Covariant derivative

The covariant derivative is a generalization of the concept of a derivative for tensor fields on manifolds . The concept of a covariant derivative is closely related to the concept of affine connection .

The covariant derivative of the tensor field $T$ ${\ displaystyle T}$ $T$ in the direction of the tangent vector ${\mathbf {v} }$ ${\ displaystyle {\ mathbf {v}}}$ ${{\ mathbf v}}$ usually indicated $\nabla _{\mathbf {v} }T$ ${\ displaystyle \ nabla _ {\ mathbf {v}} T}$ $\ nabla _ {{{\ mathbf v}}} T$ .

Content

1 Motivation
- 1.1 Notes
2 Formal definition
- 2.1 Scalar Functions
- 2.2 Vector fields
  - 2.2.1 Note
- 2.3 Covert fields
- 2.4 Tensor fields
3 Expression in coordinates
- 3.1 Examples for some types of tensor fields
4 See also
5 Literature

Motivation

The concept of a covariant derivative allows one to determine the differentiation of tensor fields in the direction of the tangent vector of a manifold. Like the directional derivative, the covariant derivative $\nabla _{\mathbf {u} }{\mathbf {v} }$ ${\ displaystyle \ nabla _ {\ mathbf {u}} {\ mathbf {v}}}$ takes as arguments: (1) the vector $\mathbf {u}$ ${\ displaystyle \ mathbf {u}}$ defined at a certain point $P$ ${\ displaystyle P}$ , and (2) the vector field $\mathbf {v}$ ${\ displaystyle \ mathbf {v}}$ defined in the neighborhood $P$ ${\ displaystyle P}$ . The result is a vector $\nabla _{\mathbf {u} }{\mathbf {v} }\left(P\right)$ ${\ displaystyle \ nabla _ {\ mathbf {u}} {\ mathbf {v}} \ left (P \ right)}$ also defined in $P$ ${\ displaystyle P}$ . The main difference from the directional derivative is that $\nabla _{\mathbf {u} }{\mathbf {v} }$ ${\ displaystyle \ nabla _ {\ mathbf {u}} {\ mathbf {v}}}$ should not depend on the choice of coordinate system .

Any vector can be represented as a set of numbers, which depends on the choice of basis . The vector as a geometric object does not change when the basis is changed, while the components of its coordinate representation change according to the covariant transformation , which depends on the basis transformation. The covariant derivative must obey the same covariant transformation.

In the case of Euclidean space, the derivative of a vector field is often defined as the limit of the difference of two vectors defined at two nearby points. In this case, one of the vectors can be moved to the beginning of another vector using parallel transfer, and then subtract. Thus, the simplest example of a covariant derivative is componentwise differentiation in an orthonormal coordinate system .

In the general case, it is necessary to take into account the change in the basis vectors during parallel transfer . Example: the covariant derivative written in the polar coordinates of a two-dimensional Euclidean space contains additional terms that describe the “rotation” of the coordinate system itself in parallel transfer. In other cases, the covariant derivative formula may include terms corresponding to compression, stretching, torsion, interlacing, and other transformations to which an arbitrary curvilinear coordinate system is subject.

As an example, consider a curve $\gamma \left(t\right)$ ${\ displaystyle \ gamma \ left (t \ right)}$ defined on the Euclidean plane. In polar coordinates, the curve can be expressed in terms of the polar angle and radius $\gamma \left(t\right)={\big (}r\left(t\right),\,\theta \left(t\right){\big )}$ ${\ displaystyle \ gamma \ left (t \ right) = {\ big (} r \ left (t \ right), \, \ theta \ left (t \ right) {\ big)}}$ . At an arbitrary point in time $t$ ${\ displaystyle t}$ radius vector can be represented in a couple $({\mathbf {e} }_{r},{\mathbf {e} }_{\theta })$ ${\ displaystyle ({\ mathbf {e}} _ {r}, {\ mathbf {e}} _ {\ theta})}$ where ${\mathbf {e} }_{r}$ ${\ displaystyle {\ mathbf {e}} _ {r}}$ and ${\mathbf {e} }_{\theta }$ ${\ displaystyle {\ mathbf {e}} _ {\ theta}}$ - unit vectors tangent to the polar coordinate system, which form the basis used to decompose the vector into radial and tangent components. When changing a parameter $t$ ${\ displaystyle t}$ a new basis arises, which is nothing but the old basis subjected to rotation. This transformation is expressed as the covariant derivative of the basis vectors, also known as Christoffel Symbols .

