The Contravariant Derivative of a Contravariant Vector
The Dual Christoffel Symbols (2nd Kind)
The Dual Christoffel Symbols (1st Kind)
The Contravariant Derivative of a Covariant Vector
A Convenient Notation for All the Christoffel Symbols
An Operator Dual to ∂
The Minus Sign
The Curl of a Vector
Beyond personal compulsion and a desire for symmetry, my motivation for pursuing this work is the following: In some problems in classical physics, such as magnetohydrodynamics, we may have the rectangular cartesian coordinates as mathematically well defined functions of general curvilinear coordinates. In this circumstance, it seems likely that we can calculate to arbitrary precision the covariant components of the metric tensor. This would enable us, if the need exists, to compute accurately the Christoffel symbols and form covariant derivatives. Because of the generally intractible nature of the inverse coordinate transformation - which gives the curvilinear coordinates as functions of the Cartesians - the calculation of the contravariant metric tensor components would be accomplished by inverting the matrix containing the covariant components. This approach does not involve any need to differentiate the contravariant metric tensor components with respect to the coordinate variables. If, however, the curvilinear coordinates are given as mathematically well defined functions of the Cartesians, then the contravariant metric tensor components are the ones we can accurately calculate and it would seem desirable to avoid differentaiting the covariant components - which is required to get the Christoffel symbols. Therefore, the dual approach seems desirable in which the contravariant metric tensor is fundamental, and in which we use dual Christoffel symbols formed with the contravariant metric tensor components, and in which we use the contravariant derivative.
Usefully we can use tensor notation with mathematical objects that are not tensors. This investigation has led me to adopt the following notation, which is an example of the idea expressed in the previous sentence. (Note that repeated indices are summed from 1 to a positive integer n which is greater than or equal to 2. The superscripted variables x are general curvilinear coordinates and the superscripted g gives the contravariant metric tensor components in the X coordinate system.)
Before continuing, it seems now desirable to quote Levi-Civita [1, page 149]:
"4. Contravariant differentiation. There is in the absolute differential calculus a kind of law of reciprocity or duality in accordance with which we can deduce from every theorem or formula a reciprocal theorem or formula, by interchanging the words covariant and contravariant, and lowering or raising the indices. We have already had several examples of this
and my main example consists of the two formulas for the curl of a vector
]; we shall now make some brief remarks on the operation of contravariant differentiation, which corresponds to that of covariant differentiation just described.
The shortest way to deduce from a system the system , which has the properties reciprocal to those of the covariant derivatives, is to find the covariant derivative of the given system and then to compound it with the system of the 's [i.e. compound it with the contravariant metric tensor components]; to make
We could find for this system an expression analogous to (4) [which is the general definition for the covariant derivative] and properties corresponding exactly to those of the covariant derivatives; or we could find these properties directly from those of the covariant derivatives, by using the foregoing formula of definition. We shall therefore not pursue the argument in detail, and shall instead resume our discussion of the fundamental properties of covariant differentiation."
Using Levi-Civita's definition, we have
where is the contravariant derivative of the vector whose contravariant components are , where the superscripted g's are the contravariant metric tensor components, where is the covariant derivative of the vector whose contravariant components are , and where
are the Christoffel symbols of the second and first kind respectively. (Note that the three bar equal sign indicates definition. The subscripted g's are the covariant metric tensor components. The last expression in the equation for the first kind Christoffel symbol, [αγ, β] , is given to emphasize the memory device for writing down one symbol in terms of the other. It appears to work like the raising or lowering of indices on tensors. One of the memory devices I use for the first kind symbol involves the fact that the negative term involves the first two indices - α and γ in this case. I have found that all the memory devices that are useful in the conventional formulation carry over into the dual formulation.) Distributing the inside the parenthesis in the above equation for , and using the definition from Eq_1 in Eq_2 and lowering the index on F, we get
where the subscripted T is simply an abbreviation for its corresponding expression - which will now be investigated. We have
Now we distribute the contravariant metric tensor terms into the parenthesis and use the following identity (which is appropriate to the first term within the parenthesis and which uses the Kronecker delta, ) to move the differentiation from the covariant metric tensor components to contravariant metric tensor components. Note from the following identity that, when there is a summation involved as there is here, the derivative is simply transferred to the other term with a change of sign. Thus we obtain the result shown in Eq_7.
If the middle term in the last expression above had a positive rather than a negative sign, this result would have the form of the Christoffel symbol of the first kind, [ij,k]. Consequently, in Eq_8 below, the appropriate quantity is added and subtracted.
