9.4.1 Some rules of tensor analysis on manifolds

For many pragmatic situations, the rules of tensor analysis can be thought of as a systematic way to apply the chain rule on curved manifolds. More fundamentally, tensor analysis provides a general formalism which efficiently exploits the linkage between analysis and geometry. In turn, it can render a deeper and more concise description of physical laws without being diverted by often cumbersome coordinate dependent statements.

One of the key reasons that tensor analysis is so useful in physics is that an equation written in a form which respects a basic set of tensor rules remains form invariant under changes in coordinates. Consequently, one can work within a simple set of coordinates, such as Cartesian, in order to establish results which are then easily generalizable to other coordinates. This result allows for much of the discussion in this chapter to employ Cartesian tensors, as in the work of Smagorinsky (1993) and Wajsowicz (1993). However, to facilitate the eventual transition to curvilinear coordinates, the approach taken here is to employ the notation of curvilinear tensor analysis (e.g., Aris 1962). The purpose of this section is to summarize salient aspects of tensor analysis. Use of the following rules and ideas will prove sufficient.

• Conservation of indices: Lower and upper tensor indices are balanced across equal signs.
• Einstein summation convention: Repeated indices are summed, unless otherwise noted.
• Metric tensor: The metric tensor provides a means to measure the distance between two points on a manifold:
  

  $$g_{mn} \, d\xi^{m} \, d\xi^{n}.$$ (ds)² = (9.66)

  
  
  In this expression, (ds)², often written ds², is the squared infinitesimal arc-length between the points, $\xi^{m}$ is the coordinate for a point, and m=1,2,3 labels the coordinate (m is not a power). The metric for spherical coordinates on a sphere is diagonal. With coordinates $(\xi^{1},\xi^{2},\xi^{3}) = (\lambda,\phi,r)$, where $\lambda$ is longitude and $\phi$ latitude, the metric is
  

  $$g_{mn} = \mbox{diag}(g_{\lambda \lambda}, g_{\phi \phi}, g_{rr}) = \mbox{diag}(r^{2}\cos^{2}\phi, r^{2}, 1).$$ (9.67)

  
  
  The inverse metric components g^mn will also be needed, and they are given by
  

  $$g^{mn} = \mbox{diag}((r \, \cos\phi)^{-2}, r^{-2}, 1).$$ (9.68)

  
  

  In Cartesian coordinates, the metric tensor is given by
  

  $$g_{mn} = \delta_{mn} = \delta^{mn} = \delta^{m}_{n},$$ (9.69)

  
  
  where $\delta$ is the unit or Kronecker delta tensor. There is no distinction between raised and lowered indices in Cartesian coordinates, hence the ability to jettison the conservation of indices rule. For curvilinear coordinates, however, this rule is essential.

• Covariant and contravariant: A lower label is often termed ``covariant'' and an upper label ``contravariant.'' The mnemonic ``co-low'' assists in remembering the terminology. Modern tensor language jettisons this terminology, yet it will be sufficient for the following. Covariant and contravariant tensors can be considered dual, where the connection is through the metric tensor. For example, the covariant components to the velocity vector u_m are related to the contravariant components through
  

  u_m = g_mn uⁿ. (9.70)

  
  
  Some examples are useful. In Cartesian coordinates, the velocity vector takes the familiar form
  

  $$(u^{1}, u^{2}, u^{3}) = (u_{1}, u_{2}, u_{3}) = \left( \frac{D x}{Dt}, \frac{D y}{Dt}, \frac{D z}{Dt} \right),$$ (9.71)

  
  
  where again there is no distinction between covariant and contravariant for Cartesian tensors. In spherical coordinates, however,
  

  $$\left( \frac{D \lambda}{Dt}, \frac{D \phi}{Dt}, \frac{D r}{Dt} \right),$$ (u¹, u², u³) = (9.72)

  
  
  whereas the covariant components are
  

  $$\left( (r \, \cos\phi)^{2} \, \frac{D \lambda}{Dt}, r^{2} \, \frac{D \phi}{Dt}, \frac{D r}{Dt} \right).$$ (u₁, u₂, u₃) = (9.73)

  
  

