Suppose we have the Lagrangian density for a triplet of real scalar fields, $$ L = \sum_{a=1}^3 \left[ \frac{1}{2}\partial_\mu\phi_a\partial^\mu\phi_a - \frac{1}{2}\phi_a\phi_a \right]. $$
How do you show that it's invariant under the infinitesimal $SO(3)$ transformation $$ \phi_a \to \phi_a + \theta \epsilon_{abc}n_b\phi_c, $$
where $\theta$ is a constant and $n_b$ a unit vector. I'm not really sure actually what to do after simply substituting $\phi_a$.
This is easiest done not using index notation. Define the column vector
$$ \vec{\phi} = \begin{pmatrix} \phi_1 \\ \phi_2 \\ \phi_3 \end{pmatrix} $$
The derivative acts on the column vector as $\mathbf{1}_3\partial_\mu$ where $\mathbf{1}_3$ is the identity matrix on the same vector space in which $\vec{\phi}$ lives. That is,
$$ \partial_\mu \vec{\phi} = \begin{pmatrix} \partial_\mu \phi_1 \\ \partial_\mu \phi_2 \\ \partial_\mu \phi_3 \end{pmatrix} $$
Then we can write the lagrangian as
$$ \mathcal{L}=\frac{1}{2}(\partial_\mu \vec{\phi})^T (\partial_\mu \vec{\phi}) - \frac{1}{2}\vec{\phi}^T \vec{\phi} $$
Now it can be seen basically by inspection that for $R\in SO(3)$ we have that
$$ \vec{\phi} \stackrel{SO(3)}{\longmapsto} R\vec{\phi} \tag{(a)}\\ \vec{\phi}^T \stackrel{SO(3)}{\longmapsto} \vec{\phi}^T R^T $$
Under this transfermation, both terms in the lagrangian both terms pick up an $R^TR = \mathbf{1}_3$ sandwiched in the middle. Hence the lagrangian is invariant.
Relating to the transformation law using index notation:
$$ \phi_a \to \phi_a + \theta \epsilon_{abc}n_b\phi_c $$
A general $SO(3)$ transformation can be written as
$$ R = e^{-iL_an_a \theta} $$
so that an infinitesimal transformation can be written as
$$ R = 1 -iL_an_a \theta + \mathcal{O}(\theta^2) $$
Now, in the fundamental representation the generators are given by
$$ (L^a)_{bc} = -\epsilon_{abc} $$
We therefore see that an infinitesimal SO(3) transformation on $\phi_a$ is given as
$$ \phi_a \to R_{ab}\phi_b = \phi_a + i\theta\epsilon_{abc}n_c \phi_b + \mathcal{O}(\theta^2) \tag{b} $$
So we see that proving the lagrangian is invariant under (b) is a special case of proving it is invariant under (a).