The last line in Eq_8 gives two possible definitions for the square bracketed element in which the indices are in line with the top of the square brackets. The definition with the plus sign is the one I chose on first obtaining these equations. The one with the negative sign was chosen later, when it became obvious that the negative sign should be included. Therefore, on a first reading the boxed signs should be ignored. Then, when you look back after seeing the full development, incorporate the boxed signs instead of the unboxed signs. It is often the case that minus signs are included in definitions in a way that seems to have no reason. This work provides an illustration of how such signs can come to be included. Using the product rule for differentiation, we now have the following formula:
Substituting Eq_9 into Eq_4, we obtain
which seems to be dual to
At first, I was expecting to be dual to . However, we see that this term cancels. Also, in the first development (without the boxed signs), it did not seem unexpected that there should be a change of sign. This is because the covariant derivative of a covariant vector (see Eq_11) has a negative sign in front of the Christoffel symbol term, while the covariant derivative of a contravariant vector has a plus sign in front of the Christoffel symbol term. This lead me to expect the plus sign in front of the Christoffel symbol term in the expression for the contravariant derivative of a contravariant vector. I was aware however that the rule might be that a negative sign is used if the two indices are on the same level (both being subscripts or both being superscripts), and a positive sign is used if the two indices are on different levels. As a result of the remainder of this work, it appears that the latter of these two possibilities is the one that is true.
In the development of the previous section, the following symbols were introduced:
These are obvious candidates to be dual to the Christoffel symbols (2nd and 1st kind):
It will now be shown that the candidate duals (of Eq_12) are obtained from the original symbols (of Eq_13) by a complete interchange of the "levels" of the indices. That is, every index which starts out "upstairs" goes "downstairs" and vice versa. In the symbol [ij,k], all three indices are to be regarded as being "downstairs". We have
We now change the form of this equation using the identities which are derived below. First, we have
This formula can be easily recalled by viewing the indices in as a "V". The two Christoffel symbols of the first kind are then written down by traversing the V twice - first forward from the i to the k to the j, and then in reverse. Second, we derive an analogous equation for the derivatives of the contravariant metric tensor components in terms of the Christoffel symbols of the second kind as follows:
From this last equation, we obtain n new equations by multiplying by . Adding these n new equations together, we get
Now making the index changes β→i and i→α we get
This formula can be recalled by remembering its general form (with two minus signs, two g's, and two Christoffel symbols of the second kind) combined with the following: (1) There is an "upstairs-downstairs" summation (on α here) between the metric tensor components and the Christoffel symbols. (2) The index which is "downstairs" in the partial derivative (k here) stays down stairs in the Christoffel symbols. (3) The remaining indices in the partial (i and j here) are placed in the remaining places - first in their original order and then in interchanged order.
Using Eq_18 to eliminate the partial derivatives of the contravariant metric tensor components in Eq_14 we get
The three line expression in parenthesis at the end of Eq_19 will be referred to by line number. Note that each line contains two terms - after and respectively are distributed within the square brackets in the first two lines. If, in the first term of the second line, we interchange the dummy summation indices β and γ and use the symmetry of the second kind Christoffel symbols with respect to the two "downstairs" indices, then the first terms in the first two lines are seen to be equal. Thus, we obtain
The second term on the first line is now seen to cancel with the second term on the third line, and the first terms on the second and third line also cancel. Consequently, we get
After interchanging the indices α and γ, we obtain
This formula shows that, apart from the issue of the sign, the new "top heavy" symbol is obtained from the old "bottom heavy" one by raising and lowering respectively all the indices that are "downstairs" or "upstairs". Similarly, we can quickly show that the old "bottom heavy" symbol is obtained from the new "top heavy" one by lowering or raising respectively all the indices which are "upstairs" or "downstairs":
Making the index changes ρ→i, σ→j, τ→k, i→α, j→β, k→γ in Eq_23, and repeating Eq_22 for summary, we have
That these symbols are duals of each other now seems obvious.
We have, repeating the definition that was made in Eq_8 and using the identity shown in Eq_18,
The first terms on the first two lines add and the remaining terms cancel. Therefore, we get
Or, using the symmetry of the 1st kind Christoffel symbols with respect to the first two indices, we have
We can invert Eq_27 as follows:
Now make the index changes i→α, j→β, k→γ, ρ→i, σ→j, τ→k in Eq_28 and repeating Eq_27 we have
The duality of these symbols now seems obvious.
