œ™๐œ– ๐‘–๐‘—๐‘˜ ๐‘‡ ๐›ผ๐‘˜ , ๐‘— ๐  ๐‘– โŠ— ๐  ๐›ผ = ๐œ– ๐‘–๐‘—๐‘˜ ๐œ™, ๐‘— ๐  ๐‘– โŠ— ๐  ๐‘˜ ๐‘‡ ๐›ผ๐›ฝ ๐  ๐›ฝ โŠ— ๐  ๐›ผ + ๐œ™๐œ– ๐‘–๐‘—๐‘˜ ๐‘‡ ๐›ผ๐‘˜ , ๐‘— ๐  ๐‘– โŠ— ๐  ๐›ผ = ๐œ™ curl ๐‘ป + grad๐œ™ ร— ๐‘ป ๐‘‡ Dept of Systems Engineering, University of Lagos 63 oafak@unilag.edu.ng 12/27/2012
  64. 64. For a second-order tensor field ๐‘ป, show that div curl ๐‘ป = curl div ๐‘ป ๐“ Define the second order tensor ๐‘† as curl ๐‘ป โ‰ก ๐œ– ๐‘–๐‘—๐‘˜ ๐‘‡ ๐›ผ๐‘˜ , ๐‘— ๐  ๐‘– โŠ— ๐  ๐›ผ = ๐‘†.๐›ผ ๐  ๐‘– โŠ— ๐  ๐›ผ ๐‘– The gradient of ๐‘บ is ๐‘†.๐›ผ , ๐›ฝ ๐  ๐‘– โŠ— ๐  ๐›ผ โŠ— ๐  ๐›ฝ = ๐œ– ๐‘–๐‘—๐‘˜ ๐‘‡ ๐›ผ๐‘˜ , ๐‘—๐›ฝ ๐  ๐‘– โŠ— ๐  ๐›ผ โŠ— ๐  ๐›ฝ ๐‘– Clearly, div curl ๐‘ป = ๐œ– ๐‘–๐‘—๐‘˜ ๐‘‡ ๐›ผ๐‘˜ , ๐‘—๐›ฝ ๐  ๐‘– โŠ— ๐  ๐›ผ โ‹… ๐  ๐›ฝ = ๐œ– ๐‘–๐‘—๐‘˜ ๐‘‡ ๐›ผ๐‘˜ , ๐‘—๐›ฝ ๐  ๐‘– ๐‘” ๐›ผ๐›ฝ = ๐œ– ๐‘–๐‘—๐‘˜ ๐‘‡ ๐›ฝ ๐‘˜ , ๐‘—๐›ฝ ๐  ๐‘– = curl div ๐‘ปTDept of Systems Engineering, University of Lagos 64 oafak@unilag.edu.ng 12/27/2012
  65. 65. If the volume ๐‘ฝ is enclosed by the surface ๐‘บ, the position vector ๐’“ = ๐’™ ๐’Š ๐  ๐’Š and ๐’ is the external unit normal to each surface ๐Ÿ element, show that ๐‘บ ๐› ๐’“ โ‹… ๐’“ โ‹… ๐’๐’…๐‘บ equals the volume ๐Ÿ” contained in ๐‘ฝ. ๐’“ โ‹… ๐’“ = ๐‘ฅ ๐‘– ๐‘ฅ ๐‘— ๐  ๐‘– โ‹… ๐  ๐‘— = ๐‘ฅ ๐‘– ๐‘ฅ ๐‘— ๐‘” ๐‘–๐‘— By the Divergence Theorem, ๐›ป ๐’“ โ‹… ๐’“ โ‹… ๐’๐‘‘๐‘† = ๐›ปโ‹… ๐›ป ๐’“โ‹… ๐’“ ๐‘‘๐‘‰ ๐‘† ๐‘‰ = ๐œ• ๐‘™ ๐œ• ๐‘˜ ๐‘ฅ ๐‘– ๐‘ฅ ๐‘— ๐‘” ๐‘–๐‘— ๐  ๐‘™ โ‹… ๐  ๐‘˜ ๐‘‘๐‘‰ ๐‘‰ = ๐œ• ๐‘™ ๐‘” ๐‘–๐‘— ๐‘ฅ ๐‘– , ๐‘˜ ๐‘ฅ ๐‘— + ๐‘ฅ ๐‘– ๐‘ฅ ๐‘— , ๐‘˜ ๐  ๐‘™ โ‹… ๐  ๐‘˜ ๐‘‘๐‘‰ ๐‘‰ ๐‘— = ๐‘” ๐‘–๐‘— ๐‘” ๐‘™๐‘˜ ๐›ฟ ๐‘–๐‘˜ ๐‘ฅ ๐‘— + ๐‘ฅ ๐‘– ๐›ฟ ๐‘˜ , ๐‘™ ๐‘‘๐‘‰ = 2๐‘” ๐‘–๐‘˜ ๐‘” ๐‘™๐‘˜ ๐‘ฅ ๐‘– , ๐‘™ ๐‘‘๐‘‰ ๐‘‰ ๐‘‰ = 2๐›ฟ ๐‘–๐‘™ ๐›ฟ ๐‘™๐‘– ๐‘‘๐‘‰ = 6 ๐‘‘๐‘‰ ๐‘‰ ๐‘‰Dept of Systems Engineering, University of Lagos 65 oafak@unilag.edu.ng 12/27/2012
