Definitions
from The American Heritage® Dictionary of the English Language, 4th Edition
 adj. Not varying; constant.
 adj. Mathematics Unaffected by a designated operation, as a transformation of coordinates.
 n. An invariant quantity, function, configuration, or system.
from Wiktionary, Creative Commons Attribution/ShareAlike License
 adj. not varying; constant
 adj. Unaffected by a specified operation (especially by a transformation)
 adj. Neither covariant nor contravariant.
 n. An invariant quantity, function etc.
from the GNU version of the Collaborative International Dictionary of English
 n. An invariable quantity; specifically, a function of the coefficients of one or more forms, which remains unaltered, when these undergo suitable linear transformations.
from The Century Dictionary and Cyclopedia
 Not varying or changing; remaining always the same.
 n. In mathematics, a function of the coefficients of a quantic such that, if the quantic is linearly transformed, the same function of the new coefficients is equal to the first function multiplied by some power of the modulus of transformation.
 n. See the adjectives.
 In physical chemistry, having a variance equal to zero.
 n. An entity compounded of constituents, some of them subject to change or variation, which, despite this change, remains itself unchanged.
from WordNet 3.0 Copyright 2006 by Princeton University. All rights reserved.
 n. a feature (quantity or property or function) that remains unchanged when a particular transformation is applied to it
 adj. unvarying in nature
 adj. unaffected by a designated operation or transformation
Etymologies
Sorry, no etymologies found.
Examples

He goes so far as to say: “I think the idea of invariant is the clue to a relational concept of reality, not only in physics but in every aspect of the world.”

At least when the spacetime background is timetranslation invariant, which is a very good approximation here in the Solar System.

Tumulka believes that a GRWlike theory may be fully Lorentzinvariant which is clearly wrong: his GRWlike "counterexample" to the KochenConway theorem has no interactions which is

In his work on symplectic geometry, Gromov found that different types of space also have unique, identifying 'invariant' characteristics.

It has allowed major progress in classical areas of algebraic geometry such as invariant theory and the moduli of curves.

More carefully, Noether proved that a physical system described by a Lagrangian invariant with respect to the symmetry transformations of a Lie group, has, in the case of a group with a finite number of independent infinitesimal generators, a conservation law for each such generator.
Special Post: Noether’s First Theorem – Emmy Noether for Ada Lovelace Day

The observable is conserved if and only if the equations of motion are invariant under the transformations generated by the corresponding [Hermitian] operator
Special Post: Noether’s First Theorem – Emmy Noether for Ada Lovelace Day

This always struck me as redundant, since if we assume the system obeys the equations of motion, the action must be invariant under ANY infinitesimal variation (since the EOM are found by assuming that the action is at an extremum).
Special Post: Noether’s First Theorem – Emmy Noether for Ada Lovelace Day

The line element, we know, is incredibly useful, as it provides us with an invariant quantity and also imparts information about causal structure.
Bad Language: Metric vs Metric Tensor vs Matrix Form vs Line Element

Greg proved two things, the space defined by labor alone and the space defined by labor and height are spectrally invariant.
Mankiw Defends his Tall Tale, Arnold Kling  EconLog  Library of Economics and Liberty
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