In curved space, such as, for example, the surface of the Earth, an unambiguous parallel transfer is not defined. Instead, the operation of parallel transfer of a vector from one point to another, which depends on the choice of the path, is defined. Indeed, imagine a vector ${\mathbf {e} }$ ${\ displaystyle {\ mathbf {e}}}$ defined at the point $Q$ ${\ displaystyle Q}$ (which lies at the equator), and directed towards the north pole. Using parallel translation, first move the vector along the equator without changing its direction, then raise ${\mathbf {e} }$ ${\ displaystyle {\ mathbf {e}}}$ along a meridian to the north pole, and let us drop back to the equator along another meridian. Obviously, such a movement of a vector along a closed path on a sphere will change its orientation. A similar phenomenon is caused by the curvature of the surface of the globe and is not observed in Euclidean space. It arises on manifolds when a vector moves along any (even infinitesimal) closed loop, which includes motion along at least two different directions. In this case, the limit of the infinitesimal increment of the vector is a measure of the curvature of the manifold.

Remarks

The definition of a covariant derivative does not use the concept of a metric. Moreover, for any choice of the space metric, there exists a unique torsion-free covariant derivative called the Levi-Civita connection . It is determined through the condition: the covariant derivative of the metric tensor is zero.
Derivative properties imply that $\nabla _{\mathbf {v} }{\mathbf {u} }$ ${\ displaystyle \ nabla _ {\ mathbf {v}} {\ mathbf {u}}}$ depends on an arbitrarily small neighborhood of the point $P$ ${\ displaystyle P}$ the same as, for example, the derivative of a scalar function along a curve at a given point $P$ ${\ displaystyle P}$ depends on an infinitesimal neighborhood of this point.
Information contained in a neighborhood of a point $P$ ${\ displaystyle P}$ , can be used to determine parallel vector translation. Also, the concepts of curvature , torsion, and geodesic lines can be introduced using only the concept of the covariant derivative and its generalization, such as linear connectivity .

Formal Definition

Scalar Functions

For scalar function $f$ ${\ displaystyle f}$ covariant derivative ${\nabla }_{\mathbf {v} }f$ ${\ displaystyle {\ nabla} _ {\ mathbf {v}} f}$ coincides with the usual derivative of the function in the direction of the vector field $\mathbf {v}$ ${\ displaystyle \ mathbf {v}}$ .

Vector Fields

Covariant derivative $\nabla$ ${\ displaystyle \ nabla}$ vector field ${\mathbf {u} }$ ${\ displaystyle {\ mathbf {u}}}$ in the direction of the vector field ${\mathbf {v} }$ ${\ displaystyle {\ mathbf {v}}}$ denoted by $\nabla _{\mathbf {v} }{\mathbf {u} }$ ${\ displaystyle \ nabla _ {\ mathbf {v}} {\ mathbf {u}}}$ determined by the following properties, for any vector $\mathbf {v}$ ${\ displaystyle \ mathbf {v}}$ , vector fields $\mathbf {u}$ ${\ displaystyle \ mathbf {u}}$ , $\mathbf {w}$ ${\ displaystyle \ mathbf {w}}$ and scalar functions $f$ ${\ displaystyle f}$ and $g$ ${\ displaystyle g}$ :

