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Transformation of Variants under Coordinate Changes

An invariant is a variant of order zero, i.e. a single number or single vector, that has the same value in all coordinate systems. Since an invariant is independent of coordinates, it must represent a meaningful geometric quantity. It is for this reason that producing an invariant is the ultimate goal of any analysis. Crucial to constructing invariants is the concept of a tensor which we will introduce over the course of this and the next Chapters. Tensors are variants of any order whose values in different coordinate systems are related by a special set of relationships known as the tensor transformation rules. Tensors are closed under addition, multiplication, and contraction. In other words, a sum of tensors is also a tensor, a product of tensors is also a tensor, and a contraction of a tensor is also a tensor. Crucially, a tensor of order zero is guaranteed to be an invariant. Thus, although tensors have different values in different coordinate systems, and therefore contain within them artifacts of the coordinate system, they may be thought of as preserving the geometric meaning of the objects that they represent in the coordinate space.
This Chapter is devoted to the analysis of the transformation rules, i.e. the relationships between the values of a variant in two different coordinate systems, for the most fundamental variants in a Euclidean space. This will lay the necessary groundwork for the exploration of tensors in the next Chapter.
In a three-dimensional Euclidean space, simultaneously introduce two coordinate systems Z1,Z2,Z3Z^{1},Z^{2},Z^{3} and Z1′,Z2′,Z3′Z^{1^{\prime}},Z^{2^{\prime} },Z^{3^{\prime}} or, collectively, ZiZ^{i} and Zi′Z^{i^{\prime}}. We will refer to these coordinates as unprimed and primed. You may be surprised to see the prime placed on the index ii rather than the letter ZZ. However, you will soon see the tremendous utility of this style, as it will allow us to use every letter twice -- once for the unprimed coordinate system and once for the primed. It will also enable us to treat the two coordinate systems in a more parallel fashion.
Although the two coordinate systems play equal roles, we will tend to think of the unprimed coordinates ZiZ^{i} as the "original" coordinate system and the primed coordinates Zi′Z^{i^{\prime}} as the "new" coordinate system. We will therefore tend to highlight the transformation from the unprimed to the primed coordinates even though the inverse transformation is equally relevant.
With each point PP in the Euclidean space we have associated two sets of coordinates: (Z1,Z2,Z3)\left( Z^{1},Z^{2},Z^{3}\right) and (Z1′,Z2′,Z3′)\left( Z^{1^{\prime} },Z^{2^{\prime}},Z^{3^{\prime}}\right) . This correspondence creates a natural analytical relationship between the two coordinate systems: with each set of unprimed coordinates (Z1,Z2,Z3)\left( Z^{1},Z^{2},Z^{3}\right) we can associate the set of primed coordinates (Z1′,Z2′,Z3′)\left( Z^{1^{\prime}},Z^{2^{\prime} },Z^{3^{\prime}}\right) that correspond to the same point PP. This relationship can be described by a set of three functions F1′F^{1^{\prime}}, F2′F^{2^{\prime}}, and F3′F^{3^{\prime}}, i.e.
Z1′=F1′(Z1,Z2,Z3)          (13.1)Z2′=F2′(Z1,Z2,Z3)          (13.2)Z3′=F3′(Z1,Z2,Z3).          (13.3)\begin{aligned}Z^{1^{\prime}} & =F^{1^{\prime}}\left( Z^{1},Z^{2},Z^{3}\right)\ \ \ \ \ \ \ \ \ \ \left(13.1\right)\\Z^{2^{\prime}} & =F^{2^{\prime}}\left( Z^{1},Z^{2},Z^{3}\right)\ \ \ \ \ \ \ \ \ \ \left(13.2\right)\\Z^{3^{\prime}} & =F^{3^{\prime}}\left( Z^{1},Z^{2},Z^{3}\right) .\ \ \ \ \ \ \ \ \ \ \left(13.3\right)\end{aligned}
The inverse relationship can likewise be described by a set of three functions F1F^{1}, F2F^{2}, and F3F^{3}, i.e.
Z1=F1(Z1′,Z2′,Z3′)          (13.4)Z2=F2(Z1′,Z2′,Z3′)          (13.5)Z3=F3(Z1′,Z2′,Z3′).          (13.6)\begin{aligned}Z^{1} & =F^{1}\left( Z^{1^{\prime}},Z^{2^{\prime}},Z^{3^{\prime}}\right)\ \ \ \ \ \ \ \ \ \ \left(13.4\right)\\Z^{2} & =F^{2}\left( Z^{1^{\prime}},Z^{2^{\prime}},Z^{3^{\prime}}\right)\ \ \ \ \ \ \ \ \ \ \left(13.5\right)\\Z^{3} & =F^{3}\left( Z^{1^{\prime}},Z^{2^{\prime}},Z^{3^{\prime}}\right) .\ \ \ \ \ \ \ \ \ \ \left(13.6\right)\end{aligned}
Collectively, we refer to these relationships as the equations of the coordinate transformation.
For an example of equations of a coordinate transformation, let ZiZ^{i} be Cartesian coordinates and Zi′Z^{i^{\prime}} be spherical coordinates aligned with the Cartesian coordinates in the usual way. Then the functions F1′F^{1^{\prime}}, F2′F^{2^{\prime}}, and F3′F^{3^{\prime}} are
F1′(Z1,Z2,Z3)=(Z1)2+(Z2)2+(Z3)2          (13.7)F2′(Z1,Z2,Z3)=arctan⁡(Z3,(Z1)2+(Z2)2)          (13.8)F3′(Z1,Z2,Z3)=arctan⁡(Z1,Z2),          (13.9)\begin{aligned}F^{1^{\prime}}\left( Z^{1},Z^{2},Z^{3}\right) & =\sqrt{\left( Z^{1}\right) ^{2}+\left( Z^{2}\right) ^{2}+\left( Z^{3}\right) ^{2}}\ \ \ \ \ \ \ \ \ \ \left(13.7\right)\\F^{2^{\prime}}\left( Z^{1},Z^{2},Z^{3}\right) & =\arctan\left( Z^{3},\sqrt{\left( Z^{1}\right) ^{2}+\left( Z^{2}\right) ^{2}}\right)\ \ \ \ \ \ \ \ \ \ \left(13.8\right)\\F^{3^{\prime}}\left( Z^{1},Z^{2},Z^{3}\right) & =\arctan\left( Z^{1},Z^{2}\right) ,\ \ \ \ \ \ \ \ \ \ \left(13.9\right)\end{aligned}
while the functions F1F^{1}, F2F^{2}, and F3F^{3} are
F1(Z1′,Z2′,Z3′)=Z1′sin⁡Z2′cos⁡Z3′          (13.10)F2(Z1′,Z2′,Z3′)=Z1′sin⁡Z2′sin⁡Z3′          (13.11)F3(Z1′,Z2′,Z3′)=Z1′cos⁡Z2′.          (13.12)\begin{aligned}F^{1}\left( Z^{1^{\prime}},Z^{2^{\prime}},Z^{3^{\prime}}\right) & =Z^{1^{\prime}}\sin Z^{2^{\prime}}\cos Z^{3^{\prime}}\ \ \ \ \ \ \ \ \ \ \left(13.10\right)\\F^{2}\left( Z^{1^{\prime}},Z^{2^{\prime}},Z^{3^{\prime}}\right) & =Z^{1^{\prime}}\sin Z^{2^{\prime}}\sin Z^{3^{\prime}}\ \ \ \ \ \ \ \ \ \ \left(13.11\right)\\F^{3}\left( Z^{1^{\prime}},Z^{2^{\prime}},Z^{3^{\prime}}\right) & =Z^{1^{\prime}}\cos Z^{2^{\prime}}.\ \ \ \ \ \ \ \ \ \ \left(13.12\right)\end{aligned}
It may be helpful to see these functions in terms of the familiar variables x,y,zx,y,z and r,θ,φr,\theta,\varphi. We have
F1′(x,y,z)=x2+y2+z2          (13.13)F2′(x,y,z)=arctan⁡(z,x2+y2)          (13.14)F3′(x,y,z)=arctan⁡(x,y)          (13.15)\begin{aligned}F^{1^{\prime}}\left( x,y,z\right) & =\sqrt{x^{2}+y^{2}+z^{2}}\ \ \ \ \ \ \ \ \ \ \left(13.13\right)\\F^{2^{\prime}}\left( x,y,z\right) & =\arctan\left( z,\sqrt{x^{2}+y^{2} }\right)\ \ \ \ \ \ \ \ \ \ \left(13.14\right)\\F^{3^{\prime}}\left( x,y,z\right) & =\arctan\left( x,y\right)\ \ \ \ \ \ \ \ \ \ \left(13.15\right)\end{aligned}
and
F1(r,θ,φ)=rsin⁡θcos⁡φ          (13.16)F2(r,θ,φ)=rsin⁡θsin⁡φ          (13.17)F3(r,θ,φ)=rcos⁡θ.          (13.18)\begin{aligned}F^{1}\left( r,\theta,\varphi\right) & =r\sin\theta\cos\varphi\ \ \ \ \ \ \ \ \ \ \left(13.16\right)\\F^{2}\left( r,\theta,\varphi\right) & =r\sin\theta\sin\varphi\ \ \ \ \ \ \ \ \ \ \left(13.17\right)\\F^{3}\left( r,\theta,\varphi\right) & =r\cos\theta.\ \ \ \ \ \ \ \ \ \ \left(13.18\right)\end{aligned}
In the spirit of the tensor notation, let us switch to a more economical way of recording the relationships between the coordinate systems. Switch from the symbols F1′F^{1^{\prime}}, F2′F^{2^{\prime}}, F3′F^{3^{\prime}} and F1F^{1}, F2F^{2}, F3F^{3} to Z1′Z^{1^{\prime}}, Z2′Z^{2^{\prime}}, Z3′Z^{3^{\prime}} and Z1Z^{1}, Z2Z^{2}, Z3Z^{3} to denote the functions and enumerate them with indices. Thus, the transformation from the unprimed to the primed coordinates reads
Zi′=Zi′(Z1,Z2,Z3)(13.19)Z^{i^{\prime}}=Z^{i^{\prime}}\left( Z^{1},Z^{2},Z^{3}\right)\tag{13.19}
while the inverse transformation reads
Zi=Zi(Z1′,Z2′,Z3′).(13.20)Z^{i}=Z^{i}\left( Z^{1^{\prime}},Z^{2^{\prime}},Z^{3^{\prime}}\right) .\tag{13.20}
Furthermore, collapse the arguments of the functions resulting in the compact equations
Zi′=Zi′(Z)(13.21)Z^{i^{\prime}}=Z^{i^{\prime}}\left( Z\right)\tag{13.21}
and
Zi=Zi(Z′).(13.22)Z^{i}=Z^{i}\left( Z^{\prime}\right) .\tag{13.22}
It must be noted that the last two equations exhibit such economy of notation, that they are nearly cryptic out of context. The reader is therefore encouraged to apply the technique of unpacking until they become comfortable with this exceedingly concise notation.
Recall that in a Euclidean space referred to a coordinate system ZiZ^{i}, the covariant basis Zi\mathbf{Z}_{i} is constructed by differentiating the position vector R\mathbf{R} as a function of the coordinates ZiZ^{i} with respect to each of the coordinates, i.e.