  66. 66. For any Euclidean coordinate system, show that div ๐ฎ ร— ๐ฏ = ๐ฏ curl ๐ฎ โˆ’ ๐ฎ curl ๐ฏ Given the contravariant vector ๐‘ข ๐‘– and ๐‘ฃ ๐‘– with their associated vectors ๐‘ข ๐‘– and ๐‘ฃ ๐‘– , the contravariant component of the above cross product is ๐œ– ๐‘–๐‘—๐‘˜ ๐‘ข ๐‘— ๐‘ฃ ๐‘˜ .The required divergence is simply the contraction of the covariant ๐‘ฅ ๐‘– derivative of this quantity: ๐œ– ๐‘–๐‘—๐‘˜ ๐‘ข ๐‘— ๐‘ฃ ๐‘˜ = ๐œ– ๐‘–๐‘—๐‘˜ ๐‘ข ๐‘—,๐‘– ๐‘ฃ ๐‘˜ + ๐œ– ๐‘–๐‘—๐‘˜ ๐‘ข ๐‘— ๐‘ฃ ๐‘˜,๐‘– ,๐‘– where we have treated the tensor ๐œ– ๐‘–๐‘—๐‘˜ as a constant under the covariant derivative. Cyclically rearranging the RHS we obtain, ๐œ– ๐‘–๐‘—๐‘˜ ๐‘ข ๐‘— ๐‘ฃ ๐‘˜ = ๐‘ฃ ๐‘˜ ๐œ– ๐‘˜๐‘–๐‘— ๐‘ข ๐‘—,๐‘– + ๐‘ข ๐‘— ๐œ– ๐‘—๐‘˜๐‘– ๐‘ฃ ๐‘˜,๐‘– = ๐‘ฃ ๐‘˜ ๐œ– ๐‘˜๐‘–๐‘— ๐‘ข ๐‘—,๐‘– + ๐‘ข ๐‘— ๐œ– ๐‘—๐‘–๐‘˜ ๐‘ฃ ๐‘˜,๐‘– ,๐‘– where we have used the anti-symmetric property of the tensor ๐œ– ๐‘–๐‘—๐‘˜ . The last expression shows clearly that div ๐ฎ ร— ๐ฏ = ๐ฏ curl ๐ฎ โˆ’ ๐ฎ curl ๐ฏ as required.Dept of Systems Engineering, University of Lagos 66 oafak@unilag.edu.ng 12/27/2012
  67. 67. Liouville Formula ๐‘‘๐“ ๐‘‘ For a scalar variable ๐›ผ, if the tensor ๐“ = ๐“(๐›ผ) and ๐“ โ‰ก ๐‘‘๐›ผ, Show that ๐‘‘๐›ผ det ๐“ = det ๐“ tr(๐“ ๐“ โˆ’๐Ÿ ) Proof: Let ๐€ โ‰ก ๐“ ๐“ โˆ’1 so that, ๐“ = ๐€๐“. In component form, we have ๐‘‡๐‘— ๐‘– = ๐ด ๐‘– ๐‘š ๐‘‡๐‘— ๐‘š . Therefore, ๐‘‘ ๐‘‘ det ๐“ = ๐œ– ๐‘–๐‘—๐‘˜ ๐‘‡๐‘–1 ๐‘‡๐‘—2 ๐‘‡ 3 = ๐œ– ๐‘–๐‘—๐‘˜ ๐‘‡๐‘–1 ๐‘‡๐‘—2 ๐‘‡ 3 + ๐‘‡๐‘–1 ๐‘‡๐‘—2 ๐‘‡ 3 + ๐‘‡๐‘–1 ๐‘‡๐‘—2 ๐‘‡ 3 ๐‘˜ ๐‘˜ ๐‘˜ ๐‘˜ ๐‘‘๐›ผ ๐‘‘๐›ผ = ๐œ– ๐‘–๐‘—๐‘˜ ๐ด1 ๐‘‡๐‘– ๐‘™ ๐‘‡๐‘—2 ๐‘‡ 3 + ๐‘‡๐‘–1 ๐ด2๐‘š ๐‘‡๐‘— ๐‘š ๐‘‡ 3 + ๐‘‡๐‘–1 ๐‘‡๐‘—2 ๐ด3๐‘› ๐‘‡ ๐‘˜๐‘› ๐‘™ ๐‘˜ ๐‘˜ = ๐œ– ๐‘–๐‘—๐‘˜ ๐ด1 ๐‘‡๐‘–1 + ๐ด1 ๐‘‡๐‘–2 + ๐ด1 ๐‘‡๐‘–3 1 2 3 ๐‘‡๐‘—2 ๐‘‡ 3 ๐‘˜ + ๐‘‡๐‘–1 ๐ด1 ๐‘‡๐‘—1 + ๐ด2 ๐‘‡๐‘—2 + ๐ด2 ๐‘‡๐‘—3 2 2 3 ๐‘‡ 3 + ๐‘‡๐‘–1 ๐‘‡๐‘—2 ๐‘˜ ๐ด1 ๐‘‡ 1 + ๐ด3 ๐‘‡ 2 + ๐ด3 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. 68. Liouville Theorem Contโ€™d (For example, the first boxed term yields, ๐œ– ๐‘–๐‘—๐‘˜ ๐ด1 ๐‘‡๐‘–2 ๐‘‡๐‘—2 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œ– ๐‘–๐‘—๐‘˜ ๐ด1 ๐‘‡๐‘–1 ๐‘‡๐‘—2 ๐‘‡ 3 + ๐‘‡๐‘–1 ๐ด2 ๐‘‡๐‘—2 ๐‘‡ 3 + ๐‘‡๐‘–1 ๐‘‡๐‘—2 ๐ด3 ๐‘‡ 3 1 ๐‘˜ 2 ๐‘˜ 3 ๐‘˜ ๐‘‘๐›ผ = ๐ด ๐‘š ๐œ– ๐‘–๐‘—๐‘˜ ๐‘‡๐‘–1 ๐‘‡๐‘—2 ๐‘‡ 3 ๐‘š ๐‘˜ = tr ๐“ ๐“ โˆ’1 det ๐“ as required.Dept of Systems Engineering, University of Lagos 68 oafak@unilag.edu.ng 12/27/2012
  69. 69. For a general tensor field ๐‘ป show that, ๐œ๐ฎ๐ซ๐ฅ ๐œ๐ฎ๐ซ๐ฅ ๐‘ป = ๐Ÿ ๐ญ๐ซ ๐‘ป โˆ’ ๐๐ข๐ฏ ๐๐ข๐ฏ ๐‘ป ๐‘ฐ + ๐ ๐ซ๐š๐ ๐๐ข๐ฏ ๐‘ป + ๐“ ๐› ๐ ๐ซ๐š๐ ๐๐ข๐ฏ ๐‘ป โˆ’ ๐ ๐ซ๐š๐ ๐ ๐ซ๐š๐ ๐ญ๐ซ๐‘ป โˆ’ ๐› ๐Ÿ ๐‘ป ๐“ curl ๐‘ป = ๐ ๐›ผ๐‘ ๐‘ก ๐‘‡ ๐›ฝ๐‘ก , ๐‘  ๐  ๐›ผ โŠ— ๐  ๐›ฝ ๐›ผ = ๐‘† .๐›ฝ ๐  ๐›ผ โŠ— ๐  ๐›ฝ ๐›ผ curl ๐‘บ = ๐œ– ๐‘–๐‘—๐‘˜ ๐‘† .๐‘˜ , ๐‘— ๐  ๐‘– โŠ— ๐  ๐›ผ so that curl ๐‘บ = curl curl ๐‘ป = ๐œ– ๐‘–๐‘—๐‘˜ ๐œ– ๐›ผ๐‘ ๐‘ก ๐‘‡ ๐‘˜๐‘ก , ๐‘ ๐‘— ๐  ๐‘– โŠ— ๐  ๐›ผ ๐‘” ๐‘–๐›ผ ๐‘” ๐‘–๐‘  ๐‘” ๐‘–๐‘ก = ๐‘” ๐‘—๐›ผ ๐‘” ๐‘—๐‘  ๐‘” ๐‘—๐‘ก ๐‘‡ ๐‘˜๐‘ก , ๐‘ ๐‘— ๐  ๐‘– โŠ— ๐  ๐›ผ ๐‘” ๐‘˜๐›ผ ๐‘” ๐‘˜๐‘  ๐‘” ๐‘˜๐‘ก ๐‘” ๐‘–๐›ผ ๐‘” ๐‘—๐‘  ๐‘” ๐‘˜๐‘ก โˆ’ ๐‘” ๐‘—๐‘ก ๐‘” ๐‘˜๐‘  + ๐‘” ๐‘–๐‘  ๐‘” ๐‘—๐‘ก ๐‘” ๐‘˜๐›ผ โˆ’ ๐‘” ๐‘—๐›ผ ๐‘” ๐‘˜๐‘ก = ๐‘–๐‘ก ๐‘—๐›ผ ๐‘˜๐‘  ๐‘—๐‘  ๐‘˜๐›ผ ๐‘‡ ๐‘˜๐‘ก , ๐‘ ๐‘— ๐  ๐‘– โŠ— ๐  ๐›ผ +๐‘” ๐‘” ๐‘” โˆ’ ๐‘” ๐‘” ๐‘ก ๐‘ก = ๐‘” ๐‘—๐‘  ๐‘‡ .๐‘ก , ๐‘ ๐‘— โˆ’๐‘‡..๐‘ ๐‘— , ๐‘ ๐‘— ๐  ๐›ผ โŠ— ๐  ๐›ผ + ๐‘‡ ..