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The covariant derivative is a generalization of the directional derivative from vector calculus.As with the directional derivative, the covariant derivative is a rule, , which takes as its inputs: (1) a vector, u, defined at a point P, and (2) a vector field v defined in a neighborhood of P. [7]
In the math branches of differential geometry and vector calculus, the second covariant derivative, or the second order covariant derivative, of a vector field is the derivative of its derivative with respect to another two tangent vector fields.
For example, for a macroscopic scalar field φ(x, t) and a macroscopic vector field A(x, t) the definition becomes: +, +. In the scalar case ∇ φ is simply the gradient of a scalar, while ∇ A is the covariant derivative of the macroscopic vector (which can also be thought of as the Jacobian matrix of A as a function of x ).
The components v i [f] are the contravariant components of the vector v in the basis f, and the components v i [f] are the covariant components of v in the basis f. The terminology is justified because under a change of basis,
A covariant derivative in differential geometry is a linear differential operator which takes the directional derivative of a section of a vector bundle in a covariant manner. It also allows one to formulate a notion of a parallel section of a bundle in the direction of a vector: a section s is parallel along a vector X {\displaystyle X} if ∇ ...
A covariant derivative is defined to be any operation (,) which mimics these properties, together with a form of the product rule. Unless the base is zero-dimensional, there are always infinitely many connections which exist on a given differentiable vector bundle, and so there is always a corresponding choice of how to differentiate sections.
The derivative with respect to a vector as discussed above can be generalized to a derivative with respect to a general multivector, called the multivector derivative. Let F {\displaystyle F} be a multivector-valued function of a multivector.
The covariant derivative incorporates the 4-gradient plus spacetime curvature effects via the Christoffel symbols The strong equivalence principle can be stated as: [ 4 ] : 184 "Any physical law which can be expressed in tensor notation in SR has exactly the same form in a locally inertial frame of a curved spacetime."