Zi=∂R(Z)∂Zi.(9.89)\mathbf{Z}_{i}=\frac{\partial\mathbf{R}\left( Z\right) }{\partial Z^{i}}. \tag{9.89}
Similarly, in the primed coordinates Zi′Z^{i^{\prime}}, the covariant basis Zi′\mathbf{Z}_{i^{\prime}} is constructed by differentiating the position vector R\mathbf{R} as a function of the coordinates Zi′Z^{i^{\prime}} with respect to each of the coordinates:
Zi′=∂R(Z′)∂Zi′.(13.23)\mathbf{Z}_{i^{\prime}}=\frac{\partial\mathbf{R}\left( Z^{\prime}\right) }{\partial Z^{i^{\prime}}}.\tag{13.23}
Also recall that the fact that the two algorithms in the two coordinate systems are identical is what makes the covariant basis a variant.
Our present goal is to determine the transformation rule for the covariant basis Zi\mathbf{Z}_{i}, i.e. the relationship between Zi\mathbf{Z} _{i} and Zi′\mathbf{Z}_{i^{\prime}}. To this end, we must first relate the functions R(Z)\mathbf{R}\left( Z\right) and R(Z′)\mathbf{R}\left( Z^{\prime }\right) . Clearly, R(Z′)\mathbf{R}\left( Z^{\prime}\right) is obtained by substituting the equations Zi(Z′)Z^{i}\left( Z^{\prime}\right) of the coordinate transformation into R(Z)\mathbf{R}\left( Z\right) . This relationship is expressed by the identity
R(Z′)=R(Z(Z′))(13.24)\mathbf{R}\left( Z^{\prime}\right) =\mathbf{R}\left( Z\left( Z^{\prime }\right) \right)\tag{13.24}
which, in the fully unpacked form, reads:
R(Z1′,Z2′,Z3′)=R(Z1(Z1′,Z2′,Z3′), Z2(Z1′,Z2′,Z3′), Z3(Z1′,Z2′,Z3′)).(13.25)\mathbf{R}\left( Z^{1^{\prime}},Z^{2^{\prime}},Z^{3^{\prime}}\right) =\mathbf{R}\left( Z^{1}\left( Z^{1^{\prime}},Z^{2^{\prime}},Z^{3^{\prime} }\right) ,\ Z^{2}\left( Z^{1^{\prime}},Z^{2^{\prime}},Z^{3^{\prime}}\right) ,\ Z^{3}\left( Z^{1^{\prime}},Z^{2^{\prime}},Z^{3^{\prime}}\right) \right) .\tag{13.25}
It is intuitively obvious why this identity holds. The function on the right translates the primed coordinates Zi′Z^{i^{\prime}} to the equivalent unprimed coordinates ZiZ^{i} by the equations of the transformation and then uses the function R(Z1,Z2,Z3)\mathbf{R}\left( Z^{1},Z^{2},Z^{3}\right) to map the resulting unprimed coordinates to the corresponding position vector -- and that is precisely the mapping that the function R(Z1′,Z2′,Z3′)\mathbf{R}\left( Z^{1^{\prime} },Z^{2^{\prime}},Z^{3^{\prime}}\right) accomplishes in a single step.
The resulting relationship
R(Z′)=R(Z(Z′))(13.24)\mathbf{R}\left( Z^{\prime}\right) =\mathbf{R}\left( Z\left( Z^{\prime }\right) \right) \tag{13.24}
is an identity in the primed coordinates Zi′Z^{i^{\prime}} and can therefore be differentiated with respect to each coordinate. An application of the chain rule yields
∂R(Z′)∂Zi′=∂R(Z)∂Zi∂Zi(Z′)∂Zi′.(13.26)\frac{\partial\mathbf{R}\left( Z^{\prime}\right) }{\partial Z^{i^{\prime}} }=\frac{\partial\mathbf{R}\left( Z\right) }{\partial Z^{i}}\frac{\partial Z^{i}\left( Z^{\prime}\right) }{\partial Z^{i^{\prime}}}.\tag{13.26}
On the left, we immediately recognize ∂R(Z′)/∂Zi′\partial\mathbf{R}\left( Z^{\prime }\right) /\partial Z^{i^{\prime}} as Zi′\mathbf{Z}_{i^{\prime}}. On the right, ∂R(Z)/∂Zi\partial\mathbf{R}\left( Z\right) /\partial Z^{i} is Zi\mathbf{Z} _{i}. Thus, the relationship between Zi′\mathbf{Z}_{i^{\prime}} and Zi\mathbf{Z}_{i} reads
Zi′=Zi∂Zi(Z′)∂Zi′.(13.27)\mathbf{Z}_{i^{\prime}}=\mathbf{Z}_{i}\frac{\partial Z^{i}\left( Z^{\prime }\right) }{\partial Z^{i^{\prime}}}.\tag{13.27}
This identity is precisely the transformation rule for the covariant basis Zi\mathbf{Z}_{i} that we have sought. It tells us how the elements Zi′\mathbf{Z}_{i^{\prime}} of the covariant basis in the "new" coordinate system can be obtained from the elements Zi\mathbf{Z}_{i} of the covariant basis in the "original" coordinate system. Looking ahead to the concept of a tensor, this is precisely the special kind of transformation described by that term.
The inverse relationship
Zi=Zi′∂Zi′(Z)∂Zi.(13.28)\mathbf{Z}_{i}=\mathbf{Z}_{i^{\prime}}\frac{\partial Z^{i^{\prime}}\left( Z\right) }{\partial Z^{i}}.\tag{13.28}
can be obtained by applying the same approach to the inverse relationship
R(Z)=R(Z′(Z))(13.29)\mathbf{R}\left( Z\right) =\mathbf{R}\left( Z^{\prime}\left( Z\right) \right)\tag{13.29}
between the functions R(Z)\mathbf{R}\left( Z\right) and R(Z′)\mathbf{R}\left( Z^{\prime}\right) .

13.3.1The definition

The collection of the partial derivatives
∂Zi′(Z)∂Zi(13.30)\frac{\partial Z^{i^{\prime}}\left( Z\right) }{\partial Z^{i}}\tag{13.30}
is known as the Jacobian of the coordinate transformation
Zi′=Zi′(Z).(13.21)Z^{i^{\prime}}=Z^{i^{\prime}}\left( Z\right) . \tag{13.21}
It is denoted by the symbol Jii′(Z)J_{i}^{i^{\prime}}\left( Z\right) , i.e.
Jii′(Z)=∂Zi′(Z)∂Zi.(13.31)J_{i}^{i^{\prime}}\left( Z\right) =\frac{\partial Z^{i^{\prime}}\left( Z\right) }{\partial Z^{i}}.\tag{13.31}
The Jacobian of the inverse transformation
Zi=Zi(Z′)(13.22)Z^{i}=Z^{i}\left( Z^{\prime}\right) \tag{13.22}
is denoted by Ji′i(Z′)J_{i^{\prime}}^{i}\left( Z^{\prime}\right) , i.e.
Ji′i(Z′)=∂Zi(Z′)∂Zi′.(13.32)J_{i^{\prime}}^{i}\left( Z^{\prime}\right) =\frac{\partial Z^{i}\left( Z^{\prime}\right) }{\partial Z^{i^{\prime}}}.\tag{13.32}
In terms of the newly defined symbols Jii′J_{i}^{i^{\prime}} and Ji′iJ_{i^{\prime} }^{i}, the transformation rules for the covariant basis read
Zi′=ZiJi′i          (13.33)Zi=Zi′Jii′.          (13.34)\begin{aligned}\mathbf{Z}_{i^{\prime}} & =\mathbf{Z}_{i}J_{i^{\prime}}^{i}\ \ \ \ \ \ \ \ \ \ \left(13.33\right)\\\mathbf{Z}_{i} & =\mathbf{Z}_{i^{\prime}}J_{i}^{i^{\prime}}.\ \ \ \ \ \ \ \ \ \ \left(13.34\right)\end{aligned}
In Chapter 3, we discovered that two bases have the same orientation when the determinant of the matrix that relates them is positive, and opposite orientations when the determinant is negative. Therefore, the coordinate transformation preserves the orientation of the basis if the determinant of the Jacobian Ji′iJ_{i^{\prime}}^{i} is positive and reverses the orientation otherwise.

13.3.2The Jacobians for the transformation between Cartesian and polar coordinates

Since this example uses two of the most common coordinate systems, let us use the standard names x,yx,y and r,θr,\theta for the variables and the corresponding functions representing the coordinate transformations. The "forward" and the "inverse" equations of the coordinate transformation read
r(x,y)=x2+y2θ(x,y)=arctan⁡(x,y)andx(r,θ)=rcos⁡θy(r,θ)=rsin⁡θ.(13.35)\begin{array} {llll} r\left( x,y\right) =\sqrt{x^{2}+y^{2}} & & & \\ \theta\left( x,y\right) =\arctan\left( x,y\right) & & & \end{array} \text{and} \begin{array} {llll} & & & x\left( r,\theta\right) =r\cos\theta\\ & & & y\left( r,\theta\right) =r\sin\theta. \end{array}\tag{13.35}
Adopting the convention that the superscript is the first index and the subscript is the second, we find that
Jii′ corresponds to [∂r(x,y)∂x∂r(x,y)∂y∂θ(x,y)∂x∂θ(x,y)∂y]=[xx2+y2yx2+y2−yx2+y2xx2+y2](13.36)J_{i}^{i^{\prime}}\text{ corresponds to }\left[ \begin{array} {cc} \frac{\partial r\left( x,y\right) }{\partial x} & \frac{\partial r\left( x,y\right) }{\partial y}\\ \frac{\partial\theta\left( x,y\right) }{\partial x} & \frac{\partial \theta\left( x,y\right) }{\partial y} \end{array} \right] =\left[ \begin{array} {cc} \frac{x}{\sqrt{x^{2}+y^{2}}} & \frac{y}{\sqrt{x^{2}+y^{2}}}\\ -\frac{y}{x^{2}+y^{2}} & \frac{x}{x^{2}+y^{2}} \end{array} \right]\tag{13.36}
and
Ji′i corresponds to [∂x(r,θ)∂r∂x(r,θ)∂θ∂y(r,θ)∂r∂y(r,θ)∂θ]=[cos⁡θ−rsin⁡θsin⁡θrcos⁡θ].(13.37)J_{i^{\prime}}^{i}\ \text{corresponds to }\left[ \begin{array} {cc} \frac{\partial x\left( r,\theta\right) }{\partial r} & \frac{\partial x\left( r,\theta\right) }{\partial\theta}\\ \frac{\partial y\left( r,\theta\right) }{\partial r} & \frac{\partial y\left( r,\theta\right) }{\partial\theta} \end{array} \right] =\left[ \begin{array} {rr} \cos\theta & -r\sin\theta\\ \sin\theta & r\cos\theta \end{array} \right] .\tag{13.37}
Note that we have already encountered these matrices in Section 8.6 when discussing the partial derivatives of inverse sets of functions.