๐›ผ๐‘— , ๐‘ ๐‘— โˆ’๐‘” ๐‘—๐›ผ ๐‘‡ .๐‘ก , ๐‘ ๐‘— ๐  ๐‘  โŠ— ๐  ๐›ผ ๐‘ . ๐›ผ. + ๐‘” ๐‘—๐›ผ ๐‘‡ .๐‘ก , ๐‘ ๐‘— โˆ’๐‘” ๐‘—๐‘  ๐‘‡ .๐‘ก , ๐‘ ๐‘— ๐  ๐‘ก โŠ— ๐  ๐›ผ T = ๐›ป 2 tr ๐‘ป โˆ’ div div ๐‘ป ๐‘ฐ + grad div ๐‘ป โˆ’ grad grad tr๐‘ป + grad div ๐‘ป โˆ’ ๐›ป 2 ๐‘ปTDept of Systems Engineering, University of Lagos 69 oafak@unilag.edu.ng 12/27/2012
  70. 70. When ๐‘ป is symmetric, show that tr(curl ๐‘ป) vanishes. curl ๐‘ป = ๐ ๐‘–๐‘—๐‘˜ ๐‘‡ ๐›ฝ๐‘˜ , ๐‘— ๐  ๐‘– โŠ— ๐  ๐›ฝ tr curl ๐‘ป = ๐œ– ๐‘–๐‘—๐‘˜ ๐‘‡ ๐›ฝ๐‘˜ , ๐‘— ๐  ๐‘– โ‹… ๐  ๐›ฝ ๐›ฝ = ๐œ– ๐‘–๐‘—๐‘˜ ๐‘‡ ๐›ฝ๐‘˜ , ๐‘— ๐›ฟ ๐‘– = ๐œ– ๐‘–๐‘—๐‘˜ ๐‘‡๐‘–๐‘˜ , ๐‘— which obviously vanishes on account of the symmetry and antisymmetry in ๐‘– and ๐‘˜. In this case, curl curl ๐‘ป = ๐›ป 2 tr ๐‘ป โˆ’ div div ๐‘ป ๐‘ฐ โˆ’ grad grad tr๐‘ป + 2 grad div ๐‘ป โˆ’ ๐›ป 2 ๐‘ป T as grad div ๐‘ป = grad div ๐‘ป 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. 71. For a scalar function ฮฆ and a vector ๐‘ฃ ๐‘– show that the divergence of the vector ๐‘ฃ ๐‘– ฮฆ is equal to, ๐ฏ โ‹… gradฮฆ + ฮฆ div ๐ฏ ๐‘ฃ ๐‘– ฮฆ ,๐‘– = ฮฆ๐‘ฃ ๐‘– ,๐‘– + ๐‘ฃ ๐‘– ฮฆ,i Hence the result.Dept of Systems Engineering, University of Lagos 71 oafak@unilag.edu.ng 12/27/2012
  72. 72. Show that ๐’„๐’–๐’“๐’ ๐ฎ ร— ๐ฏ = ๐ฏ โˆ™ ๐›๐ฎ + ๐ฎ โ‹… div ๐ฏ โˆ’ ๐ฏ โ‹… div ๐ฎ โˆ’ ๐ฎโˆ™ ๐› ๐ฏ 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œ– ๐‘™๐‘š๐‘– ๐œ– ๐‘–๐‘—๐‘˜ ๐‘ข ๐‘— ๐‘ฃ ๐‘˜ ,๐‘š Expanding in the usual way, noting the relation between the alternating tensors and the Kronecker deltas, ๐‘™๐‘š๐‘– ๐œ– ๐‘™๐‘š๐‘– ๐œ– ๐‘–๐‘—๐‘˜ ๐‘ข ๐‘— ๐‘ฃ ๐‘˜ = ๐›ฟ ๐‘—๐‘˜๐‘– ๐‘ข ๐‘— ,๐‘š ๐‘ฃ ๐‘˜ โˆ’ ๐‘ข ๐‘— ๐‘ฃ ๐‘˜ ,๐‘š ,๐‘š ๐‘™๐‘š ๐›ฟ ๐‘—๐‘™ ๐›ฟ๐‘— ๐‘š = ๐›ฟ ๐‘—๐‘˜ ๐‘ข ๐‘— ,๐‘š ๐‘ฃ ๐‘˜ โˆ’ ๐‘ข ๐‘— ๐‘ฃ ๐‘˜ ,๐‘š = ๐‘ข ๐‘— ,๐‘š ๐‘ฃ ๐‘˜ โˆ’ ๐‘ข ๐‘— ๐‘ฃ ๐‘˜ ,๐‘š ๐›ฟ ๐‘™๐‘˜ ๐›ฟ ๐‘˜๐‘š = ๐›ฟ ๐‘—๐‘™ ๐›ฟ ๐‘˜๐‘š โˆ’ ๐›ฟ ๐‘™๐‘˜ ๐›ฟ ๐‘— ๐‘š ๐‘ข ๐‘— ,๐‘š ๐‘ฃ ๐‘˜ โˆ’ ๐‘ข ๐‘— ๐‘ฃ ๐‘˜ ,๐‘š = ๐›ฟ ๐‘—๐‘™ ๐›ฟ ๐‘˜๐‘š ๐‘ข ๐‘— ,๐‘š ๐‘ฃ ๐‘˜ โˆ’ ๐›ฟ ๐‘—๐‘™ ๐›ฟ ๐‘˜๐‘š ๐‘ข ๐‘— ๐‘ฃ ๐‘˜ ,๐‘š + ๐›ฟ ๐‘™๐‘˜ ๐›ฟ ๐‘— ๐‘š ๐‘ข ๐‘— ,๐‘š ๐‘ฃ ๐‘˜ โˆ’ ๐›ฟ ๐‘™๐‘˜ ๐›ฟ ๐‘— ๐‘š ๐‘ข ๐‘— ๐‘ฃ ๐‘˜ ,๐‘š = ๐‘ข ๐‘™ ,๐‘š ๐‘ฃ ๐‘š โˆ’ ๐‘ข ๐‘š ,๐‘š ๐‘ฃ ๐‘™ + ๐‘ข ๐‘™ ๐‘ฃ ๐‘š ,๐‘š โˆ’ ๐‘ข ๐‘š ๐‘ฃ ๐‘™ ,๐‘š 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. 73. Differentiation of Fields Given a vector point function ๐’– ๐‘ฅ 1 , ๐‘ฅ 2 , ๐‘ฅ 3 and its covariant components, ๐‘ข ๐›ผ ๐‘ฅ 1 , ๐‘ฅ 2 , ๐‘ฅ 3 , ๐›ผ = 1,2,3, with ๐  ๐›ผ as the reciprocal basis vectors, Let ๐œ• ๐œ•๐‘ข ๐›ผ ๐›ผ ๐œ•๐  ๐›ผ ๐’– = ๐‘ข ๐›ผ ๐  ๐›ผ , then ๐‘‘๐’– = ๐‘ข ๐›ผ ๐  ๐›ผ ๐‘‘๐‘ฅ ๐‘˜ = ๐  + ๐‘ข ๐›ผ ๐‘‘๐‘ฅ ๐‘˜ ๐œ•๐‘ฅ ๐‘˜ ๐œ•๐‘ฅ ๐‘˜ ๐œ•๐‘ฅ ๐‘˜ Clearly, ๐œ•๐’– ๐œ•๐‘ข ๐›ผ ๐›ผ ๐œ•๐  ๐›ผ ๐‘˜ = ๐‘˜ ๐  + ๐‘ข ๐‘˜ ๐›ผ ๐œ•๐‘ฅ ๐œ•๐‘ฅ ๐œ•๐‘ฅ And the projection of this quantity on the ๐  ๐‘– direction is, ๐œ•๐’– ๐œ•๐‘ข ๐›ผ ๐›ผ ๐œ•๐  ๐›ผ ๐œ•๐‘ข ๐›ผ ๐›ผ ๐œ•๐  ๐›ผ โˆ™ ๐  = ๐  + ๐‘ข โˆ™ ๐ ๐‘– = ๐  โˆ™ ๐ ๐‘– + โˆ™ ๐  ๐‘ข ๐œ•๐‘ฅ ๐‘˜ ๐‘– ๐œ•๐‘ฅ ๐‘˜ ๐œ•๐‘ฅ ๐‘˜ ๐›ผ ๐œ•๐‘ฅ ๐‘˜ ๐œ•๐‘ฅ ๐‘˜ ๐‘– ๐›ผ ๐œ•๐‘ข ๐›ผ ๐›ผ ๐œ•๐  ๐›ผ = ๐›ฟ + ๐‘˜ ๐‘– ๐‘˜ โˆ™ ๐ ๐‘– ๐‘ข ๐›ผ ๐œ•๐‘ฅ ๐œ•๐‘ฅDept of Systems Engineering, University of Lagos 73 oafak@unilag.edu.ng 12/27/2012
  74. 74. Christoffel Symbols ๐‘— ๏€ช Now, ๐  ๐‘– โˆ™ ๐  ๐‘— = ๐›ฟ ๐‘– so that ๐œ•๐  ๐‘– ๐‘–โˆ™ ๐œ•๐  ๐‘— ๐‘˜ โˆ™ ๐ ๐‘— + ๐  ๐‘˜ = 0. ๐œ•๐‘ฅ ๐œ•๐‘ฅ ๐œ•๐  ๐‘– ๐‘– ๐œ•๐  ๐‘— ๐‘– โˆ™ ๐  ๐‘— = โˆ’๐  โˆ™ โ‰กโˆ’ . ๐œ•๐‘ฅ ๐‘˜ ๐œ•๐‘ฅ ๐‘˜ ๐‘—๐‘˜ 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. 