The contravarian derivative of a covariant vector is dual to
and therefore I guess that
Using Levi-Civita's definition of the contravariant derivative (see the Introduction), we have
If we use Eq_29 to replace the Christoffel symbol [β i, k] with its dual, we will obtain
Using the identity of Eq_15 and the definition (see Eq_12) of the dual Christoffel symbol of the second kind we get
and the guess made in Eq_31 is proven correct. The two equations for contravariant derivatives are sumarized in Eq_35 below along with their more conventional duals.
The lack of symmetry in the above equations prompted me to contemplate an operator dual to the ordinary partial differentiation symbol, ∂. My choice is presented in the section after the next one. The next section introduces a single notation for all four of the Christoffel symbols. Beyond the desirability of using a single notation for four objects, this new notation also encorporates two standardly used features in the tensor calculus. Additionally, the new notation is better with a view to application programming.
Sokolnikoff [2, page 79] points out that an important alternative notation - generally used by "followers of the Princeton school" - for the Christoffel symbol of the second kind is for . Furthermore, when they present the process of raising and lowering indices on tensors in a Riemannian space, Synge and Schild [3, page 31] establish the following convention: "We shall in future refrain from writing a subscript and a superscript on the same vertical line; in vacant spaces we shall write dots, thus: ." Since the four Christoffel symbols that appear in this work are interrelated through the raising and lowering of indices using the metric tensor components, and in view of the "Princeton school" use of the upper case gamma, the following notation seems highly desirable:
1st kind Christoffel symbol:
2nd kind Christoffel symbol:
Dual 2nd kind Christoffel symbol:
Dual 1st kind Christoffel symbol:
This notation has an added advantage for me with respect to programming. In programming, there is no provision for subscripts and superscripts and all three indices are always on one line. The different symbols are then distinguished using variable names - such as cs1[i,j,k], cs2[i,j,k], dualCS1[i,j,k], and dualCS2[i,j,k]. In the above notation, note that the lone index is always the third one, and note that the symmetrical indices are always the first two.
I will provide an example in which we can see how this notation conveniently encompases objects that are closely related to but different from all of the above symbols. In passing from the third to the fourth line of Eq_34 above, the following equality was asserted:
In the new notation, we would have
If we perform the "raising" and "lowering" of indices on both sides, we get
which strongly suggests that the answer to the question mark is in the affirmative. That this is the case is seen if we substitute Eq_24 in Eq_37 (I ignore the issue of sign choice for this calculation):
Finally, the contravariant and covariant derivatives using the new notation are as follows:
When I contemplated what symbol should be used for an operator dual to ∂, the introduction of which would give Eq_41 the complete symmetry of duality, I settled on turning ∂ upside down, and the numeral nine, seems to be a passable likeness: 9. Since the symbol, ∂, already has the well established name, "partial", I will continue to use that name. To choose a name for the dual operator, 9, I was guided by the following thoughts: Ordinary partial differentiation, when applied to a tensor of rank greater than or equal to one, "almost" produces an additional index of covariance. Similarly, the operator, 9, "almost" produces an additional index of contravariance. Therefore, I will refer to the operator, 9, by the name "qc-partial", which is short for "quasi-contravariant partial differentiation". Using the formulas of Eq_41 to define the operator, we have
The following table illustrates how one can easily write down the various expressions involving this new operator. The table shows three steps in writing down the formulas which connect the two partial differential opperators. Thus, obviously, it is not intended that the three expressions in each row should be written down, but that the second column entry would be written down, and then the formula would be filled in as shown in the last two columns. The partial derivative parts, for example and in row one, at the two ends of each row are essentially duals of each other. Therefore, every index that is downstairs on the right must be raised and every index that is upstairs on the right must be lowered. Thus, the number of g's involved must be equal to the total number of indices involved. In this case, there is one index on the vector component and one index on the coordinate variable with respect to which the differentiation is done. The first metric tensor to be "filled in" is the one associated with the coordinate variable. The index on this variable is always written as a superscript, but it is to be regarded as being "downstairs" if the differentiation is ordinary - because ordinary partial differentiation "almost" produces an index of covariance - and it is to be regarded as being "upstairs" if the differentiation is "qc" - because qc-partial differentiation "almost" produces an index of contravariance. The indices on the metric tensor, g, are placed as superscripts or subscripts so as to effect the required "change of floor". Each remaining index on the right has its corresponding "on the other floor" index on the left and an associated g on the right. Again, indices are placed on the g so as to effect the required "raising" or "lowering".
