# Geometric Calculus - Covariant Derivative

Covariant Derivative

A sufficiently smooth k-surface in an n-dimensional space is deemed a manifold. To each point on the manifold, we may attach a k-blade B that is tangent to the manifold. Locally, B acts as a pseudoscalar of the k-dimensional space. This blade defines a projection of vectors onto the manifold:

Just as the geometric derivative is defined over the entire n-dimensional space, we may wish to define an intrinsic derivative, locally defined on the manifold:

If a is a vector tangent to the manifold, then indeed both the geometric derivative and intrinsic derivative give the same directional derivative:

Although this operation is perfectly valid, it is not always useful because itself is not necessarily on the manifold. Therefore we define the covariant derivative to be the forced projection of the intrinsic derivative back onto the manifold:

Since any general multivector can be expressed as a sum of a projection and a rejection, in this case

we introduce a new function, the shape tensor, which satisfies

where is the commutator product. In a local coordinate basis spanning the tangent surface, the shape tensor is given by

Importantly, on a general manifold, the covariant derivative does not commute. In particular, the commutator is related to the shape tensor by

Clearly the term is of interest. However it, like the intrinsic derivative, is not necessarily on the manifold. Therefore we can define the Riemann tensor to be the projection back onto the manifold:

Lastly, if F is of grade r, then we can define interior and exterior covariant derivatives as

and likewise for the intrinsic derivative.

### Other articles related to "derivative, covariant derivative, derivatives, covariant derivatives":

Mathematics Of General Relativity - Tensorial Derivatives - The Lie Derivative
... Another important tensorial derivative is the Lie derivative ... Unlike the covariant derivative, the Lie derivative is independent of the metric, although in general relativity one usually uses an expression that seemingly depends on the metric ... Whereas the covariant derivative required an affine connection to allow comparison between vectors at different points, the Lie derivative uses a congruence from a vector ...
Covariant Derivative - Relation To Lie Derivative
... A covariant derivative introduces an extra geometric structure on a manifold which allows vectors in neighboring tangent spaces to be compared ... from different vector spaces, as is necessary for this generalization of the directional derivative ... is however another generalization of directional derivatives which is canonical the Lie derivative ...
Riemannian Connection On A Surface - Covariant Derivative
... The assignment defines an operator on (M) called the covariant derivative, satisfying the following properties is C∞(M)-linear in X (Leibniz rule for derivation of a module) (compatibi ...
Mathematics Of General Relativity - Tensorial Derivatives - The Covariant Derivative
... The formula for a covariant derivative of along associated with connection turns out to give curve-independent results and can be used as a "physical definition" of a ... The expression in brackets, called a covariant derivative of (with respect to the connection) and denoted by, is more often used in calculations A covariant derivative of X ... By definition, a covariant derivative of a scalar field is equal to the regular derivative of the field ...
List Of Formulas In Riemannian Geometry - Christoffel Symbols, Covariant Derivative
... The covariant derivative of a vector field with components is given by and similarly the covariant derivative of a -tensor field with components is given by For a -tensor field with components this becomes and ... The covariant derivative of a function (scalar) is just its usual differential Because the Levi-Civita connection is metric-compatible, the covariant derivatives of ...

### Famous quotes containing the word derivative:

Poor John Field!—I trust he does not read this, unless he will improve by it,—thinking to live by some derivative old-country mode in this primitive new country.... With his horizon all his own, yet he a poor man, born to be poor, with his inherited Irish poverty or poor life, his Adam’s grandmother and boggy ways, not to rise in this world, he nor his posterity, till their wading webbed bog-trotting feet get talaria to their heels.
Henry David Thoreau (1817–1862)