75. Derivatives in General Coordinates Using this, we can now write that, ๐œ•๐’– ๐œ•๐‘ข ๐›ผ ๐›ผ ๐œ•๐  ๐›ผ ๐œ•๐‘ข ๐‘– ๐›ผ โˆ™ ๐ ๐‘– = ๐›ฟ๐‘– + โˆ™ ๐ ๐‘– ๐‘ข ๐›ผ = โˆ’ ๐‘ข ๐œ•๐‘ฅ ๐‘˜ ๐œ•๐‘ฅ ๐‘˜ ๐œ•๐‘ฅ ๐‘˜ ๐œ•๐‘ฅ ๐‘˜ ๐‘–๐‘˜ ๐›ผ ๏€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, ๐œ•๐‘ข ๐‘– ๐œ•๐  ๐‘– ๐‘– ๐œ•๐‘ข ๐‘– ๐›ผ ๐‘‘๐’– = ๐ ๐‘– + ๐‘ข ๐‘‘๐‘ฅ ๐‘˜ = ๐ ๐‘– + ๐  ๐‘ข ๐‘– ๐‘‘๐‘ฅ ๐‘˜ ๐œ•๐‘ฅ ๐‘˜ ๐œ•๐‘ฅ ๐‘˜ ๐œ•๐‘ฅ ๐‘˜ ๐‘–๐‘˜ ๐›ผ ๏€ช So that, ๐œ•๐’– ๐œ•๐‘ข ๐›ผ ๐œ•๐  ๐›ผ ๐›ผ = ๐ ๐›ผ+ ๐‘ข ๐œ•๐‘ฅ ๐‘˜ ๐œ•๐‘ฅ ๐‘˜ ๐œ•๐‘ฅ ๐‘˜ ๏€ช The components of this in the direction of ๐  ๐‘– can be obtained by taking a dot product as before: ๐œ•๐’– ๐‘– = ๐œ•๐‘ข ๐›ผ ๐œ•๐  ๐›ผ ๐›ผ ๐‘– = ๐œ•๐‘ข ๐›ผ ๐‘– ๐œ•๐  ๐›ผ ๐‘– ๐‘ข๐›ผ = ๐œ•๐‘ข ๐‘– ๐‘– โˆ™ ๐  ๐ ๐›ผ+ ๐‘ข โˆ™ ๐  ๐›ฟ๐›ผ + โˆ™ ๐  + ๐‘ข๐›ผ ๐œ•๐‘ฅ ๐‘˜ ๐œ•๐‘ฅ ๐‘˜ ๐œ•๐‘ฅ ๐‘˜ ๐œ•๐‘ฅ ๐‘˜ ๐œ•๐‘ฅ ๐‘˜ ๐œ•๐‘ฅ ๐‘˜ ๐›ผ๐‘˜Dept of Systems Engineering, University of Lagos 75 oafak@unilag.edu.ng 12/27/2012
  76. 76. Covariant Derivatives The two results above are represented symbolically as, ๐œ•๐‘ข ๐‘– ๐›ผ ๐‘– ๐œ•๐‘ข ๐›ผ ๐‘– ๐‘ข ๐‘–,๐‘˜ = โˆ’ ๐‘ข ๐›ผ and ๐‘ข,๐‘˜ = + ๐‘ข๐›ผ ๐œ•๐‘ฅ ๐‘˜ ๐‘–๐‘˜ ๐œ•๐‘ฅ ๐‘˜ ๐›ผ๐‘˜ ๏€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
  77. 77. Christoffel Symbols ๐œ•๐ซ ๏€ช We observe that the derivative of the covariant basis, ๐  ๐‘– = ๐‘– , ๐œ•๐‘ฅ ๐œ•๐  ๐‘– ๐œ• 2๐ซ ๐œ• 2๐ซ ๐œ•๐  ๐‘— = = = ๐œ•๐‘ฅ ๐‘— ๐œ•๐‘ฅ ๐‘— ๐œ•๐‘ฅ ๐‘– ๐œ•๐‘ฅ ๐‘– ๐œ•๐‘ฅ ๐‘— ๐œ•๐‘ฅ ๐‘– ๏€ช Taking the dot product with ๐  ๐‘˜ , ๐œ•๐  ๐‘– 1 ๐œ•๐  ๐‘— ๐œ•๐  ๐‘– ๐‘— โˆ™ ๐ ๐‘˜= ๐‘– โˆ™ ๐ ๐‘˜+ ๐‘— โˆ™ ๐ ๐‘˜ ๐œ•๐‘ฅ 2 ๐œ•๐‘ฅ ๐œ•๐‘ฅ 1 ๐œ• ๐œ• ๐œ•๐  ๐‘˜ ๐œ•๐  ๐‘˜ = ๐  โˆ™ ๐ ๐‘˜ + ๐  โˆ™ ๐ ๐‘˜ โˆ’ ๐ ๐‘— โˆ™ โˆ’ ๐ ๐‘– โˆ™ 2 ๐œ•๐‘ฅ ๐‘– ๐‘— ๐œ•๐‘ฅ ๐‘— ๐‘– ๐œ•๐‘ฅ ๐‘– ๐œ•๐‘ฅ ๐‘— 1 ๐œ• ๐œ• ๐œ•๐  ๐‘– ๐œ•๐  ๐‘— = ๐ ๐‘— โˆ™ ๐ ๐‘˜ + ๐ ๐‘– โˆ™ ๐  ๐‘˜ โˆ’ ๐ ๐‘— โˆ™ ๐‘˜ โˆ’ ๐ ๐‘– โˆ™ 2 ๐œ•๐‘ฅ ๐‘– ๐œ•๐‘ฅ ๐‘— ๐œ•๐‘ฅ ๐œ•๐‘ฅ ๐‘˜ 1 ๐œ• ๐œ• ๐œ• = ๐  โˆ™ ๐ ๐‘˜ + ๐  โˆ™ ๐ ๐‘˜ โˆ’ ๐  โˆ™ ๐ ๐‘— 2 ๐œ•๐‘ฅ ๐‘– ๐‘— ๐œ•๐‘ฅ ๐‘— ๐‘– ๐œ•๐‘ฅ ๐‘˜ ๐‘– 1 ๐œ•๐‘” ๐‘—๐‘˜ ๐œ•๐‘” ๐‘˜๐‘– ๐œ•๐‘” ๐‘–๐‘— = + โˆ’Dept of Systems Engineering, ๐œ•๐‘ฅ ๐‘– 2 University of Lagos ๐œ•๐‘ฅ ๐‘— ๐œ•๐‘ฅ ๐‘˜ 77 oafak@unilag.edu.ng 12/27/2012
  78. 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, ๐œ•๐  ๐‘– ๐œ•๐  ๐‘— 1 ๐œ•๐‘” ๐‘—๐‘˜ ๐œ•๐‘” ๐‘˜๐‘– ๐œ•๐‘” ๐‘–๐‘— ๐‘–๐‘—, ๐‘˜ โ‰ก โˆ™ ๐ ๐‘˜ = โˆ™ ๐ ๐‘˜ = + โˆ’ ๐œ•๐‘ฅ ๐‘— ๐œ•๐‘ฅ ๐‘– 2 ๐œ•๐‘ฅ ๐‘– ๐œ•๐‘ฅ ๐‘— ๐œ•๐‘ฅ 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. 79. The Second Kind ๏€ช Contract the above equation with the conjugate metric tensor, we have, ๐‘˜๐›ผ ๐‘˜๐›ผ ๐œ•๐  ๐‘– ๐‘˜๐›ผ ๐œ•๐  ๐‘— ๐œ•๐  ๐‘– ๐‘˜ ๐‘” ๐‘–๐‘—, ๐›ผ โ‰ก ๐‘” โˆ™ ๐ ๐›ผ= ๐‘” โˆ™ ๐ ๐›ผ= โˆ™ ๐ ๐‘˜= ๐œ•๐‘ฅ ๐‘— ๐œ•๐‘ฅ ๐‘– ๐œ•๐‘ฅ ๐‘— ๐‘–๐‘— ๐œ•๐  ๐‘˜ =โˆ’ ๐‘— โˆ™ ๐ ๐‘– ๐œ•๐‘ฅ Which connects the common definition of the second Christoffel symbol with the one defined in the above derivation. The relationship, ๐‘” ๐‘˜๐›ผ ๐‘–๐‘—, ๐›ผ = ๐‘˜ ๐‘–๐‘— 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. 80. Two Christoffel Symbols Related We contract the above equation with ๐‘” ๐‘˜๐›ฝ and obtain, ๐‘˜๐›ผ ๐‘–๐‘—, ๐›ผ = ๐‘” ๐‘˜ ๐‘” ๐‘˜๐›ฝ ๐‘” ๐‘˜๐›ฝ ๐‘–๐‘— ๐›ผ ๐‘˜ ๐›ฟ ๐›ฝ ๐‘–๐‘—, ๐›ผ = ๐‘–๐‘—, ๐›ฝ = ๐‘” ๐‘˜๐›ฝ ๐‘–๐‘— so that, ๐›ผ ๐‘” ๐‘˜๐›ผ ๐‘–๐‘— = ๐‘–๐‘—, 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. 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“ = ๐‘‡๐‘–๐‘— ๐  ๐‘– โŠ— ๐  ๐‘— = ๐‘‡ ๐‘–๐‘— ๐  ๐‘– โจ‚๐  ๐‘— = ๐‘‡.๐‘— ๐  ๐‘– โŠ— ๐  ๐‘— = ๐‘‡๐‘—.๐‘– ๐  ๐‘— โจ‚๐  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