Finally, note that the first and last rows in Eq_43 are duals of one another - as are the middle two rows.
In Eq_8, the dual Christoffel symbol was introduced, and, in Eq_12, its definition was summarized as shown in Eq_44 immediately below.
I will now derive the equation which is dual to Eq_44, which involves the original Christoffel symbol of the first kind and qc-partial differentiation, 9. Following the procedure described in the last section, we write
Now lower the indices i, j, and k - and refer to Eq_29, which is essentially repeated immediately below as Eq_46:
Now repeating the second line of Eq_44, we had
and we see that if we include the minus sign in the original definition of the dual Christoffel symbol of the first kind, then we have duality without a change of sign. That this is what we want, is seen in the following derivation in which we begin with the definition of the original Christoffel symbol of the first kind:
Now raise the indices i, j, and k:
And now it seems obvious that we should include the negative sign in the original definition of the dual Christoffel symbol of the first kind in Eq_8. With this done, the equations are summarized in the next section.
Contravariant and Covariant Derivatives
Christoffel Symbols - Defined
Christoffel Symbols - Relationship
In terms of the covariant components of a vector in three dimensions, the contravariant components of its curl are given by
where g is the determinant of the matrix containing covariant metric tensor components, where is what Mathematica calls the Signature function - which is equal to 1 if ijk are an even permutation of 123, equal to -1 if ijk are an odd permutation of 123, and equal to 0 otherwise, and where is the covariant derivative. If we substitute into Eq_53 the last of the formulas in Eq_50, the Christoffel symbols conveniently cancel and we obtain
where ijk are 123, 231, or 312.
My interest in duality prompted me to surmise that the covariant components of the curl are given by
Using Levi-Civita's definition of the contravariant derivative (see the Introduction), I was able to verify Eq_55 as follows:
Thus we have
Then, letting G be the determinant of the contravariant metric tensor component matrix, the question reduces to the following:
and the answer is yes. Thus, we have the dual equations
Before doing this work, I was not able to find for the second formula in Eq_59 any reasonably simple expression comparable to Eq_54. With the aid of this work, however, I have been able to proceed as follows:
where ijk are 123, 231, or 312. But, using the first formula in Eq_50, we have
where ijk are 123, 231, or 312. Using the qc-partial differentiation operator, 9, we obtain the following completely dual expressions:
If we carry the concept of duality to its logical conclusion, it would seem to be desirable to explore the possibility of developing algorithms for the direct performance of qc-partial differentiation. I would begin with the observation that the qc-partial derivative of a scalar function, φ, is the following:
This gives simply the contravariant components of the gradient of φ, which suggests that qc-partial differentiation in some sense involves differentiation combined with a switch of basis vectors from the ones that are tangent to the coordinate curves to the ones that are normal to the coordinate surfaces. I leave this subject for a future date.
[*] As presented in the introduction, Levi-Civita makes a few remarks on the subject of duality in his 1923 book on the Absolute Differential Calculus. The subject is not treated in any of the books that I have, but certainly - in the 100+ years since Ricci and Levi-Civita published their exposition (G. Ricci & T. Levi-Civita, Methodes du calcul differentiel absolu et leurs applications, Mathematische Annalen, 54 (1900) 125-201.) somebody has explored the subject in detail. Since my circumstances do not permit me to do library searches, I would like to hear from anyone who knows what has been done on this subject. I have done an internet search on "contravariant derivative". The term is used in special relativity, but that application seems to be the special case in which the coordinates are components of a vector - which is not true in the general case which is of interest to me. There was one interesting looking item in connection to Poisson manifolds and a contravariant derivative operator, but details were not available. In any case, I don't know topology so the details would no doubt have been unintelligible to me. Of more use to me would be a source contemporary to Levi-Civita or work done in the first half of the twentieth century.
 The Absolute Differential Calculus, Dover, 1977 unabridged and unaltered republication of the English translation by Marjorie Long, first published by Blackie & Son Limited, London and Glasgow, in 1926.
 I. S. Sokolnikoff, Tensor Analysis Theory and Applications to Geometry and Mechanics of Continua, Wiley, 2nd Edition, 1964.
 J. L. Synge & A. Schild, Tensor Calculus, Dover, 1978 unabridged republication of the work originally published by University of Toronto Press in 1949, as No. 5 in its series, Mathematical Expositions. This edition is reprinted from the corrected edition published in 1969.