Also note that each Jacobian has its own natural variables. For example, Jii′J_{i}^{i^{\prime}} is naturally a function of xx and yy and Ji′iJ_{i^{\prime}}^{i} is naturally a function of rr and θ\theta. In general, Jii′J_{i}^{i^{\prime}} is naturally a function of the unprimed coordinates ZiZ^{i} and Ji′iJ_{i^{\prime}}^{i} is naturally a function of the primed coordinates Zi′Z^{i^{\prime}}. However, nothing is preventing us from converting either Jacobian to the other set of variables by substituting the equations of the coordinate transformation. In the case of the interplay between Cartesian and polar coordinates, we may state that
Jii′ corresponds to [xx2+y2yx2+y2−yx2+y2xx2+y2]=[cos⁡θsin⁡θ−sin⁡θrcos⁡θr](13.38)J_{i}^{i^{\prime}}\text{ corresponds to }\left[ \begin{array} {cc} \frac{x}{\sqrt{x^{2}+y^{2}}} & \frac{y}{\sqrt{x^{2}+y^{2}}}\\ -\frac{y}{x^{2}+y^{2}} & \frac{x}{x^{2}+y^{2}} \end{array} \right] =\left[ \begin{array} {cc} \cos\theta & \sin\theta\\ -\frac{\sin\theta}{r} & \frac{\cos\theta}{r} \end{array} \right]\tag{13.38}
and
Ji′i corresponds to [cos⁡θ−rsin⁡θsin⁡θrcos⁡θ]=[xx2+y2−yyx2+y2x].(13.39)J_{i^{\prime}}^{i}\ \text{corresponds to }\left[ \begin{array} {rr} \cos\theta & -r\sin\theta\\ \sin\theta & r\cos\theta \end{array} \right] =\left[ \begin{array} {rr} \frac{x}{\sqrt{x^{2}+y^{2}}} & -y\\ \frac{y}{\sqrt{x^{2}+y^{2}}} & x \end{array} \right] .\tag{13.39}
This observation will take on significant relevance often, including in the next Section as well as later in this Chapter when we are faced with the need to differentiate the Jacobians with respect to both natural and "unnatural" variables.
Interestingly, we have already discussed the conversion of the matrix corresponding to Ji′iJ_{i^{\prime}}^{i} to the variables xx and yy in Section 8.6. However, at that point, the substitution r=x2+y2r=\sqrt{x^{2}+y^{2}} and θ=arctan⁡(x,y)\theta=\arctan\left( x,y\right) was a formal analytical matter dictated by the chain rule. Specifically, in the chain rule
G(F(x))′=G′(F(x))F′(x),(13.40)G\left( F\left( x\right) \right) ^{\prime}=G^{\prime}\left( F\left( x\right) \right) F^{\prime}\left( x\right) ,\tag{13.40}
the derivative G′G^{\prime} is evaluated at F(x)F\left( x\right) , i.e. F(x)F\left( x\right) is substituted into G′G^{\prime}. In the present context, on the other hand, the substitution is much more intuitive: since all analytical objects are associated with the physical point at which they are being evaluated, we are free to express them in any coordinates we wish. And, if a particular object happens to be expressed in terms of rr and θ\theta, we can re-express it in terms of xx and yy by utilizing the equations of the coordinate transformation
r(x,y)=x2+y2          (13.41)θ(x,y)=arctan⁡(x,y)          (13.42)\begin{aligned}r\left( x,y\right) & =\sqrt{x^{2}+y^{2}}\ \ \ \ \ \ \ \ \ \ \left(13.41\right)\\\theta\left( x,y\right) & =\arctan\left( x,y\right)\ \ \ \ \ \ \ \ \ \ \left(13.42\right)\end{aligned}
since these questions translate the polar coordinates of a point to its Cartesian coordinates.

13.3.3Confirming the identity Zi′=ZiJi′i\mathbf{Z}_{i^{\prime}}=\mathbf{Z} _{i}J_{i^{\prime}}^{i}

As a sanity check, let us confirm that the covariant bases Zi\mathbf{Z}_{i} and Zi′\mathbf{Z}_{i^{\prime}} for Cartesian and polar coordinates are indeed related by these matrices. Denote the covariant basis in Cartesian coordinates by the familiar symbols i\mathbf{i} and j\mathbf{j}, and in polar coordinates by the symbols er\mathbf{e}_{r} and eθ\mathbf{e}_{\theta}. The identity
Zi′=ZiJi′i(13.33)\mathbf{Z}_{i^{\prime}}=\mathbf{Z}_{i}J_{i^{\prime}}^{i} \tag{13.33}
in matrix form reads
[ereθ]=[ij][cos⁡θ−rsin⁡θsin⁡θrcos⁡θ].(13.43)\left[ \begin{array} {cc} \mathbf{e}_{r} & \mathbf{e}_{\theta} \end{array} \right] =\left[ \begin{array} {cc} \mathbf{i} & \mathbf{j} \end{array} \right] \left[ \begin{array} {rr} \cos\theta & -r\sin\theta\\ \sin\theta & r\cos\theta \end{array} \right] .\tag{13.43}
In other words
er=cos⁡θ i+sin⁡θ j          (13.44)eθ=−rsin⁡θ i+rcos⁡θ j          (13.45)\begin{aligned}\mathbf{e}_{r} & =\cos\theta\ \mathbf{i}+\sin\theta\ \mathbf{j}\ \ \ \ \ \ \ \ \ \ \left(13.44\right)\\\mathbf{e}_{\theta} & =-r\sin\theta\ \mathbf{i}+r\cos\theta\ \mathbf{j}\ \ \ \ \ \ \ \ \ \ \left(13.45\right)\end{aligned}
An inspection of the geometric arrangement of the vectors i\mathbf{i}, j\mathbf{j}, eθ\mathbf{e}_{\theta}, and er\mathbf{e}_{r} confirms the correctness of the above identities:
(13.46)

13.3.4The inverse relationship between Jii′J_{i}^{i^{\prime}} and Ji′iJ_{i^{\prime}}^{i}

Consider the matrices associated with the systems Ji′iJ_{i^{\prime}}^{i} and Jii′J_{i}^{i^{\prime}} arranged in such a way that the superscripts refers to the row and the subscripts to the column. As we demonstrated in Section 8.3, matrices that represent opposite transformations between two bases are the inverses of each other, i.e.
Ji′iJji′=δji(13.47)J_{i^{\prime}}^{i}J_{j}^{i^{\prime}}=\delta_{j}^{i}\tag{13.47}
and
Jii′Jj′i=δj′i′.(13.48)J_{i}^{i^{\prime}}J_{j^{\prime}}^{i}=\delta_{j^{\prime}}^{i^{\prime}}.\tag{13.48}
Note that these relationships also follow from the discussion in Section 8.6, where we showed that the matrices of partial derivatives of inverse sets of functions are inverses of each other.
Looking ahead, the inverse relationship between the Jacobians Jii′J_{i} ^{i^{\prime}} and Ji′iJ_{i^{\prime}}^{i} will prove key to achieving invariance. In a nutshell, when a variant that transforms by Jii′J_{i} ^{i^{\prime}} is contracted with another that transforms by Ji′iJ_{i^{\prime} }^{i}, the two transformations cancel each other leading to invariance.
In order to confirm the above identities for the interplay between Cartesian and polar coordinates, the two Jacobians need to be expressed in the same variables. When expressed in the same coordinates, the two matrices can be multiplied to confirm they are the inverses of each other. For example, in polar coordinates, the identity
Ji′iJji′=δji(13.49)J_{i^{\prime}}^{i}J_{j}^{i^{\prime}}=\delta_{j}^{i}\tag{13.49}
reads
[cos⁡θ−rsin⁡θsin⁡θrcos⁡θ][cos⁡θsin⁡θ−1rsin⁡θ1rcos⁡θ]=[1001],(13.50)\left[ \begin{array} {rr} \cos\theta & -r\sin\theta\\ \sin\theta & r\cos\theta \end{array} \right] \left[ \begin{array} {rr} \cos\theta & \sin\theta\\ -\frac{1}{r}\sin\theta & \frac{1}{r}\cos\theta \end{array} \right] =\left[ \begin{array} {cc} 1 & 0\\ 0 & 1 \end{array} \right] ,\tag{13.50}
while in Cartesian coordinates, it reads
[xx2+y2−yyx2+y2x][xx2+y2yx2+y2−yx2+y2xx2+y2]=[1001].(13.51)\left[ \begin{array} {rr} \frac{x}{\sqrt{x^{2}+y^{2}}} & -y\\ \frac{y}{\sqrt{x^{2}+y^{2}}} & x \end{array} \right] \left[ \begin{array} {cc} \frac{x}{\sqrt{x^{2}+y^{2}}} & \frac{y}{\sqrt{x^{2}+y^{2}}}\\ -\frac{y}{x^{2}+y^{2}} & \frac{x}{x^{2}+y^{2}} \end{array} \right] =\left[ \begin{array} {cc} 1 & 0\\ 0 & 1 \end{array} \right] .\tag{13.51}

13.3.5The operation of inverting the Jacobian

Note that we derived the transformation rules
Zi′=ZiJi′i and          (13.33)Zi=Zi′Jii′,          (13.34)\begin{aligned}\mathbf{Z}_{i^{\prime}} & =\mathbf{Z}_{i}J_{i^{\prime}}^{i}\text{ and}\ \ \ \ \ \ \ \ \ \ \left(13.33\right)\\\mathbf{Z}_{i} & =\mathbf{Z}_{i^{\prime}}J_{i}^{i^{\prime}}, \ \ \ \ \ \ \ \ \ \ \left(13.34\right)\end{aligned}
for the covariant basis independently from each other. But, of course, given our experience with inverting the metric tensor, first introduced in Section 9.5.2 and subsequently used on a number of occasions, we recognize that the two identities above are equivalent and can be derived from each other on the basis of the matrix inverse relationship between the two Jacobians. As a refresher, let us give the details of the analytical tactic that will show the equivalence of the above identities. Naturally, it follows the Linear Algebra logic of multiplying both sides of Ax=bAx=b by A−1A^{-1} to yield x=A−1bx=A^{-1}b.
Start with the equation
Zi′=ZiJi′i(13.33)\mathbf{Z}_{i^{\prime}}=\mathbf{Z}_{i}J_{i^{\prime}}^{i} \tag{13.33}
and contract both sides with Jji′J_{j}^{i^{\prime}}, i.e.
Zi′Jji′=ZiJi′iJji′(13.52)\mathbf{Z}_{i^{\prime}}J_{j}^{i^{\prime}}=\mathbf{Z}_{i}J_{i^{\prime}} ^{i}J_{j}^{i^{\prime}}\tag{13.52}
Since Ji′iJji′=δjiJ_{i^{\prime}}^{i}J_{j}^{i^{\prime}}=\delta_{j}^{i}, we have ZiJi′iJji′=Ziδji=Zj\mathbf{Z}_{i}J_{i^{\prime}}^{i}J_{j}^{i^{\prime}}=\mathbf{Z}_{i}\delta _{j}^{i}=\mathbf{Z}_{j} and therefore
Zj=Zi′Jji′.(13.53)\mathbf{Z}_{j}=\mathbf{Z}_{i^{\prime}}J_{j}^{i^{\prime}}.\tag{13.53}
Renaming jj into ii, we arrive at the inverse identity
Zi=Zi′Jii′,(13.34)\mathbf{Z}_{i}=\mathbf{Z}_{i^{\prime}}J_{i}^{i^{\prime}}, \tag{13.34}
as we set out to show.
Of course, the same approach applies to any variant TiT_{i}, i.e.
Ti′=TiJi′i    implies    Ti=Ti′Jii′(13.54)T_{i^{\prime}}=T_{i}J_{i^{\prime}}^{i}\text{ \ \ \ implies\ \ \ \ } T_{i}=T_{i^{\prime}}J_{i}^{i^{\prime}}\tag{13.54}
Naturally, the operation can be reversed leading to the conclusion that
Ti=Ti′Jii′    implies    Ti′=TiJi′i.(13.55)T_{i}=T_{i^{\prime}}J_{i}^{i^{\prime}}\text{ \ \ \ implies\ \ \ \ } T_{i^{\prime}}=T_{i}J_{i^{\prime}}^{i}.\tag{13.55}
We will refer to this type of operation as inverting the Jacobian.