  82. 82. Higher Order Tensors We have: ๐œ•๐“ ๐œ•๐‘‡ ๐‘–๐‘— ๐‘–๐‘— ๐œ•๐  ๐‘– ๐‘–๐‘— ๐œ•๐  ๐‘— ๐‘˜ = ๐‘˜ ๐‘– ๐  โจ‚๐  ๐‘— + ๐‘‡ ๐‘˜ โจ‚๐  ๐‘— + ๐‘‡ ๐  ๐‘– โจ‚ ๐œ•๐‘ฅ ๐œ•๐‘ฅ ๐œ•๐‘ฅ ๐œ•๐‘ฅ ๐‘˜ ๐œ•๐  ๐‘— Recall that, ๐‘–๐‘˜ โˆ™ ๐  ๐‘— = . It follows therefore that, ๐œ•๐‘ฅ ๐‘–๐‘˜ ๐œ•๐  ๐‘– ๐‘—โˆ’ ๐›ผ ๐›ฟ๐‘— = ๐œ•๐  ๐‘– ๐›ผ โˆ™ ๐  โˆ™ ๐ ๐‘— โˆ’ ๐  ๐›ผ โˆ™ ๐ ๐‘— ๐œ•๐‘ฅ ๐‘˜ ๐‘–๐‘˜ ๐›ผ ๐œ•๐‘ฅ ๐‘˜ ๐‘–๐‘˜ ๐œ•๐  ๐‘– ๐›ผ = โˆ’ ๐  ๐›ผ โˆ™ ๐  ๐‘— = 0. ๐œ•๐‘ฅ ๐‘˜ ๐‘–๐‘˜ ๐œ•๐  ๐‘– ๐›ผ Clearly, ๐‘˜ = ๐ ๐›ผ ๐œ•๐‘ฅ ๐‘–๐‘˜ (Obviously since ๐  ๐‘— is a basis vector it cannot vanish)Dept of Systems Engineering, University of Lagos 82 oafak@unilag.edu.ng 12/27/2012
  83. 83. Higher Order Tensors ๐œ•๐“ ๐œ•๐‘‡ ๐‘–๐‘— ๐‘–๐‘— ๐œ•๐  ๐‘– ๐‘–๐‘— ๐  โจ‚ ๐œ•๐  ๐‘— ๐œ•๐‘ฅ ๐‘˜ = ๐œ•๐‘ฅ ๐‘˜ ๐  ๐‘– โจ‚๐  ๐‘— + ๐‘‡ ๐œ•๐‘ฅ ๐‘˜ โจ‚๐  ๐‘— + ๐‘‡ ๐‘– ๐œ•๐‘ฅ ๐‘˜ ๐œ•๐‘‡ ๐‘–๐‘— ๐‘–๐‘— ๐›ผ ๐‘–๐‘— ๐  โจ‚ ๐›ผ = ๐  โจ‚๐  ๐‘— + ๐‘‡ ๐  โจ‚๐  ๐‘— + ๐‘‡ ๐‘– ๐‘—๐‘˜ ๐  ๐›ผ ๐œ•๐‘ฅ ๐‘˜ ๐‘– ๐‘–๐‘˜ ๐›ผ ๐œ•๐‘‡ ๐‘–๐‘— ๐‘– ๐‘— = ๐‘˜ ๐  ๐‘– โจ‚๐  ๐‘— + ๐‘‡ ๐›ผ๐‘— ๐  ๐‘– โจ‚๐  ๐‘— + ๐‘‡ ๐‘–๐›ผ ๐  ๐‘– โจ‚ ๐ ๐‘— ๐œ•๐‘ฅ ๐›ผ๐‘˜ ๐›ผ๐‘˜ ๐œ•๐‘‡ ๐‘–๐‘— ๐‘– ๐‘— ๐‘–๐‘— = +๐‘‡ ๐›ผ๐‘— + ๐‘‡ ๐‘–๐›ผ ๐  ๐‘– โจ‚๐  ๐‘— = ๐‘‡ ,๐‘˜ ๐  ๐‘– โจ‚๐  ๐‘— ๐œ•๐‘ฅ ๐‘˜ ๐›ผ๐‘˜ ๐›ผ๐‘˜ Where ๐œ•๐‘‡ ๐‘–๐‘— ๐‘–๐‘— ๐‘– ๐‘— ๐œ•๐‘‡ ๐‘–๐‘— ๐‘– ๐‘— = ๐‘‡,๐‘˜ +๐‘‡ ๐›ผ๐‘— + ๐‘‡ ๐‘–๐›ผ or +๐‘‡ ๐›ผ๐‘— ฮ“ ๐›ผ๐‘˜ + ๐‘‡ ๐‘–๐›ผ ฮ“ ๐›ผ๐‘˜ ๐œ•๐‘ฅ ๐‘˜ ๐›ผ๐‘˜ ๐›ผ๐‘˜ ๐œ•๐‘ฅ ๐‘˜ are the components of the covariant derivative of the tensor ๐“ in terms of contravariant components on the product covariant bases as shown.Dept of Systems Engineering, University of Lagos 83 oafak@unilag.edu.ng 12/27/2012

ร—