• Notation: As the above indicates, for curvilinear tensor analysis the difference between a raised and lowered label is important. Additionally, in order to avoid confusion, partial derivatives will be denoted with a comma:
  

  $$u_{m \, ,n} = \frac{\partial u_{m}}{\partial \xi^{n}}.$$ (9.74)

  
  

• Covariant derivative: In order to account for nonconstant unit vectors on a curved manifold, it is necessary to generalize partial derivatives to so-called covariant derivatives. In particular, the strain tensor (described later) has components
  

  $$e_{m n} = \frac{1}{2} \, (u_{m \, ; \, n} + u_{n \, ; \, m}),$$ (9.75)

  
  
  where the comma has been generalized to a semi-colon. For a ``torsionless'' manifold, such as a sphere, each component of the metric tensor has a vanishing covariant derivative
  

  g_{mn ; p} = 0. (9.76)

  
  
  This is a trivial property for Cartesian coordinates on a plane, in which case the metric is the constant unit tensor and the covariant derivative a partial derivative
  

  $$\delta_{mn ; p} = \delta_{mn , p} = 0.$$ (9.77)

  
  
  However, for curvilinear coordinates g_{mn ; p} = 0 is quite useful. For example, it provides for the convenient relation
  

  $$(g_{mp} \, u^{p})_{\, ; n}$$ u_{m ; n} =

  $$g_{mp} \, u^{p}_{\, ; n}.$$ = (9.78)

  
  
  This result brings the strain tensor to the form
  
   

  $$2 \, e_{mn} g_{mp} \, u^{p}_{\, ; n} + g_{np} \, u^{p}_{\, ; m}.$$ = (9.79)

  
  
  In general, the covariant derivative of a vector on a torsionless manifold is given by
  
   

  $$u^{p}_{\, ; n} u^{p}_{, \, n} + \Gamma^{p}_{mn} \, u^{n},$$ = (9.80)

  
  
  where $\Gamma^{p}_{mn}$ are components to the Christoffel symbol, which is given by
  
   

  $$\Gamma^{p}_{mn} \frac{1}{2} \, g^{pq} \, (g_{qm,n} + g_{qn,m} - g_{mn,q}).$$ = (9.81)

  
  
  A more geometric means of understanding the Christoffel symbol is to note that they form the expansion coefficients of the partial derivative of the basis vectors for a manifold
  
   

  $$\vec{e}_{a,b} = \Gamma^{m}_{ab} \, \vec{e}_{m}.$$ (9.82)

  
  
  That is, the Christoffel symbol accounts for the nonzero changes in the basis vectors on a curved manifold. Note that it is symmetric on the lower two labels:
  

  $$\Gamma^{p}_{mn} = \Gamma^{p}_{nm},$$ (9.83)

  
  
  which is the defining property of torsionless manifolds.
• Transformation rules: Under a coordinate transformation
  

  $$\xi^{\overline{m}} = \xi^{\overline{m}}(\xi^{m}),$$ (9.84)

  
  
  tensors transform as, for example,
  

  $$e_{\overline{m} \overline{n}} \Lambda^{m}_{\; \overline{m}} \, \Lambda^{n}_{\; \overline{n}} \, e_{mn},$$ = (9.85)

  
  
  where the transformation matrix is given by the partial derivatives
  

  $$\Lambda^{m}_{\; \overline{m}} \frac{\partial \xi^{m}}{\partial \xi^{\overline{m}}}.$$ = (9.86)

  
  
  Sometimes it is useful to write the transformation matrix in traditional matrix form. The convention is that the index which is placed a bit closer to the $\Lambda$ denotes the row (m in $\Lambda^{m}_{\; \overline{m}}$), and the one pushed away a bit is the column ( $\overline{m}$ in $\Lambda^{m}_{\; \overline{m}}$). The inverse transformation of a tensor takes the form
  

  $$\Lambda^{\overline{m}}_{\; m} \, \Lambda^{\overline{n}}_{\; n} \, e_{\overline{m} \overline{n}},$$ e_mn = (9.87)

  
  
  where the inverse transformation matrix is given by
  

  $$\Lambda^{\overline{m}}_{\; m} \frac{\partial \xi^{\overline{m}}}{\partial \xi^{m}}.$$ = (9.88)

  
  
  Generalizations to tensors of different order follow analogously.