Definitions
from The American Heritage® Dictionary of the English Language, 4th Edition
 adj. Physics Expressing, exhibiting, or relating to covariant theory.
 adj. Statistics Varying with another variable quantity in a manner that leaves a specified relationship unchanged.
from Wiktionary, Creative Commons Attribution/ShareAlike License
 adj. (Of a functor) which preserves composition
 adj. Using or relating to covariance.
from the GNU version of the Collaborative International Dictionary of English
 n. A function involving the coefficients and the variables of a quantic, and such that when the quantic is lineally transformed the same function of the new variables and coefficients shall be equal to the old function multiplied by a factor. An invariant is a like function involving only the coefficients of the quantic.
from The Century Dictionary and Cyclopedia
 n. In mathematics, a function which stands in the same relation to the primitive function from which it is derived as any of its linear transforms to a similarly derived transform of its primitive; a function of the coefficients and variables of a given quantic, such that when the quantic is linearly transformed, the same function of the new variables and coefficients is equal to the old function multiplied by some power of the modulus of transformation. Covariants were discovered by Cayley, and so named by Sylvester, 1852.
from WordNet 3.0 Copyright 2006 by Princeton University. All rights reserved.
 adj. changing so that interrelations with another variable quantity or set of quantities remain unchanged
Etymologies
Sorry, no etymologies found.
Examples

Over the previous year, he had been determined to find a gravitation theory that was generally covariant, that is, whose equations were unchanged by arbitrary transformation of the spacetime coordinates.

The methods of the SimpleJDBCTemplate class take advantage of the new Java 1.5 features such as covariant return types.

The covariant form of the metric tensor is expressed in terms of three parameters, m, e, and a by ds2 = ρ2dθ2 – 2a sin2θdrdφ + 2drdu + …
Bad Language: Metric vs Metric Tensor vs Matrix Form vs Line Element

An example would be a terse equation from higher mathematics, such as from Einstein's explanation of covariant tensors.

The main problem for the latter is the general covariance of the field equations of General Relativity: any spacetime model and its image under a diffeomorphism (a infinitely differentiable, oneone and onto mapping of the model to itself) are in all observable respects equivalent to one another; all physical properties are expressed in terms of generally covariant relationships between geometrical objects.

So, as you say in your book, covariant derivative of is:

But then you take the commutator of two covariant derivatives acting on this spinor field and get a relation involving the Riemann tensor.

Being that the supersymmetry variation of the gravitino is the covariant derivative of the local spinor parameterizing the supersymmetry transformation, this would seem to always imply that the condition for supersymmetry is that there must exist a covariantly constant spinor field.

This equation is essentially a covariant form of the Dirac equation (or really the Weyl equation).

Later in 1913 he sought to transform his failure into a victory of sorts: he thought he could show that no generally covariant theory at all is admissible.
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