One of the consequences of our ability to invert the Jacobians is that for some variants -- namely, tensors -- we will only need to formulate the transformation rules from the original to the new coordinates since the inverse transformation follows easily.
From the transformation rule for the covariant basis Zi\mathbf{Z}_{i}, we can easily derive the transformation rule for the metric tensor ZijZ_{ij}. In the primed coordinates, the expression for Zi′j′Z_{i^{\prime}j^{\prime}} reads
Zi′j′=Zi′⋅Zj′.(13.56)Z_{i^{\prime}j^{\prime}}=\mathbf{Z}_{i^{\prime}}\cdot\mathbf{Z}_{j^{\prime}}.\tag{13.56}
Dotting the identities
Zi′=ZiJi′i  and  Zj′=ZjJj′j,(13.57)\mathbf{Z}_{i^{\prime}}=\mathbf{Z}_{i}J_{i^{\prime}}^{i}\text{ \ and \ }\mathbf{Z}_{j^{\prime}}=\mathbf{Z}_{j}J_{j^{\prime}}^{j},\tag{13.57}
we find
Zi′⋅Zj′=Zi⋅ZjJi′iJj′j.(13.58)\mathbf{Z}_{i^{\prime}}\cdot\mathbf{Z}_{j^{\prime}}=\mathbf{Z}_{i} \cdot\mathbf{Z}_{j}J_{i^{\prime}}^{i}J_{j^{\prime}}^{j}.\tag{13.58}
The dot product on the left yields Zi′j′Z_{i^{\prime}j^{\prime}} and the one on the right yields ZijZ_{ij}. Therefore, we arrive at the following rule for the transformation of the metric tensor ZijZ_{ij}
Zi′j′=ZijJi′iJj′j.(13.59)Z_{i^{\prime}j^{\prime}}=Z_{ij}J_{i^{\prime}}^{i}J_{j^{\prime}}^{j}.\tag{13.59}
The inverse transformation rule, of course, reads
Zij=Zi′j′Jii′Jjj′.(13.60)Z_{ij}=Z_{i^{\prime}j^{\prime}}J_{i}^{i^{\prime}}J_{j}^{j^{\prime}}.\tag{13.60}
In these identities, we begin to see the tendency of each index to be contracted with its own Jacobian in the coordinate transformation rule. We also see that the placements of indices, which were determined by somewhat informal (albeit intuitive) rules in our earlier explorations effectively predict the manner in which the corresponding variants transform. Generally speaking, we will observe that variants with subscripts change by Ji′iJ_{i^{\prime}}^{i} while variants with superscripts change by Jii′J_{i} ^{i^{\prime}}. Of course, this is by design: the rules for placing indices were specifically formulated so that those placements signal the correct transformation rule.
To close out this Section, let us confirm the rule
Zi′j′=ZijJi′iJj′j(13.59)Z_{i^{\prime}j^{\prime}}=Z_{ij}J_{i^{\prime}}^{i}J_{j^{\prime}}^{j} \tag{13.59}
for the Cartesian (unprimed) to polar (primed) transformation. Recall that
Zij corresponds to [1001],          (13.61)Zi′j′ corresponds to [100r2],          (13.62)Ji′i corresponds to [cos⁡θ−rsin⁡θsin⁡θrcos⁡θ].          (13.63)\begin{aligned}& Z_{ij}\text{ corresponds to }\left[ \begin{array} {cc} 1 & 0\\ 0 & 1 \end{array} \right] ,\ \ \ \ \ \ \ \ \ \ \left(13.61\right)\\& Z_{i^{\prime}j^{\prime}}\text{ corresponds to }\left[ \begin{array} {cc} 1 & 0\\ 0 & r^{2} \end{array} \right] ,\ \ \ \ \ \ \ \ \ \ \left(13.62\right)\\& J_{i^{\prime}}^{i}\text{ corresponds to }\left[ \begin{array} {rr} \cos\theta & -r\sin\theta\\ \sin\theta & r\cos\theta \end{array} \right] .\ \ \ \ \ \ \ \ \ \ \left(13.63\right)\end{aligned}
Thus, the identity
Zi′j′=ZijJi′iJj′j(13.59)Z_{i^{\prime}j^{\prime}}=Z_{ij}J_{i^{\prime}}^{i}J_{j^{\prime}}^{j} \tag{13.59}
corresponds to
[100r2]=[cos⁡θ−rsin⁡θsin⁡θrcos⁡θ]T[1001][cos⁡θ−rsin⁡θsin⁡θrcos⁡θ],(13.64)\left[ \begin{array} {cc} 1 & 0\\ 0 & r^{2} \end{array} \right] =\left[ \begin{array} {rr} \cos\theta & -r\sin\theta\\ \sin\theta & r\cos\theta \end{array} \right] ^{T}\left[ \begin{array} {cc} 1 & 0\\ 0 & 1 \end{array} \right] \left[ \begin{array} {rr} \cos\theta & -r\sin\theta\\ \sin\theta & r\cos\theta \end{array} \right] ,\tag{13.64}
which is easily confirmed to be true.
The derivation of the transformation rule for the contravariant coordinates UiU^{i} of a vector U\mathbf{U} is particularly satisfying. By definition, the components UiU^{i} are the decomposition coefficients of U\mathbf{U} with respect to Zi\mathbf{Z}_{i}, i.e.
U=UiZi.(13.65)\mathbf{U}=U^{i}\mathbf{Z}_{i}.\tag{13.65}
In the primed coordinate system, the components Ui′U^{i\prime} are, similarly, the coefficients of U\mathbf{U} with respect to Zi′\mathbf{Z}_{i^{\prime}}, i.e.
U=Ui′Zi′.(13.66)\mathbf{U}=U^{i^{\prime}}\mathbf{Z}_{i^{\prime}}.\tag{13.66}
The key to the relationship between UiU^{i} and Ui′U^{i^{\prime}} is, of course, that U\mathbf{U} is the same vector in both equations. Therefore, we can equate the two right sides of those equations, i.e.
Ui′Zi′=UiZi.(13.67)U^{i^{\prime}}\mathbf{Z}_{i^{\prime}}=U^{i}\mathbf{Z}_{i}.\tag{13.67}
Next, substitute the transformation rule Zi=Zi′Jii′\mathbf{Z}_{i}=\mathbf{Z} _{i^{\prime}}J_{i}^{i^{\prime}} for Zi\mathbf{Z}_{i} on the right:
Ui′Zi′=UiJii′Zi′.(13.68)U^{i^{\prime}}\mathbf{Z}_{i^{\prime}}=U^{i}J_{i}^{i^{\prime}}\mathbf{Z} _{i^{\prime}}.\tag{13.68}
Both sides of this identity represent linear decomposition with respect to the basis Zi′\mathbf{Z}_{i^{\prime}}. Matching up the coefficients, we arrive at the desired transformation rule for UiU^{i}:
Ui′=UiJii′.(13.69)U^{i^{\prime}}=U^{i}J_{i}^{i^{\prime}}.\tag{13.69}
Note, crucially, that the components UiU^{i} transform by an opposite rule compared to the basis Zi\mathbf{Z}_{i}. Indeed, Zi\mathbf{Z}_{i} transforms by Ji′iJ_{i^{\prime}}^{i}, while UiU^{i} transforms by Jii′J_{i}^{i^{\prime}}, as we observe by inspecting the two transformation rules side by side:
Zi′=ZiJi′i          (13.34)Ui′=UiJii′.          (13.69)\begin{aligned}\mathbf{Z}_{i^{\prime}} & =\mathbf{Z}_{i}J_{i^{\prime}}^{i}\ \ \ \ \ \ \ \ \ \ \left(13.34\right)\\U^{i^{\prime}} & =U^{i}J_{i}^{i^{\prime}}. \ \ \ \ \ \ \ \ \ \ \left(13.69\right)\end{aligned}
These two modes of transformation are called covariant -- i.e. changing in the same manner as the basis -- and contravariant -- i.e. changing in the opposite way. When we combine the two types of objects in a contraction, the inverse matrices meet and cancel each other, leading to invariance.
To illustrate this point further, consider the combination
A=UiZi.(13.70)\mathbf{A}=U^{i}\mathbf{Z}_{i}.\tag{13.70}
In primed coordinates, the corresponding variant A′\mathbf{A}^{\prime} is
A′=Ui′Zi′.(13.71)\mathbf{A}^{\prime}=U^{i^{\prime}}\mathbf{Z}_{i^{\prime}}.\tag{13.71}
Using the transformation rules above, we find that
A′=UiJii′ZjJi′j.(13.72)\mathbf{A}^{\prime}=U^{i}J_{i}^{i^{\prime}}\mathbf{Z}_{j}J_{i^{\prime}}^{j}.\tag{13.72}
Note that we had to use the letter jj in the contraction ZjJi′j\mathbf{Z} _{j}J_{i^{\prime}}^{j} since ii was already used in UiJii′U^{i}J_{i}^{i^{\prime} }. Now, observe that Jii′J_{i}^{i^{\prime}} and Ji′jJ_{i^{\prime}}^{j} participate in a contraction on i′i^{\prime}. Recalling that Jii′Ji′j=δijJ_{i} ^{i^{\prime}}J_{i^{\prime}}^{j}=\delta_{i}^{j}, we have
A′=UiZjδij.(13.73)\mathbf{A}^{\prime}=U^{i}\mathbf{Z}_{j}\delta_{i}^{j}.\tag{13.73}
Since Zjδij=Zi\mathbf{Z}_{j}\delta_{i}^{j}=\mathbf{Z}_{i}, we have
A′=UiZi.(13.74)\mathbf{A}^{\prime}=U^{i}\mathbf{Z}_{i}.\tag{13.74}
We therefore conclude that A′\mathbf{A}^{\prime} and A\mathbf{A} are the same vector:
A′=A(13.75)\mathbf{A}^{\prime}=\mathbf{A}\tag{13.75}
In other words, the combination UiZiU^{i}\mathbf{Z}_{i} is an invariant.
Given the transformation rules that we have discovered so far, we fully expect that Zi\mathbf{Z}^{i} transforms according to the equation
Zi′=ZiJii′.(13.84)\mathbf{Z}^{i^{\prime}}=\mathbf{Z}^{i}J_{i}^{i^{\prime}}. \tag{13.84}
This is indeed the case. However, demonstrating this relationship requires more inventiveness than the earlier ones. What makes this derivation more challenging is the fact that all concise definitions for the contravariant basis Zi\mathbf{Z}^{i} are implicit. According to one definition, Zi\mathbf{Z}^{i} is obtained by contracting the covariant basis Zi\mathbf{Z} _{i} with the contravariant metric tensor ZijZ^{ij}:
Zi=ZjZij.(9.89)\mathbf{Z}^{i}=\mathbf{Z}_{j}Z^{ij}. \tag{9.89}
This definition appears explicit, but recall ZijZ^{ij} is defined implicitly as the matrix inverse of ZijZ_{ij}
ZijZjk=δki,(9.68)Z^{ij}Z_{jk}=\delta_{k}^{i}, \tag{9.68}
and we will not have an explicit expression for the matrix inverse until Chapter 16. Thus, if we wanted to establish the transformation rule for ZijZ^{ij} first and, from it, the rule for Zi\mathbf{Z}^{i}, we would be faced with the same lack of an explicit expression for ZijZ^{ij}.
