# Invariance, covariance and symmetry

Though often heard, often read, often felt being overused, I wonder what are the precise definitions of invariance and covariance. Could you please give me an example from field theory?

In the context of field theory, one can make these notions precise as follows. Consider a theory of fields $\phi$. Let a transformation $T$ $$\phi \to\phi_T$$ on fields be given. Let a functional $F[\phi]$ of the fields be given (consider the action functional for example). The functional is said to be invariant under the transformation $T$ of the fields provided $$F[\phi_T] = F[\phi]$$ for all fields $\phi$. One the other hand, the equations of motion of the theory are said to be covariant with respect to the transformation $T$ provided if the fields $\phi$ satisfy the equations, then so do the fields $\phi_T$; the form of the equations is left the same by $T$.
For example, the action of a single real Klein-Gordon scalar $\phi$ is Lorentz-invariant meaning that it doesn't change under the transformation $$\phi(x)\to\phi_\Lambda(x) = \phi(\Lambda^{-1}x),$$ and the equations of motion of the theory are Lorentz-covariant in the sense that if $\phi$ satisfies the Klein-Gordon equation, then so does $\phi_\Lambda$.
Example: Vectors in $R^{2}$, under rotation $R_{ij}$, transform covariantly since $v'_{i}=R_{ij}v_{j}$, but it's length is invariant since $|v'|^{2}=v'_{a}v'_{a}=R_{am}v_{m}R_{an}v_{n}=v_{m}R^{t}_{ma}R_{an}v_{n}=v_{m} \delta_{mn} v_{n}=v_{n}v_{n}=|v|^{2}$. This means that Newton's second Law transforms covariantly under rotations and the magnitude of the force is invariant.