$\nabla _{\mathbf {v} }{\mathbf {u} }$ ${\ displaystyle \ nabla _ {\ mathbf {v}} {\ mathbf {u}}}$ linearly with ${\mathbf {v} }$ ${\ displaystyle {\ mathbf {v}}}$ , i.e $\nabla _{f{\mathbf {v} }+g{\mathbf {w} }}{\mathbf {u} }=f\nabla _{\mathbf {v} }{\mathbf {u} }+g\nabla _{\mathbf {w} }{\mathbf {u} }$ ${\ displaystyle \ nabla _ {f {\ mathbf {v}} + g {\ mathbf {w}}} {\ mathbf {u}} = f \ nabla _ {\ mathbf {v}} {\ mathbf {u} } + g \ nabla _ {\ mathbf {w}} {\ mathbf {u}}}$
$\nabla _{\mathbf {v} }{\mathbf {u} }$ ${\ displaystyle \ nabla _ {\ mathbf {v}} {\ mathbf {u}}}$ additive relative ${\mathbf {u} }$ ${\ displaystyle {\ mathbf {u}}}$ , i.e $\nabla _{\mathbf {v} }({\mathbf {u} }+{\mathbf {w} })=\nabla _{\mathbf {v} }{\mathbf {u} }+\nabla _{\mathbf {v} }{\mathbf {w} }$ ${\ displaystyle \ nabla _ {\ mathbf {v}} ({\ mathbf {u}} + {\ mathbf {w}} = \ nabla _ {\ mathbf {v}} {\ mathbf {u}} + \ nabla _ {\ mathbf {v}} {\ mathbf {w}}}$
$\nabla _{\mathbf {v} }{\mathbf {u} }$ ${\ displaystyle \ nabla _ {\ mathbf {v}} {\ mathbf {u}}}$ obeys the rule of the product , i.e. $\nabla _{\mathbf {v} }f{\mathbf {u} }=f\nabla _{\mathbf {v} }{\mathbf {u} }+{\mathbf {u} }\nabla _{\mathbf {v} }f$ ${\ displaystyle \ nabla _ {\ mathbf {v}} f {\ mathbf {u}} = f \ nabla _ {\ mathbf {v}} {\ mathbf {u}} + {\ mathbf {u}} \ nabla _ {\ mathbf {v}} f}$ where $\nabla _{\mathbf {v} }f$ ${\ displaystyle \ nabla _ {\ mathbf {v}} f}$ defined above.

Note

notice, that $\nabla _{\mathbf {v} }{\mathbf {u} }$ ${\ displaystyle \ nabla _ {\ mathbf {v}} {\ mathbf {u}}}$ at the point $p$ ${\ displaystyle p}$ depends only on the value $\mathbf {v}$ ${\ displaystyle \ mathbf {v}}$ at the point $p$ ${\ displaystyle p}$ and from the values $\mathbf {u}$ ${\ displaystyle \ mathbf {u}}$ in her neighborhood. In particular, the covariant derivative operator is not a tensor (despite the fact that its value on each tensor field is a tensor).

Covert fields

If the field of covectors is specified (i.e., once covariant tensors, also called 1-forms ) $\alpha$ ${\ displaystyle \ alpha}$ , its covariant derivative $\nabla _{\mathbf {v} }\alpha$ ${\ displaystyle \ nabla _ {\ mathbf {v}} \ alpha}$ can be determined using the following identity, which is satisfied for all vector fields $\mathbf {u}$ ${\ displaystyle \ mathbf {u}}$

\nabla _{\mathbf {v} }(\alpha ({\mathbf {u} }))=(\nabla _{\mathbf {v} }\alpha )({\mathbf {u} })+\alpha (\nabla _{\mathbf {v} }{\mathbf {u} }).

{\ displaystyle \ nabla _ {\ mathbf {v}} (\ alpha ({\ mathbf {u}})) = (\ nabla _ {\ mathbf {v}} \ alpha) ({\ mathbf {u}}) + \ alpha (\ nabla _ {\ mathbf {v}} {\ mathbf {u}}).}

The covariant derivative of the covector field along the vector field $\mathbf {v}$ ${\ displaystyle \ mathbf {v}}$ - also a covector field.

It is also possible to independently determine the covariant derivative of a covector field, not related to the derivative of vector fields. Then, in the general case, derivatives of scalars depend on their origin, and they speak of the nonmetricity of the affine connection associated with this covariant derivative. In the above definition, nonmetricity is zero.