We will instead base our argument on the definition of the contravariant basis contained in the identity
Zi⋅Zj=δji.(9.93)\mathbf{Z}^{i}\cdot\mathbf{Z}_{j}=\delta_{j}^{i}. \tag{9.93}
To discover the transformation rule for Zi\mathbf{Z}^{i}, recall the corresponding rule for the covariant basis Zi\mathbf{Z}_{i} expressed in the form
Zj=Zj′Jjj′.(13.34)\mathbf{Z}_{j}=\mathbf{Z}_{j^{\prime}}J_{j}^{j^{\prime}}. \tag{13.34}
Substituting this relationship into the identity
Zi⋅Zj=δji(9.93)\mathbf{Z}^{i}\cdot\mathbf{Z}_{j}=\delta_{j}^{i} \tag{9.93}
yields
Zi⋅Zj′Jjj′=δji.(13.76)\mathbf{Z}^{i}\cdot\mathbf{Z}_{j^{\prime}}J_{j}^{j^{\prime}}=\delta_{j}^{i}.\tag{13.76}
In order to eliminate Jjj′J_{j}^{j^{\prime}} on the left, contract both sides with its matrix inverse Jk′jJ_{k^{\prime}}^{j}.
Zi⋅Zj′Jjj′Jk′j=δjiJk′j.(13.77)\mathbf{Z}^{i}\cdot\mathbf{Z}_{j^{\prime}}J_{j}^{j^{\prime}}J_{k^{\prime}} ^{j}=\delta_{j}^{i}J_{k^{\prime}}^{j}.\tag{13.77}
Since Jjj′Jk′j=δk′j′J_{j}^{j^{\prime}}J_{k^{\prime}}^{j}=\delta_{k^{\prime}}^{j^{\prime}}, Zj′δk′j′=Zk′\mathbf{Z}_{j^{\prime}}\delta_{k^{\prime}}^{j^{\prime}}=\mathbf{Z} _{k^{\prime}}, and δjiJk′j=Jk′i\delta_{j}^{i}J_{k^{\prime}}^{j}=J_{k^{\prime}}^{i}, we have
Zi⋅Zk′=Jk′i.(13.78)\mathbf{Z}^{i}\cdot\mathbf{Z}_{k^{\prime}}=J_{k^{\prime}}^{i}.\tag{13.78}
Rename k′k^{\prime} into j′j^{\prime}:
Zi⋅Zj′=Jj′i.(13.79)\mathbf{Z}^{i}\cdot\mathbf{Z}_{j^{\prime}}=J_{j^{\prime}}^{i}.\tag{13.79}
This intermediate identity is of interest in its own right as it relates elements from two different coordinate systems. It also gives an elegant expression for the Jacobian Ji′iJ_{i^{\prime}}^{i}:
Ji′i=Zi⋅Zi′.(13.80)J_{i^{\prime}}^{i}=\mathbf{Z}^{i}\cdot\mathbf{Z}_{i^{\prime}}.\tag{13.80}
Returning to the form
Zi⋅Zj′=Jj′i,(13.79)\mathbf{Z}^{i}\cdot\mathbf{Z}_{j^{\prime}}=J_{j^{\prime}}^{i}, \tag{13.79}
we next wish to "attach" the Jacobian Jii′J_{i}^{i^{\prime}} to Zi\mathbf{Z}^{i} in anticipation that the combination Jii′ZiJ_{i}^{i^{\prime}}\mathbf{Z}^{i} corresponds to Zi′\mathbf{Z}^{i^{\prime}}. To this end, contract both sides of the above identity with Jii′J_{i}^{i^{\prime}}
Jii′Zi⋅Zj′=Jii′Jj′i.(13.81)J_{i}^{i^{\prime}}\mathbf{Z}^{i}\cdot\mathbf{Z}_{j^{\prime}}=J_{i}^{i^{\prime }}J_{j^{\prime}}^{i}.\tag{13.81}
Since Jii′Jj′i=δj′i′J_{i}^{i^{\prime}}J_{j^{\prime}}^{i}=\delta_{j^{\prime}}^{i^{\prime}}, we have
Jii′Zi⋅Zj′=δj′i′.(13.82)J_{i}^{i^{\prime}}\mathbf{Z}^{i}\cdot\mathbf{Z}_{j^{\prime}}=\delta _{j^{\prime}}^{i^{\prime}}.\tag{13.82}
This identity is sufficient to conclude that Jii′ZiJ_{i}^{i^{\prime}}\mathbf{Z} ^{i} equals Zi′\mathbf{Z}^{i^{\prime}}. Recall that Zi′\mathbf{Z}^{i^{\prime}} is defined by the identity
Zi′⋅Zj′=δj′i′.(13.83)\mathbf{Z}^{i^{\prime}}\cdot\mathbf{Z}_{j^{\prime}}=\delta_{j^{\prime} }^{i^{\prime}}.\tag{13.83}
Therefore, a comparison of the last two identities immediately yields
Zi′=Jii′Zi,(13.84)\mathbf{Z}^{i^{\prime}}=J_{i}^{i^{\prime}}\mathbf{Z}^{i},\tag{13.84}
as we set out to prove.
As was the case for the covariant metric tensor ZijZ_{ij}, the transformation rule for the contravariant metric tensor ZijZ^{ij} follows readily from that for the contravariant basis Zi\mathbf{Z}^{i}. Recall that ZijZ^{ij} is given by
Zij=Zi⋅Zj.(9.91)Z^{ij}=\mathbf{Z}^{i}\cdot\mathbf{Z}^{j}. \tag{9.91}
Thus, Zi′j′Z^{i^{\prime}j^{\prime}} in the primed coordinates is given by
Zi′j′=Zi′⋅Zj′.(13.85)Z^{i^{\prime}j^{\prime}}=\mathbf{Z}^{i^{\prime}}\cdot\mathbf{Z}^{j^{\prime}}.\tag{13.85}
Since Zi′=ZiJii′\mathbf{Z}^{i^{\prime}}=\mathbf{Z}^{i}J_{i}^{i^{\prime}} while Zj′=ZjJjj′\mathbf{Z}^{j^{\prime}}=\mathbf{Z}^{j}J_{j}^{j^{\prime}}, we have
Zi′j′=ZiJii′⋅ZjJjj′.(13.86)Z^{i^{\prime}j^{\prime}}=\mathbf{Z}^{i}J_{i}^{i^{\prime}}\cdot\mathbf{Z} ^{j}J_{j}^{j^{\prime}}.\tag{13.86}
Finally, we recognize the dot product Zi⋅Zj\mathbf{Z}^{i}\cdot\mathbf{Z}^{j} on the right as ZijZ^{ij}, and therefore arrive at the following transformation rule for Zi′j′Z^{i^{\prime}j^{\prime}}:
Zi′j′=ZijJii′Jjj′.(13.87)Z^{i^{\prime}j^{\prime}}=Z^{ij}J_{i}^{i^{\prime}}J_{j}^{j^{\prime}}.\tag{13.87}
An alternative derivation of this identity can be constructed directly on the basis of the original definition of the contravariant metric tensor
ZijZjk=δki.(9.68)Z^{ij}Z_{jk}=\delta_{k}^{i}. \tag{9.68}
In the alternative coordinate system, Zi′j′Z^{i^{\prime}j^{\prime}} is given by the analogous identity
Zi′j′Zj′k′=δk′i′.(13.88)Z^{i^{\prime}j^{\prime}}Z_{j^{\prime}k^{\prime}}=\delta_{k^{\prime} }^{i^{\prime}}.\tag{13.88}
Utilizing the transformation rule
Zj′k′=ZjkJj′jJk′k(13.59)Z_{j^{\prime}k^{\prime}}=Z_{jk}J_{j^{\prime}}^{j}J_{k^{\prime}}^{k} \tag{13.59}
for the covariant basis, we find
Zi′j′ZjkJj′jJk′k=δk′i′.(13.89)Z^{i^{\prime}j^{\prime}}Z_{jk}J_{j^{\prime}}^{j}J_{k^{\prime}}^{k} =\delta_{k^{\prime}}^{i^{\prime}}.\tag{13.89}
Our goal is, of course, to isolate Zi′j′Z^{i^{\prime}j^{\prime}} on the left. This can be accomplished by successive inversions of the two Jacobians and the covariant metric tensor on the left, although, importantly, not in that order. Inverting Jk′kJ_{k^{\prime}}^{k} yields
Zi′j′ZjkJj′j=δk′i′Jkk′,(13.90)Z^{i^{\prime}j^{\prime}}Z_{jk}J_{j^{\prime}}^{j}=\delta_{k^{\prime} }^{i^{\prime}}J_{k}^{k^{\prime}},\tag{13.90}
or
Zi′j′ZjkJj′j=Jki′.(13.91)Z^{i^{\prime}j^{\prime}}Z_{jk}J_{j^{\prime}}^{j}=J_{k}^{i^{\prime}}.\tag{13.91}
Next, inverting ZjkZ_{jk} yields
Zi′j′Jj′j=ZjkJki′(13.92)Z^{i^{\prime}j^{\prime}}J_{j^{\prime}}^{j}=Z^{jk}J_{k}^{i^{\prime}}\tag{13.92}
and subsequently inverting Jj′jJ_{j^{\prime}}^{j} yields
Zi′j′=ZjkJki′Jjj′.(13.93)Z^{i^{\prime}j^{\prime}}=Z^{jk}J_{k}^{i^{\prime}}J_{j}^{j^{\prime}}.\tag{13.93}
Finally, renaming kk into ii on the right and taking advantage of the symmetry of the metric tensor yields
Zi′j′=ZijJii′Jjj′.(13.87)Z^{i^{\prime}j^{\prime}}=Z^{ij}J_{i}^{i^{\prime}}J_{j}^{j^{\prime}}. \tag{13.87}
This derivation of the transformation rule for the contravariant metric tensor has great significance. Note that it is conducted strictly in the coordinate space -- or, more accurately, in two coordinate spaces -- and does not make any references to geometric objects in the Euclidean space. This insight will play an important role in the development of Riemannian spaces.
The tensor property of variants is the subject of the next Chapter. However, it is important to give the definition of a tensor at this time in order to illustrate the importance of the relationships we have derived in this Chapter.
A tensor is a variant whose transformation rule consists precisely of contractions with the appropriate Jacobian for each index. For example, a variant TiT_{i} of order one is a covariant tensor if
Ti′=TiJi′i,(13.94)T_{i^{\prime}}=T_{i}J_{i^{\prime}}^{i},\tag{13.94}
and a variant TiT^{i} of order one is a contravariant tensor if
Ti′=TiJii′(13.95)T^{i^{\prime}}=T^{i}J_{i}^{i^{\prime}}\tag{13.95}
Since the covariant basis Zi\mathbf{Z}_{i} transforms according to the rule
Zi′=ZiJi′i(13.33)\mathbf{Z}_{i^{\prime}}=\mathbf{Z}_{i}J_{i^{\prime}}^{i} \tag{13.33}
it is, indeed a covariant tensor and thus the term covariant in its name is finally justified. Since the contravariant basis Zi\mathbf{Z}^{i} transforms according to the rule
Zi′=ZiJii′,(13.84)\mathbf{Z}^{i^{\prime}}=\mathbf{Z}^{i}J_{i}^{i^{\prime}}, \tag{13.84}
it is, indeed, a contravariant tensor and thus the term contravariant in its name is finally justified. Similar statements can be made of the contravariant components UiU^{i} and the covariant components UiU_{i} of a vector U\mathbf{U}.
