I have been reading Introduction to Conformal Field Theory by Blumenhagen and Plauschinn. Equation (2.19) on page 19 states that if our theory is invariant under a general conformal transformation $x^\mu \rightarrow x'^\mu = x^\mu + \epsilon^\mu(x)$, then the conserved current is given by
$$ j^\mu = \epsilon^\nu T^\mu_{\ \nu}\tag{2.19}.$$
I have been trying to show this myself but I can't. The general procedure of Noether's theorem is to evaluate the off-shell variation and see whether it is given by the integral of a total derivative. We then equate this with the on-shell variation to yield the conserved current.
My attempt
Let the action be of the form
$$ S[\phi] = \int \mathrm{d}^3 x \mathcal{L}(\phi,\partial \phi).$$
We know that if the theory is translationally invariant under $x^\mu \rightarrow x'^\mu = x^\mu + \epsilon^\mu$, where $\epsilon^\mu$ here is a constant, the corresponding conserved currents are given by the energy-momentum tensor $T^\mu_{\ \nu}$ which takes the form
$$ T^\mu_{\ \nu} = \frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi)} \partial_\nu \phi - \delta^\mu_{\nu} \mathcal{L}, $$
where $\partial_\mu T^\mu_{\ \nu} = 0$. Now I promote $\epsilon \rightarrow \epsilon(x)$. First, I compute the on-shell variation:
$$ \delta S_\text{on-shell} = \int \mathrm{d}^3 x \frac{\partial \mathcal{L}}{\partial \phi} \delta \phi + \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)} \partial_\mu(\delta \phi) \\ =\int \mathrm{d}^3 x \bigg( \frac{\partial \mathcal{L}}{\partial \phi}- \partial_\mu \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)} \bigg)\delta \phi + \partial_\mu \bigg( \frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi)}\delta \phi \bigg) \\ = \int \mathrm{d}^3 x \partial_\mu \bigg(\frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi)}\delta \phi \bigg).$$
The translation when viewed as an active transformation on the fields means $\delta \phi = - \epsilon^\mu(x) \partial_\mu \phi(x)$. Plugging this into the on-shell variation yields
$$ \delta S_\text{on-shell}= - \int \mathrm{d}^3 x \frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi)} (\partial_\mu \epsilon^\nu)( \partial_\nu \phi) + \epsilon^\nu \partial_\mu \bigg( \frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi)} \partial_\nu \phi \bigg) . $$
The first term in the integrand would vanish if $\epsilon^\mu$ was a constant. Now we evaluate the off-shell variation:
$$ \delta S_\text{off-shell} = \int \mathrm{d}^3 x \frac{\partial \mathcal{L}}{\partial \phi} \delta \phi + \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)} \partial_\mu(\delta \phi) \\ = - \int \mathrm{d}^3 x\frac{\partial \mathcal{L}}{\partial \phi} \epsilon^\nu \partial_\nu \phi + \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)} (\partial_\mu \epsilon^\nu \partial_\nu \phi + \epsilon^\nu \partial_\nu \partial_\mu \phi) \\ = - \int \mathrm{d}^3 x \epsilon^\nu \bigg( \frac{\partial \mathcal{L}}{\partial \phi} \partial_\nu \phi + \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)} \partial_\nu \partial_\mu \phi \bigg) + \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)} \partial_\mu \epsilon^\nu \partial_\nu \phi \\ = -\int \mathrm{d}^3 x \epsilon^\nu \partial_\nu \mathcal{L} + \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)} \partial_\mu \epsilon^\nu \partial_\nu \phi. $$
Equating the on and off-shell variations, we find the terms with $\partial_\mu \epsilon^\nu$ cancel, yielding
$$\int \mathcal{d}^3 x \epsilon^\nu \partial_\mu \bigg( \frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi)} \partial_\nu \phi - \delta^\mu_{\nu} \mathcal{L} \bigg) = 0, $$
or
$$ \int \mathcal{d}^3 x \epsilon^\nu \partial_\mu T^\mu_{\ \nu} = 0. $$
The integrand is not of the form $\partial_\mu j^\mu$ where $j^\mu = \epsilon^\nu T^\mu_{\ \nu}$. In fact, we already know that $\partial_\mu T^\mu_{\ \nu} = 0$ from our assumption that the theory was translationally invariant already so this statement is trivially true. I am unable to pass the $\epsilon^\nu$ into the derivative too because it depends on the coordinates.
What am I missing? Have I made a mistake in my analysis?