Tensor Fields

Once the covariant derivative is defined for vector and covector fields, it can be easily generalized to arbitrary tensor fields using the Leibniz rule ( $\varphi$ ${\ displaystyle \ varphi}$ and ${\psi }$ ${\ displaystyle {\ psi}}$ - arbitrary tensors):

\nabla _{\mathbf {v} }(\varphi \otimes \psi )=(\nabla _{\mathbf {v} }\varphi )\otimes \psi +\varphi \otimes (\nabla _{\mathbf {v} }\psi ),

{\ displaystyle \ nabla _ {\ mathbf {v}} (\ varphi \ otimes \ psi) = (\ nabla _ {\ mathbf {v}} \ varphi) \ otimes \ psi + \ varphi \ otimes (\ nabla _ { \ mathbf {v}} \ psi),}

If $\varphi$ ${\ displaystyle \ varphi}$ and $\psi$ ${\ displaystyle \ psi}$ - tensor fields from the same tensor bundle, they can be added:

\nabla _{\mathbf {v} }(\varphi +\psi )=\nabla _{\mathbf {v} }\varphi +\nabla _{\mathbf {v} }\psi .

{\ displaystyle \ nabla _ {\ mathbf {v}} (\ varphi + \ psi) = \ nabla _ {\ mathbf {v}} \ varphi + \ nabla _ {\ mathbf {v}} \ psi.}

Coordinate expression

Let a tensor field of type $(p,q)$ ${\ displaystyle (p, q)}$ set by its components ${T^{i_{1}i_{2}\ldots i_{p}}}_{j_{1}j_{2}\ldots j_{q}}(\mathbf {x} )$ ${\ displaystyle {T ^ {i_ {1} i_ {2} \ ldots i_ {p}}} _ {j_ {1} j_ {2} \ ldots j_ {q}} (\ mathbf {x})}$ in some local coordinate system $x^{k}$ ${\ displaystyle x ^ {k}}$ , and the components are differentiable functions . Then the covariant derivative of the tensor field is a tensor of the type $(p,q+1)$ ${\ displaystyle (p, q + 1)}$ , which is determined by the formula:

$\nabla _{\ell }{T^{i_{1}i_{2}\ldots i_{p}}}_{j_{1}j_{2}\ldots j_{q}}={\frac {\partial {T^{i_{1}i_{2}\ldots i_{p}}}_{j_{1}j_{2}\ldots j_{q}}}{\partial x^{\ell }}}+\sum _{k=1}^{p}{T^{i_{1}\ldots k\ldots i_{p}}}_{j_{1}j_{2}\ldots j_{q}}\Gamma ^{i_{k}}{}_{\ell k}-\sum _{m=1}^{q}{T^{i_{1}i_{2}\ldots i_{p}}}_{j_{1}\ldots m\ldots j_{q}}\Gamma ^{m}{}_{\ell j_{m}}$ ${\ displaystyle \ nabla _ {\ ell} {T ^ {i_ {1} i_ {2} \ ldots i_ {p}}} _ {j_ {1} j_ {2} \ ldots j_ {q}} = {\ frac {\ partial {T ^ {i_ {1} i_ {2} \ ldots i_ {p}}} _ {j_ {1} j_ {2} \ ldots j_ {q}}} {\ partial x ^ {\ ell }}} + \ sum _ {k = 1} ^ {p} {T ^ {i_ {1} \ ldots k \ ldots i_ {p}}} _ {j_ {1} j_ {2} \ ldots j_ {q }} \ Gamma ^ {i_ {k}} {} _ {\ ell k} - \ sum _ {m = 1} ^ {q} {T ^ {i_ {1} i_ {2} \ ldots i_ {p} }} _ {j_ {1} \ ldots m \ ldots j_ {q}} \ Gamma ^ {m} {} _ {\ ell j_ {m}}}$

Where $\Gamma ^{k}{}_{ij}$ ${\ displaystyle \ Gamma ^ {k} {} _ {ij}}$ - Christoffel symbols expressing the connectedness of a curved variety.