A variant of TjiT_{j}^{i} of order two is a tensor if
Tj′i′=TjiJii′Jj′j.(13.96)T_{j^{\prime}}^{i^{\prime}}=T_{j}^{i}J_{i}^{i^{\prime}}J_{j^{\prime}}^{j}.\tag{13.96}
It may be described as a tensor of one contravariant and one covariant order or, if the superscript is treated as the first index, a contravariant-covariant tensor. Similarly a variant TijT_{ij} is a tensor if
Ti′j′=TijJi′iJj′j(13.97)T_{i^{\prime}j^{\prime}}=T_{ij}J_{i^{\prime}}^{i}J_{j^{\prime}}^{j}\tag{13.97}
and may be described as a covariant tensor, a doubly-covariant tensor, or a covariant-covariant tensor. Finally, a variant TijT^{ij} is a tensor if
Ti′j′=TijJii′Jjj′(13.98)T^{i^{\prime}j^{\prime}}=T^{ij}J_{i}^{i^{\prime}}J_{j}^{j^{\prime}}\tag{13.98}
and may be described as a contravariant tensor, a doubly-contravariant tensor, or a contravariant-contravariant tensor. Clearly, these descriptions are redundant when the indicial signature of the variant is available. In this, most common, situation, it is easier to describe the variant as a tensor of the type indicated by the indicial signature or a tensor of the type indicated by the flavors of the indices.
Since the covariant metric tensor ZijZ_{ij} transforms according to the rule
Zi′j′=ZijJi′iJj′j(13.99)Z_{i^{\prime}j^{\prime}}=Z_{ij}J_{i^{\prime}}^{i}J_{j^{\prime}}^{j}\tag{13.99}
and the contravariant metric tensor ZijZ^{ij} transforms according to the rule
Zi′j′=ZijJii′Jjj′,(13.87)Z^{i^{\prime}j^{\prime}}=Z^{ij}J_{i}^{i^{\prime}}J_{j}^{j^{\prime}}, \tag{13.87}
the terms covariant and contravariant in the names of these central objects are finally justified.
It should now be clear how the definition of a tensor can be extended to a variant with an arbitrary indicial signature. For example, a variant TjkliT_{jkl}^{i} is a tensor of the type indicated by the flavors of its indices if
Tj′k′l′i′=TjkliJii′Jj′jJk′kJl′l.(13.100)T_{j^{\prime}k^{\prime}l^{\prime}}^{i^{\prime}}=T_{jkl}^{i}J_{i}^{i^{\prime} }J_{j^{\prime}}^{j}J_{k^{\prime}}^{k}J_{l^{\prime}}^{l}.\tag{13.100}
Interestingly, all variants analyzed so far in this Chapter are tensors. It would, thus, be quite insightful to see an example of a variant that is not a tensor. An essential example of a non-tensor is the Christoffel symbol Γjki\Gamma_{jk}^{i}. Therefore, we will devote the remainder of this Chapter to studying the transformation rule of the Christoffel symbol.
Recall that the Jacobians Ji′iJ_{i^{\prime}}^{i} and Jii′J_{i}^{i^{\prime}} represent the collections of first-order derivatives
Ji′i(Z′)=∂Zi(Z′)∂Zi′(13.32)J_{i^{\prime}}^{i}\left( Z^{\prime}\right) =\frac{\partial Z^{i}\left( Z^{\prime}\right) }{\partial Z^{i^{\prime}}} \tag{13.32}
and
Jii′(Z)=∂Zi′(Z)∂Zi.(13.31)J_{i}^{i^{\prime}}\left( Z\right) =\frac{\partial Z^{i^{\prime}}\left( Z\right) }{\partial Z^{i}}. \tag{13.31}
of the coordinate transformations.
In order to establish the transformation rule for the Christoffel symbol, we must consider the collections, denoted by symbols Ji′j′iJ_{i^{\prime}j^{\prime} }^{i} and Jiji′J_{ij}^{i^{\prime}}, of the second-order derivatives, i.e.
Ji′j′i(Z′)=∂2Zi(Z′)∂Zi′∂Zj′(13.101)J_{i^{\prime}j^{\prime}}^{i}\left( Z^{\prime}\right) =\frac{\partial ^{2}Z^{i}\left( Z^{\prime}\right) }{\partial Z^{i^{\prime}}\partial Z^{j^{\prime}}}\tag{13.101}
and
Jiji′(Z)=∂2Zi′(Z)∂Zi∂Zj.(13.102)J_{ij}^{i^{\prime}}\left( Z\right) =\frac{\partial^{2}Z^{i^{\prime}}\left( Z\right) }{\partial Z^{i}\partial Z^{j}}.\tag{13.102}
Note that Ji′j′iJ_{i^{\prime}j^{\prime}}^{i} and Jiji′J_{ij}^{i^{\prime}} are third-order systems. Therefore, in an nn-dimensional space, they represent collections of n3n^{3} elements. Since, generally speaking, partial derivatives commute, the systems Ji′j′iJ_{i^{\prime}j^{\prime}}^{i} and Jiji′J_{ij}^{i^{\prime}} are symmetric in their subscripts, i.e.
Ji′j′i=Jj′i′i   and   Jiji′=Jjii′.(13.103)J_{i^{\prime}j^{\prime}}^{i}=J_{j^{\prime}i^{\prime}}^{i}\text{ \ \ and \ \ }J_{ij}^{i^{\prime}}=J_{ji}^{i^{\prime}}.\tag{13.103}
Thus, in actuality, Ji′j′iJ_{i^{\prime}j^{\prime}}^{i} and Jiji′J_{ij}^{i^{\prime}} have n2(n+1)/2n^{2}\left( n+1\right) /2 degrees of freedom.
For the Cartesian to polar transformation, we have
Ji′j′1 corresponds to [0−sin⁡θ−sin⁡θ−rcos⁡θ] and          (13.104)Ji′j′2 corresponds to [0cos⁡θcos⁡θ−rsin⁡θ].          (13.105)\begin{aligned}& J_{i^{\prime}j^{\prime}}^{1}\text{ corresponds to }\left[ \begin{array} {cc} 0 & -\sin\theta\\ -\sin\theta & -r\cos\theta \end{array} \right] \text{ and}\ \ \ \ \ \ \ \ \ \ \left(13.104\right)\\& J_{i^{\prime}j^{\prime}}^{2}\text{ corresponds to }\left[ \begin{array} {cc} 0 & \cos\theta\\ \cos\theta & -r\sin\theta \end{array} \right] .\ \ \ \ \ \ \ \ \ \ \left(13.105\right)\end{aligned}
Similar to the identities
Ji′iJji′=δji    and   Jii′Jj′i=δj′i′(13.47)J_{i^{\prime}}^{i}J_{j}^{i^{\prime}}=\delta_{j}^{i}\text{ \ \ \ and\ \ \ } J_{i}^{i^{\prime}}J_{j^{\prime}}^{i}=\delta_{j^{\prime}}^{i^{\prime}} \tag{13.47}
relating the first-order Jacobians, there exists identities relating the second-order Jacobians. The latter can be obtained by differentiating the first-order identities. For example, consider the identity
Ji′iJji′=δji.(13.47)J_{i^{\prime}}^{i}J_{j}^{i^{\prime}}=\delta_{j}^{i}. \tag{13.47}
In order to differentiate it, we must refer all of the elements in this identity to a consistent set of coordinates, either the unprimed coordinates ZiZ^{i} or the primed coordinates Zi′Z^{i^{\prime}}. Let us choose Zi′Z^{i^{\prime}}.
As we discussed above, the natural variables for the Jacobian Ji′iJ_{i^{\prime} }^{i} are the primed coordinates Zi′Z^{i^{\prime}}. Thus, the natural function Ji′i(Z′)J_{i^{\prime}}^{i}\left( Z^{\prime}\right) is ready for differentiation. However, for the Jacobian Jii′J_{i}^{i^{\prime}}, the natural coordinates are the unprimed coordinates ZiZ^{i}. Thus, in order to refer Jii′J_{i}^{i^{\prime}} to the primed coordinates Zi′Z^{i^{\prime}}, we must substitute the coordinate transformation
Zi=Zi(Z′)(13.106)Z^{i}=Z^{i}\left( Z^{\prime}\right)\tag{13.106}
into the natural function Jii′(Z)J_{i}^{i^{\prime}}\left( Z\right) , i.e.
Jii′(Z′)=Jii′(Z(Z′)).(13.107)J_{i}^{i^{\prime}}\left( Z^{\prime}\right) =J_{i}^{i^{\prime}}\left( Z\left( Z^{\prime}\right) \right) .\tag{13.107}
The Kronecker delta δji\delta_{j}^{i} has constant values and therefore its derivatives will equal zero regardless of the independent variables that we choose. However, for the sake of consistency, let us discuss its functional dependence on the coordinates. Since, in our present analysis, the Kronecker delta δji\delta_{j}^{i} appears with unprimed indices, it is more natural to think of it as a function δji(Z)\delta_{j}^{i}\left( Z\right) of the unprimed coordinates. Thus, the function δji(Z′)\delta_{j}^{i}\left( Z^{\prime}\right) of the primed coordinates can be obtained by the composition
δji(Z′)=δji(Z(Z′)).(13.108)\delta_{j}^{i}\left( Z^{\prime}\right) =\delta_{j}^{i}\left( Z\left( Z^{\prime}\right) \right) .\tag{13.108}
Combining all the elements, we find that, when referred to the primed coordinates Zi′Z^{i^{\prime}}, the identity
Ji′iJji′=δji(13.47)J_{i^{\prime}}^{i}J_{j}^{i^{\prime}}=\delta_{j}^{i} \tag{13.47}
reads
Ji′i(Z′)Jji′(Z(Z′))=δji(Z(Z′)).(13.109)J_{i^{\prime}}^{i}\left( Z^{\prime}\right) J_{j}^{i^{\prime}}\left( Z\left( Z^{\prime}\right) \right) =\delta_{j}^{i}\left( Z\left( Z^{\prime}\right) \right) .\tag{13.109}
Differentiating with respect to Zk′Z^{k^{\prime}}, we find
∂Ji′i(Z′)∂Zk′Jji′(Z(Z′))+Ji′i(Z′)∂Jji′(Z(Z′))∂Zk′=∂δji(Z(Z′))∂Zk′.(13.110)\frac{\partial J_{i^{\prime}}^{i}\left( Z^{\prime}\right) }{\partial Z^{k^{\prime}}}J_{j}^{i^{\prime}}\left( Z\left( Z^{\prime}\right) \right) +J_{i^{\prime}}^{i}\left( Z^{\prime}\right) \frac{\partial J_{j}^{i^{\prime }}\left( Z\left( Z^{\prime}\right) \right) }{\partial Z^{k^{\prime}} }=\frac{\partial\delta_{j}^{i}\left( Z\left( Z^{\prime}\right) \right) }{\partial Z^{k^{\prime}}}.\tag{13.110}
An application of the chain rule on the second term on the left and the term on the right yields
∂Ji′i(Z′)∂Zk′Jji′(Z(Z′))+Ji′i(Z′)∂Jji′(Z)∂Zk∂Zk(Z′)∂Zk′=∂δji(Z)∂Zk∂Zk(Z′)∂Zk′.(13.111)\frac{\partial J_{i^{\prime}}^{i}\left( Z^{\prime}\right) }{\partial Z^{k^{\prime}}}J_{j}^{i^{\prime}}\left( Z\left( Z^{\prime}\right) \right) +J_{i^{\prime}}^{i}\left( Z^{\prime}\right) \frac{\partial J_{j}^{i^{\prime }}\left( Z\right) }{\partial Z^{k}}\frac{\partial Z^{k}\left( Z^{\prime }\right) }{\partial Z^{k^{\prime}}}=\frac{\partial\delta_{j}^{i}\left( Z\right) }{\partial Z^{k}}\frac{\partial Z^{k}\left( Z^{\prime}\right) }{\partial Z^{k^{\prime}}}.\tag{13.111}
Since
∂Ji′i(Z′)∂Zk′=Ji′k′i,    ∂Jji′(Z)∂Zk=Jjki′,    and    ∂δji(Z)∂Zk=0,(13.112)\frac{\partial J_{i^{\prime}}^{i}\left( Z^{\prime}\right) }{\partial Z^{k^{\prime}}}=J_{i^{\prime}k^{\prime}}^{i}\text{,\ \ \ \ }\frac{\partial J_{j}^{i^{\prime}}\left( Z\right) }{\partial Z^{k}}=J_{jk}^{i^{\prime} }\text{,\ \ \ \ and \ \ \ }\frac{\partial\delta_{j}^{i}\left( Z\right) }{\partial Z^{k}}=0,\tag{13.112}
we have
Ji′k′iJji′+Ji′iJjki′Jk′k=0.(13.113)J_{i^{\prime}k^{\prime}}^{i}J_{j}^{i^{\prime}}+J_{i^{\prime}}^{i} J_{jk}^{i^{\prime}}J_{k^{\prime}}^{k}=0.\tag{13.113}
Renaming the dummy index i′i^{\prime} into j′j^{\prime} in the first term and changing the order of the Jacobians in the second term yields a more elegant form of the same relationship
Jj′k′iJjj′+Jjki′Ji′iJk′k=0.(13.114)J_{j^{\prime}k^{\prime}}^{i}J_{j}^{j^{\prime}}+J_{jk}^{i^{\prime}} J_{i^{\prime}}^{i}J_{k^{\prime}}^{k}=0.\tag{13.114}
This is one of many equivalent forms that this relationship can take. Other forms can be obtained by inverting the first-order Jacobians present in the identity. For example, inverting Jjj′J_{j}^{j^{\prime}} yields
Jj′k′i+Jjki′Ji′iJj′jJk′k=0.(13.115)J_{j^{\prime}k^{\prime}}^{i}+J_{jk}^{i^{\prime}}J_{i^{\prime}}^{i} J_{j^{\prime}}^{j}J_{k^{\prime}}^{k}=0.\tag{13.115}
The advantage of this form is that it gives the Jacobian Jj′k′iJ_{j^{\prime }k^{\prime}}^{i} explicitly in terms of Jjki′J_{jk}^{i^{\prime}}. Alternatively, inverting Jk′kJ_{k^{\prime}}^{k} in the penultimate identity yields
Jj′k′iJjj′Jkk′+Jjki′Ji′i=0.(13.116)J_{j^{\prime}k^{\prime}}^{i}J_{j}^{j^{\prime}}J_{k}^{k^{\prime}} +J_{jk}^{i^{\prime}}J_{i^{\prime}}^{i}=0.\tag{13.116}
The advantage of this form is that all live indices are unprimed and thus correspond to the same coordinate system. Note that there are 23=82^{3}=8 equivalent forms of this relationship depending on which second-order Jacobian each first-order Jacobian is contracted with.
