Here one considers a modification of the directional derivative by a certain linear operator, whose components are called the Christoffel symbols, which involves no derivatives on the vector field itself.
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More precisely, when is differentiable, the dot product of the gradient of with a given unit vector is equal to the directional derivative of in the direction of that unit vector.
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The covariant formulation of the directional derivative of any tensor field along a vector v ^ \ gamma may be expressed as its contraction with the covariant derivative, e . g .:
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Where \ partial _ X and \ partial _ Y denote the operations of taking the directional derivatives with respect to " X " and " Y ", respectively.
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This form has 64 coefficients a _ { ijk }; requiring the function to have a given value or given directional derivative at a point places one linear constraint on the 64 coefficients.
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Both a directional derivative and a functional derivative are expressed as real numbers, the first in case of a finite number of variables and the second for an infinite number so to say.
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The primary difference from the usual directional derivative is that the covariant derivative must, in a certain precise sense, be independent of the manner in which it is expressed in a coordinate system.
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That means if you take a directional derivative in any direction, it will agree with what you'd get for the tangent space . talk ) 05 : 15, 17 December 2009 ( UTC)
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This above definition can be generalized to real-valued functions " f " defined on subsets of "'R " "'n " using a weaker version of the directional derivative.
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The constant C is easy to determine in the important special case described in terms of the directional derivative of X at a given point ( of the sphere ) in a given direction ( tangential to the sphere ).
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