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    LECTURE NOTE Covariant Derivative

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    Rivu Saha
    The document discusses covariant derivatives. [1] In curved coordinate systems, taking the derivative of a vector field is complicated because the basis vectors change direction at different points. [2] The Christoffel symbol describes how basis vectors change and can be related to the metric tensor. [3] The covariant derivative accounts for changes in basis vectors when taking derivatives, allowing vectors to be parallel transported consistently across a curved space.

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    © All Rights Reserved
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    LECTURE NOTE Covariant Derivative

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    Rivu Saha
    19 views4 pages
    The document discusses covariant derivatives. [1] In curved coordinate systems, taking the derivative of a vector field is complicated because the basis vectors change direction at different points. [2] The Christoffel symbol describes how basis vectors change and can be related to the metric tensor. [3] The covariant derivative accounts for changes in basis vectors when taking derivatives, allowing vectors to be parallel transported consistently across a curved space.

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    LECTURE_NOTE_covariant_derivative

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    The document discusses covariant derivatives. [1] In curved coordinate systems, taking the derivative of a vector field is complicated because the basis vectors change direction at different points. [2] The Christoffel symbol describes how basis vectors change and can be related to the metric tensor. [3] The covariant derivative accounts for changes in basis vectors when taking derivatives, allowing vectors to be parallel transported consistently across a curved space.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    19 views4 pages

    LECTURE NOTE Covariant Derivative

    Uploaded by

    Rivu Saha
    The document discusses covariant derivatives. [1] In curved coordinate systems, taking the derivative of a vector field is complicated because the basis vectors change direction at different points. [2] The Christoffel symbol describes how basis vectors change and can be related to the metric tensor. [3] The covariant derivative accounts for changes in basis vectors when taking derivatives, allowing vectors to be parallel transported consistently across a curved space.

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    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 4

    Lecture Note: Covariant Derivative

    KalyanmoyChatterjee
    March 29, 2023

    1 Introduction
    Taking Derivative of a vector or a vector filed is prettymuch straight forward
    in cartesian co-ordinate, where a set of basis vectors is constant through
    out the space (e, g, î, ĵ, k̂ is same in magnitude and direction). However,
    in case of curved coordinate system, such as spherical-polar or cylindracal
    coordinate system, it is not so simple, because the basis vectors point to
    different directions in different points in that space.
    Consider a vector expressed in terms of general co-ordinate x1 , x2 , x3 and
    basis e~1 , e~2 , e~3 :
    ~ = A1 e~1 + A2 e~2 + A3 e~3
    A
    = Ai ei

    ~ with respect to any coordinate xj is:


    Derivative of A

    ~
    ∂A ∂Ai e~i
    =
    ∂xj ∂xj
    ∂Ai i ∂e
    ~i
    = e
    ~i + A (1)
    ∂xj ∂xj
    The second term in this equation goes to zero in cartesian coordinate system,
    but makes the life complecated in curvilinear coordinate system. Similar
    problem occure for higher rank tensor aswell. Covariant derivative, takes
    care for such change in the basis vectors and ensures that the derivative of a
    tensor is always another tensor.

    1
    2 Christofell symbol
    To understand the covariant derivative, first we have to understand the
    Christofell symbol. Keeping the fact in mind that the derivative of a ba-
    sis vectror will also be a vector, and could be written in terms of weighted
    sum of the basis vecrotrs, we can write:
    ∂ e~i
    = Γkij .e~k (2)
    ∂xj
    where the weight factor Γkij is the Christofell symbol. Index i specifies the
    basis vector for which the derivative is being taken, the index j denotes the
    coordinate being varied to induce this change in the ith basis vector, and
    the index k identifies the direction in which this component of the derivative
    points.

    Example: if e~r and e~θ are two basises in 2D, then:


    ∂ e~r
    = Γkrθ .e~k
    ∂ e~θ
    = Γrrθ .~
    er + Γθrθ .e~θ

    2.1 Relation between metric tensor and Christofell sym-


    bol
    It takes a bit algebra to find the realtion between metric tensor and the
    Christofell symbol, which is given by :
     
    l 1 kl ∂gik ∂gjk ∂gij
    Γij = g + − k (3)
    2 ∂xj ∂xi ∂x
    We can aslo write:
     
    1 ∂gik ∂gjk ∂gij
    Γijk = + − k (4)
    2 ∂xj ∂xi ∂x
    Γlij = g kl Γijk (5)
    Γlij is called the Christofell symbol of second kind where as Γijk is known as
    the Christofell symbol of first kind.

    Example In cylindrical coordinate system, differential lenth is given by


    ds2 = dr2 + r2 dφ2 + dz 2 . Find out the the covariant metric tensor
    and the contravariant metric tensor. From there, find out Γ111 and Γ122 .

    2
    3 Covariant Derivative
    In Euclidean space, two vectors at different locations may be compared and
    combined by dragging one of the vectors to the location of the other without
    changing its magnitude or its direction. If the vector is expanded using
    Cartesian coordinates, such “parallel transport” is accomplished simply by
    keeping each of its components the same (because the Cartesian basis vectors
    have the same magnitude and direction everywhere). But if the vector is
    expressed in non-Cartesian coordinates, the length and direction of the basis
    vectors may be different at the two locations. In such cases, the covariant
    derivative provides a means of parallel-transporting one of the vectors to the
    location of the other. [?]
    You can understand the role of the covariant derivative by considering
    a two-dimensional spherical surface embedded in a three- dimensional Eu-
    clidean space. Imagine a series of tangent planes just touching the sphere at
    each location, and picture a vector lying in one of those tangent planes. If
    that vector is moved to a different location on the sphere while holding its
    direction constant (as viewed in the larger three-dimensional space), it will
    not lie in the tangent plane at the new location (you can think of the vector
    as “sticking out” of the two-dimensional space of the sphere). In such cases,
    the covariant derivative serves to project the derivative of the vector into the
    tangent space of the sphere. [?].
    Equation 1 and 2 could be writen as:
    ∂A~ ∂Ai ~i
    = e + Ai Γkij e~k
    ∂xj ∂xj
    Interchanging the dummy variable i and k:
    ~
    ∂A ∂Ai ~i
    = e + Ak Γikj e~i
    ∂xj ∂xj
    ∂Ai
    
    k i
    = + A Γkj e~i (6)
    ∂xj
    The covariant derivative is defined as the combination of the terms inside the
    parentheses and denoed by asemicolon (;) in front of the index with respect
    to which the covariant derivative is being taken.
    ∂Ai
    Ai;j = + Ak Γikj (7)
    ∂xj
    From a similar analysis, co-variant derivative of covariant components
    will be:
    ∂Ai
    Ai;j = j
    − Ak Γkij (8)
    ∂x
    3
    Example In a cylindracal corodinate system a vecrotrs A~ is expressed by
    the coordinates x = r, x = φ, x = z. find (a) A;φ and (b) Aφ;φ
    1 2 3 r

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