Let us now establish the transformation rule for the Christoffel symbol Γjki\Gamma_{jk}^{i}. Our present analysis will be based on the equation
Γijk=Zk⋅∂Zi∂Zj.(12.24)\Gamma_{ij}^{k}=\mathbf{Z}^{k}\cdot\frac{\partial\mathbf{Z}_{i}}{\partial Z^{j}}. \tag{12.24}
Let us first determine the transformation rule for the vector variant
Γij=∂Zi∂Zj.(12.2)\mathbf{\Gamma}_{ij}=\frac{\partial\mathbf{Z}_{i}}{\partial Z^{j}}. \tag{12.2}
Consider the primed version of the same variant
Γi′j′=∂Zi′∂Zj′.(13.117)\mathbf{\Gamma}_{i^{\prime}j^{\prime}}=\frac{\partial\mathbf{Z}_{i^{\prime}} }{\partial Z^{j^{\prime}}}.\tag{13.117}
We will relate Γi′j′\mathbf{\Gamma}_{i^{\prime}j^{\prime}} to Γij\mathbf{\Gamma }_{ij} by differentiating the identity
Zi′=ZiJi′i,(13.33)\mathbf{Z}_{i^{\prime}}=\mathbf{Z}_{i}J_{i^{\prime}}^{i}, \tag{13.33}
In order to differentiate this identity, refer all elements in it to the primed coordinates Zi′Z^{i^{\prime}}, i.e.
Zi′(Z′)=Zi(Z(Z′))Ji′i(Z′).(13.118)\mathbf{Z}_{i^{\prime}}\left( Z^{\prime}\right) =\mathbf{Z}_{i}\left( Z\left( Z^{\prime}\right) \right) J_{i^{\prime}}^{i}\left( Z^{\prime }\right) .\tag{13.118}
Now, differentiate both sides with respect to Zj′Z^{j^{\prime}}. By an application of the product rule, we have
∂Zi′(Z′)∂Zj′=∂Zi(Z(Z′))∂Zj′Ji′i+Zi∂Ji′i(Z′)∂Zj′.(13.119)\frac{\partial\mathbf{Z}_{i^{\prime}}\left( Z^{\prime}\right) }{\partial Z^{j^{\prime}}}=\frac{\partial\mathbf{Z}_{i}\left( Z\left( Z^{\prime }\right) \right) }{\partial Z^{j^{\prime}}}J_{i^{\prime}}^{i}+\mathbf{Z} _{i}\frac{\partial J_{i^{\prime}}^{i}\left( Z^{\prime}\right) }{\partial Z^{j^{\prime}}}.\tag{13.119}
A subsequent application of the chain rule to the first term on the right yields
∂Zi′(Z′)∂Zj′=∂Zi(Z)∂Zj∂Zj(Z′)∂Zj′Ji′i+Zi∂Ji′i(Z′)∂Zj′.(13.120)\frac{\partial\mathbf{Z}_{i^{\prime}}\left( Z^{\prime}\right) }{\partial Z^{j^{\prime}}}=\frac{\partial\mathbf{Z}_{i}\left( Z\right) }{\partial Z^{j}}\frac{\partial Z^{j}\left( Z^{\prime}\right) }{\partial Z^{j^{\prime} }}J_{i^{\prime}}^{i}+\mathbf{Z}_{i}\frac{\partial J_{i^{\prime}}^{i}\left( Z^{\prime}\right) }{\partial Z^{j^{\prime}}}.\tag{13.120}
Since
∂Zi′(Z′)∂Zj′=Γi′j′,    ∂Zi(Z)∂Zj=Γij,    ∂Zj(Z′)∂Zj′=Jjj,    and    ∂Ji′i(Z′)∂Zj′=Ji′j′i,(13.121)\frac{\partial\mathbf{Z}_{i^{\prime}}\left( Z^{\prime}\right) }{\partial Z^{j^{\prime}}}=\mathbf{\Gamma}_{i^{\prime}j^{\prime}}\text{, \ \ \ } \frac{\partial\mathbf{Z}_{i}\left( Z\right) }{\partial Z^{j}}=\mathbf{\Gamma }_{ij}\text{, \ \ \ }\frac{\partial Z^{j}\left( Z^{\prime}\right) }{\partial Z^{j^{\prime}}}=J_{j}^{j}\text{, \ \ \ and\ \ \ \ }\frac{\partial J_{i^{\prime}}^{i}\left( Z^{\prime}\right) }{\partial Z^{j^{\prime}} }=J_{i^{\prime}j^{\prime}}^{i},\tag{13.121}
the transformation rule for Γi′j′j\mathbf{\Gamma}_{i^{\prime}j^{\prime}}^{j} reads
Γi′j′=ΓijJi′iJj′j+ZiJi′j′i.(13.122)\mathbf{\Gamma}_{i^{\prime}j^{\prime}}=\mathbf{\Gamma}_{ij}J_{i^{\prime}} ^{i}J_{j^{\prime}}^{j}+\mathbf{Z}_{i}J_{i^{\prime}j^{\prime}}^{i}.\tag{13.122}
We must now make the observation that Γij\mathbf{\Gamma}_{ij} is not a tensor, which would have required the above identity to read Γi′j′=ΓijJi′iJj′j\mathbf{\Gamma }_{i^{\prime}j^{\prime}}=\mathbf{\Gamma}_{ij}J_{i^{\prime}}^{i}J_{j^{\prime} }^{j}, which it does not. This is our first example of a variant that is not a tensor. It is, therefore, not surprising that the Christoffel symbol Γijk\Gamma_{ij}^{k}, so closely related to Γij\mathbf{\Gamma}_{ij}, is also not a tensor. We will now show this.
The Christoffel symbol
Γijk=Zk⋅∂Zi∂Zj(12.24)\Gamma_{ij}^{k}=\mathbf{Z}^{k}\cdot\frac{\partial\mathbf{Z}_{i}}{\partial Z^{j}} \tag{12.24}
in the primed coordinate system is given by
Γi′j′k′=Zk′⋅∂Zi′∂Zj′(13.123)\Gamma_{i^{\prime}j^{\prime}}^{k^{\prime}}=\mathbf{Z}^{k^{\prime}}\cdot \frac{\partial\mathbf{Z}_{i^{\prime}}}{\partial Z^{j^{\prime}}}\tag{13.123}
Since
Zk′=ZkJkk′(13.124)\mathbf{Z}^{k^{\prime}}=\mathbf{Z}^{k}J_{k}^{k^{\prime}}\tag{13.124}
and, as we just derived,
∂Zi′∂Zj′=∂Zi∂ZjJi′iJj′j+ZiJi′j′i,(13.122)\frac{\partial\mathbf{Z}_{i^{\prime}}}{\partial Z^{j^{\prime}}}=\frac {\partial\mathbf{Z}_{i}}{\partial Z^{j}}J_{i^{\prime}}^{i}J_{j^{\prime}} ^{j}+\mathbf{Z}_{i}J_{i^{\prime}j^{\prime}}^{i}, \tag{13.122}
we have
Γi′j′k′=ZkJkk′⋅(∂Zi∂ZjJi′iJj′j+ZiJi′j′i).(13.125)\Gamma_{i^{\prime}j^{\prime}}^{k^{\prime}}=\mathbf{Z}^{k}J_{k}^{k^{\prime} }\cdot\left( \frac{\partial\mathbf{Z}_{i}}{\partial Z^{j}}J_{i^{\prime}} ^{i}J_{j^{\prime}}^{j}+\mathbf{Z}_{i}J_{i^{\prime}j^{\prime}}^{i}\right) .\tag{13.125}
Multiplying out the expression on the right yields
Γi′j′k′=ZkJkk′⋅∂Zi∂ZjJi′iJj′j+ZkJkk′⋅ZiJi′j′i.(13.126)\Gamma_{i^{\prime}j^{\prime}}^{k^{\prime}}=\mathbf{Z}^{k}J_{k}^{k^{\prime} }\cdot\frac{\partial\mathbf{Z}_{i}}{\partial Z^{j}}J_{i^{\prime}} ^{i}J_{j^{\prime}}^{j}+\mathbf{Z}^{k}J_{k}^{k^{\prime}}\cdot\mathbf{Z} _{i}J_{i^{\prime}j^{\prime}}^{i}.\tag{13.126}
The first term equals ΓijkJi′iJj′jJkk′\Gamma_{ij}^{k}J_{i^{\prime}}^{i}J_{j^{\prime}} ^{j}J_{k}^{k^{\prime}}. Since Zk⋅Zi=δik\mathbf{Z}^{k}\cdot\mathbf{Z}_{i}=\delta _{i}^{k}, the second term equals Ji′j′kJkk′J_{i^{\prime}j^{\prime}}^{k}J_{k} ^{k^{\prime}}. Thus, the transformation rule for the Christoffel symbol Γijk\Gamma_{ij}^{k} reads
Γi′j′k′=ΓijkJi′iJj′jJkk′+Ji′j′kJkk′.(13.127)\Gamma_{i^{\prime}j^{\prime}}^{k^{\prime}}=\Gamma_{ij}^{k}J_{i^{\prime}} ^{i}J_{j^{\prime}}^{j}J_{k}^{k^{\prime}}+J_{i^{\prime}j^{\prime}}^{k} J_{k}^{k^{\prime}}.\tag{13.127}
Therefore, the Christoffel symbol Γijk\Gamma_{ij}^{k} is not a tensor.