Examples for some types of tensor fields

Covariant derivative of a vector field $V^{m}\$ ${\ displaystyle V ^ {m} \}$ compared with the partial derivative has an additional term,

\nabla _{\ell }V^{m}={\frac {\partial V^{m}}{\partial x^{\ell }}}+\Gamma ^{m}{}_{k\ell }V^{k}.\

{\ displaystyle \ nabla _ {\ ell} V ^ {m} = {\ frac {\ partial V ^ {m}} {\ partial x ^ {\ ell}}} + \ Gamma ^ {m} {} _ { k \ ell} V ^ {k}. \}

Covariant derivative of a scalar field $\varphi \$ ${\ displaystyle \ varphi \}$ coincides with the partial derivative,

\nabla _{i}\varphi ={\frac {\partial \varphi }{\partial x^{i}}}\

{\ displaystyle \ nabla _ {i} \ varphi = {\ frac {\ partial \ varphi} {\ partial x ^ {i}}} \}

and the covariant derivative of the covector field $\omega _{m}\$ ${\ displaystyle \ omega _ {m} \}$ -

\nabla _{\ell }\omega _{m}={\frac {\partial \omega _{m}}{\partial x^{\ell }}}-\Gamma ^{k}{}_{\ell m}\omega _{k}.\

{\ displaystyle \ nabla _ {\ ell} \ omega _ {m} = {\ frac {\ partial \ omega _ {m}} {\ partial x ^ {\ ell}}} - \ Gamma ^ {k} {} _ {\ ell m} \ omega _ {k}. \}

For torsion-free connectivity, the Christoffel symbols are symmetric, and the covariant derivatives of the scalar field commute:

\nabla _{i}\nabla _{j}\varphi =\nabla _{j}\nabla _{i}\varphi \

{\ displaystyle \ nabla _ {i} \ nabla _ {j} \ varphi = \ nabla _ {j} \ nabla _ {i} \ varphi \}

In the general case, covariant derivatives of tensors do not commute (see the curvature tensor ).

The covariant derivative of a tensor field of type $(2,0)$ ${\ displaystyle (2.0)}$ $A^{ik}\$ ${\ displaystyle A ^ {ik} \}$ is equal to

\nabla _{\ell }A^{ik}={\frac {\partial A^{ik}}{\partial x^{\ell }}}+\Gamma ^{i}{}_{m\ell }A^{mk}+\Gamma ^{k}{}_{m\ell }A^{im},\

{\ displaystyle \ nabla _ {\ ell} A ^ {ik} = {\ frac {\ partial A ^ {ik}} {\ partial x ^ {\ ell}}} + \ Gamma ^ {i} {} _ { m \ ell} A ^ {mk} + \ Gamma ^ {k} {} _ {m \ ell} A ^ {im}, \}

i.e

A^{ik}{}_{;\ell }=A^{ik}{}_{,\ell }+A^{mk}\Gamma ^{i}{}_{m\ell }+A^{im}\Gamma ^{k}{}_{m\ell }.\

{\ displaystyle A ^ {ik} {} _ {; \ ell} = A ^ {ik} {} _ {, \ ell} + A ^ {mk} \ Gamma ^ {i} {} _ {m \ ell} + A ^ {im} \ Gamma ^ {k} {} _ {m \ ell}. \}

For a tensor field with one upper, one lower index, the covariant derivative is

A^{i}{}_{k;\ell }=A^{i}{}_{k,\ell }+A^{m}{}_{k}\Gamma ^{i}{}_{m\ell }-A^{i}{}_{m}\Gamma ^{m}{}_{k\ell },\

{\ displaystyle A ^ {i} {} _ {k; \ ell} = A ^ {i} {} _ {k, \ ell} + A ^ {m} {} _ {k} \ Gamma ^ {i} {} _ {m \ ell} -A ^ {i} {} _ {m} \ Gamma ^ {m} {} _ {k \ ell}, \}

finally, for a twice covariant tensor field, that is, a field of type $(0,2)$ ${\ displaystyle (0,2)}$ ,

A_{ik;\ell }=A_{ik,\ell }-A_{mk}\Gamma ^{m}{}_{i\ell }-A_{im}\Gamma ^{m}{}_{k\ell }.\

{\ displaystyle A_ {ik; \ ell} = A_ {ik, \ ell} -A_ {mk} \ Gamma ^ {m} {} _ {i \ ell} -A_ {im} \ Gamma ^ {m} {} _ {k \ ell}. \}

Literature

Rashevsky P.K. Rimanova geometry and tensor analysis. - Any edition.