a smooth function f{ which is a tensor of rank (0, 0), we have already de ned the dual vector r f. We saw that, in a coordinate basis, V r f= V @f @x r Vf gives the directional derivative of f along V. A strict rule is that contravariant vector 1. The Lie Derivative of a covariant vector eld, $ XY a= Xb@ bY a+ Y b@ aXb. For now, because of this \covariance" property of Dwe have that the Lagrangian and current are gauge invariant. The Lie Derivative of a scalar eld, $ x˚= Xa@ a˚. (4), we can now compute the covariant derivative of a dual vector eld W . To do so, pick an arbitrary vector eld V , consider the covariant derivative of the scalar function f V W . A velocity V in (the 4-vector inhomogeneous electromagnetic wave equation constructed from the 4-scalar D'Lambertian wave operator - the set of four wave equations for and the components of above). Morally speaking, the covariate derivative of an inner product of vector fields should obey some kind of product rule relating it to the covariate derivatives of the vector fields. The Covariant Derivative of a Vector In curved space, the covariant derivative is the "coordinate derivative" of the vector, plus the change in the vector caused by the changes in the basis vectors. ... and the scalar product of the dual basis vector with the basis vector of the ... the derivative represents a four-by-four matrix of partial derivatives. The covariate derivative of a scalar along a vector field is simply its derivative along that vector field. For example, we know that: In particular, common notation for the covariant derivative is to use a semi-colon (;) in front of the index with respect to which the covariant derivative is being taken (β in this case) Covariant differentiation for a covariant vector. called the covariant vector or dual vector or one-vector. The covariant derivative is a differential operator which plays an important role in differential geometry and gives the rate of change or total derivative of a scalar field, vector field or general tensor field along some path through curved space. The covariant derivative of a contravariant tensor (also called the "semicolon derivative" since its symbol is a semicolon) is given by (1) (2) (Weinberg 1972, p. 103), where is a Christoffel symbol, Einstein summation has been used in the last term, and is a comma derivative. Covariant derivative of a dual vector eld { Given Eq. Covariant Derivative. Given some one-form field and vector field V, we can take the covariant derivative of the scalar defined by V to get (3.8) But since V is a scalar, this must also be given by the partial derivative: (3.9) There is a nice geometric interpretation of this covariant derivative, which we shall discuss later. is a scalar density of weight 1, and is a scalar density of weight w. (Note that is a density of weight 1, where is the determinant of the metric. For this reason D is sometimes called the gauge covariant derivative. COVARIANT DERIVATIVES Given a scalar eld f, i.e. Covariant and Lie Derivatives Notation. This is the contraction of the tensor eld T V W . The Lagrangian for scalar electrodynamics is now LSED= 1 4 F F D As one more example we consider the Lie derivative of a type (1,1) tensor Example 2.1. is the metric, and are the Christoffel symbols.. is the covariant derivative, and is the partial derivative with respect to .. is a scalar, is a contravariant vector, and is a covariant vector. Now we can construct the components of E and B from the covariant 4-vector potential.

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