The above formula provides a convenient way of calculating the elements of the Christoffel symbol by relating them to the elements of the Christoffel symbol in another coordinate system. In particular, if the latter system is affine where, if you recall, the Christoffel symbol vanishes, then Γi′j′k′\Gamma _{i^{\prime}j^{\prime}}^{k^{\prime}} is given by
Γi′j′k′=Ji′j′kJkk′.(13.128)\Gamma_{i^{\prime}j^{\prime}}^{k^{\prime}}=J_{i^{\prime}j^{\prime}}^{k} J_{k}^{k^{\prime}}.\tag{13.128}
It is left as an exercise to calculate the elements of the Christoffel symbol in polar and spherical coordinates using this formula.
Exercise 13.1On the basis of the equation
Zi′=Jii′Zi,(13.33)\mathbf{Z}^{i^{\prime}}=J_{i}^{i^{\prime}}\mathbf{Z}^{i}, \tag{13.33}
show that the covariant components UiU_{i} of a vector U\mathbf{U} transform according to the rule
Ui′=UiJi′i.(13.129)U_{i^{\prime}}=U_{i}J_{i^{\prime}}^{i}.\tag{13.129}
Exercise 13.2Calculate the elements of Ji′j′iJ_{i^{\prime}j^{\prime}}^{i} for a transformation from one affine coordinate system to another.
Exercise 13.3Confirm that
Ji′j′1 corresponds to [0−sin⁡θ−sin⁡θ−rcos⁡θ] and          (13.104)Ji′j′2 corresponds to [0cos⁡θcos⁡θ−rsin⁡θ]          (13.105)\begin{aligned}& J_{i^{\prime}j^{\prime}}^{1}\text{ corresponds to }\left[ \begin{array} {cc} 0 & -\sin\theta\\ -\sin\theta & -r\cos\theta \end{array} \right] \text{ and}\ \ \ \ \ \ \ \ \ \ \left(13.104\right)\\& J_{i^{\prime}j^{\prime}}^{2}\text{ corresponds to }\left[ \begin{array} {cc} 0 & \cos\theta\\ \cos\theta & -r\sin\theta \end{array} \right] \ \ \ \ \ \ \ \ \ \ \left(13.105\right)\end{aligned}
for the transformation between Cartesian and polar coordinates.
Exercise 13.4Let Jj′k′l′iJ_{j^{\prime}k^{\prime}l^{\prime}}^{i} denote the third-order Jacobian
Jj′k′l′i=∂3Zi(Z′)∂Zj′∂Zk′∂Zl′.(13.130)J_{j^{\prime}k^{\prime}l^{\prime}}^{i}=\frac{\partial^{3}Z^{i}\left( Z^{\prime}\right) }{\partial Z^{j^{\prime}}\partial Z^{k^{\prime}}\partial Z^{l^{\prime}}}.\tag{13.130}
Show that
Jj′k′l′i+Jjkli′Ji′iJj′jJk′kJl′l+Jjki′Ji′l′iJj′jJk′k+Jjki′Ji′iJj′l′jJk′k+Jjki′Ji′iJj′jJk′l′k=0.(13.131)J_{j^{\prime}k^{\prime}l^{\prime}}^{i}+J_{jkl}^{i^{\prime}}J_{i^{\prime}} ^{i}J_{j^{\prime}}^{j}J_{k^{\prime}}^{k}J_{l^{\prime}}^{l}+J_{jk}^{i^{\prime} }J_{i^{\prime}l^{\prime}}^{i}J_{j^{\prime}}^{j}J_{k^{\prime}}^{k} +J_{jk}^{i^{\prime}}J_{i^{\prime}}^{i}J_{j^{\prime}l^{\prime}}^{j} J_{k^{\prime}}^{k}+J_{jk}^{i^{\prime}}J_{i^{\prime}}^{i}J_{j^{\prime}} ^{j}J_{k^{\prime}l^{\prime}}^{k}=0.\tag{13.131}
Exercise 13.5Consider a circular change of variables: from ZiZ^{i} to Zi′Z^{i^{\prime}}, from Zi′Z^{i^{\prime}} to Zi′′Z^{i^{\prime\prime}}, and from Zi′′Z^{i^{\prime\prime }} back to ZiZ^{i}. Show that
Ji′iJi′′i′Jji′′=δji.(13.132)J_{i^{\prime}}^{i}J_{i^{\prime\prime}}^{i^{\prime}}J_{j}^{i^{\prime\prime} }=\delta_{j}^{i}.\tag{13.132}
Exercise 13.6Consider the transformation between the Cartesian and spherical coordinates in the three-dimensional Euclidean space:
r(x,y,z)=x2+y2+z2          (13.133)θ(x,y,z)=arctan⁡(z,x2+y2)          (13.134)φ(x,y,z)=arctan⁡(x,y)          (13.135)\begin{aligned}r\left( x,y,z\right) & =\sqrt{x^{2}+y^{2}+z^{2}}\ \ \ \ \ \ \ \ \ \ \left(13.133\right)\\\theta\left( x,y,z\right) & =\arctan\left( z,\sqrt{x^{2}+y^{2}}\right)\ \ \ \ \ \ \ \ \ \ \left(13.134\right)\\\varphi\left( x,y,z\right) & =\arctan\left( x,y\right)\ \ \ \ \ \ \ \ \ \ \left(13.135\right)\end{aligned}
and
x(r,θ,φ)=rsin⁡θcos⁡φ          (13.136)y(r,θ,φ)=rsin⁡θsin⁡φ          (13.137)z(r,θ,φ)=rcos⁡θ.          (13.138)\begin{aligned}x\left( r,\theta,\varphi\right) & =r\sin\theta\cos\varphi\ \ \ \ \ \ \ \ \ \ \left(13.136\right)\\y\left( r,\theta,\varphi\right) & =r\sin\theta\sin\varphi\ \ \ \ \ \ \ \ \ \ \left(13.137\right)\\z\left( r,\theta,\varphi\right) & =r\cos\theta.\ \ \ \ \ \ \ \ \ \ \left(13.138\right)\end{aligned}
Show that
Jii′ corresponds to [xx2+y2+z2yx2+y2+z2zx2+y2+z2xzx2+y2(x2+y2+z2)yzx2+y2(x2+y2+z2)−x2+y2x2+y2+z2−yx2+y2xx2+y20](13.139)J_{i}^{i^{\prime}}\text{ corresponds to }\left[ \begin{array} {ccc} \frac{x}{\sqrt{x^{2}+y^{2}+z^{2}}} & \frac{y}{\sqrt{x^{2}+y^{2}+z^{2}}} & \frac{z}{\sqrt{x^{2}+y^{2}+z^{2}}}\\ \frac{xz}{\sqrt{x^{2}+y^{2}}\left( x^{2}+y^{2}+z^{2}\right) } & \frac {yz}{\sqrt{x^{2}+y^{2}}\left( x^{2}+y^{2}+z^{2}\right) } & -\frac {\sqrt{x^{2}+y^{2}}}{x^{2}+y^{2}+z^{2}}\\ -\frac{y}{x^{2}+y^{2}} & \frac{x}{x^{2}+y^{2}} & 0 \end{array} \right]\tag{13.139}
and
Ji′i corresponds to [sin⁡θcos⁡φ−rcos⁡θcos⁡φ−rsin⁡θsin⁡φsin⁡θsin⁡φ−rcos⁡θsin⁡φ−rsin⁡θcos⁡φcos⁡θ−rsin⁡θ0].(13.140)J_{i^{\prime}}^{i}\text{ corresponds to }\left[ \begin{array} {lll} \sin\theta\cos\varphi & \phantom{-} r\cos\theta\cos\varphi & -r\sin\theta\sin\varphi\\ \sin\theta\sin\varphi & \phantom{-} r\cos\theta\sin\varphi & \phantom{-} r\sin\theta\cos\varphi\\ \cos\theta & -r\sin\theta & 0 \end{array} \right] .\tag{13.140}
Also show that in terms of r,θ,φr,\theta,\varphi,
Jii′ corresponds to [sin⁡θcos⁡φsin⁡θsin⁡φcos⁡θcos⁡θcos⁡φrcos⁡θsin⁡φr−sin⁡θr−sin⁡φrsin⁡θcos⁡φrsin⁡θ0](13.141)J_{i}^{i^{\prime}}\text{ corresponds to }\left[ \begin{array} {ccc} \sin\theta\cos\varphi & \sin\theta\sin\varphi & \cos\theta\\ \frac{\cos\theta\cos\varphi}{r} & \frac{\cos\theta\sin\varphi}{r} & -\frac{\sin\theta}{r}\\ -\frac{\sin\varphi}{r\sin\theta} & \frac{\cos\varphi}{r\sin\theta} & 0 \end{array} \right]\tag{13.141}
and use this form to confirm that the matrices corresponding to Ji′iJ_{i^{\prime }}^{i} and Jii′J_{i}^{i^{\prime}} are the inverses of each other. Repeat the task by instead converting Ji′iJ_{i^{\prime}}^{i} to Cartesian coordinates.
Exercise 13.7 Derive the following rule for the transformation of the Christoffel symbol Γk′,i′j′\Gamma_{k^{\prime},i^{\prime}j^{\prime}}:
Γk′,i′j′=Γk,ijJi′iJj′jJk′k+Jk′lJi′j′kZkl.(13.142)\Gamma_{k^{\prime},i^{\prime}j^{\prime}}=\Gamma_{k,ij}J_{i^{\prime}} ^{i}J_{j^{\prime}}^{j}J_{k^{\prime}}^{k}+J_{k^{\prime}}^{l}J_{i^{\prime }j^{\prime}}^{k}Z_{kl}.\tag{13.142}
This is one of the few relationships where the metric tensor appears in a contraction and does not get absorbed by index juggling. Of course, we could introduce a new symbol Jk,i′j′J_{k,i^{\prime}j^{\prime}} for the combination Ji′j′kZklJ_{i^{\prime}j^{\prime}}^{k}Z_{kl} but there is not a sufficiently compelling reason for doing so.
Exercise 13.8Use the equation
Γi′j′k′=Ji′j′kJkk′(13.128)\Gamma_{i^{\prime}j^{\prime}}^{k^{\prime}}=J_{i^{\prime}j^{\prime}}^{k} J_{k}^{k^{\prime}} \tag{13.128}
for the transformation of the Christoffel symbol from affine coordinates to calculate the elements of the Christoffel symbol in polar coordinates and cylindrical coordinates.
Exercise 13.9Repeat the previous exercise for spherical coordinates.
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