Below we clarify the conventions used for normalization in our paper (Punk, Landreman & Herlander Reference Punk, Landreman and Herlander2019), and correct some associated errors in the equations. The validity of the numerical solutions and the main conclusions of the paper are unaffected by these corrections.

Following Garren & Boozer (Reference Garren and Boozer1991), we define the expansion parameter as $\epsilon = \sqrt {\psi }$ so that the magnetic field can be expressed to first order as $B(\epsilon , \theta , \varphi ) \approx B_{{a}}(\varphi ) (1 + \epsilon \sqrt {2/B_{{a}}(\varphi )}\kappa ^s(\varphi ) \eta _{\mathrm {GB}}(\varphi ) \cos [\theta - \alpha (\varphi ) ])$, where the ‘$\eta$’ of Garren & Boozer (Reference Garren and Boozer1991) is here denoted $\eta _{\mathrm {GB}}$; see their (79). For simplicity, we introduced the quantity $d$, related to $\eta _{\mathrm {GB}}$ by

(0.1)\begin{equation} d(\varphi) = \sqrt{\frac{2}{B_{{a}}(\varphi)}}\;\kappa^s(\varphi)\eta_{\mathrm{GB}}(\varphi), \end{equation}

so that the magnetic field to first order becomes

(0.2)\begin{equation} B(\epsilon, \theta, \varphi) \approx B_{{a}}(\varphi) \left(1 + \epsilon d(\varphi) \cos[\theta - \alpha(\varphi) ] \right), \end{equation}

as correctly written in (6.1) of our paper.

Our definitions for $d$ and $\epsilon$ affect the forms of the first order components of the coordinate mapping, $X_1$ and $Y_1$. These quantities are introduced in the text at the beginning of § 7, where the the coordinate mapping ${\boldsymbol {x}}$ to first order should read

(0.3)\begin{equation} {\boldsymbol{x}}\approx {\boldsymbol{r}}_0 + \epsilon(X_1 {\boldsymbol{n}}^s + Y_1 {\boldsymbol{t}}^s). \end{equation}

Note the factor of $\epsilon$ is missing in the paper. The form of $X_1$ was correctly given by (7.1), but that for $Y_1$, (7.2), should read

(0.4)\begin{equation} Y_1 = \frac{2}{B_{{a}}(\varphi)\bar{d}(\varphi)} \{ \sin[\theta - \alpha(\varphi)] + \sigma(\varphi) \cos[\theta - \alpha(\varphi)] \},\end{equation}

with the correction being the factor of $1/B_{{a}}(\varphi )$.

Our forms of $B$, $X_1$ and $Y_1$ can be compared with (79)–(81) of Garren & Boozer (Reference Garren and Boozer1991), and can be seen to agree, given our definitions of $\epsilon$, $d$ and $\bar {d}$, and the substitution $\kappa \rightarrow \kappa ^s$. A related error was introduced into the definition of $P$ immediately following (7.7), which should read

(0.5)\begin{equation} P = 1 + B_{{a}}^2\bar{d}^4/4. \end{equation}

The solutions presented in § 8 remain valid, but we note that $d$ defined in (8.4) (also depicted in figure 2b) misses a factor of $\sqrt {2}$, and should read

(0.6)\begin{equation} d(\varphi) = \sqrt{2}\left[ 1.08 \sin(\varphi) + 0.26 \sin(2\varphi) + 0.46 \sin(3 \varphi)\right]. \end{equation}

Finally, there is a typo, unrelated to the preceding issues: following (7.10) it should read $\Delta \varphi (\varphi ) = \varphi - \varphi _b(\varphi )$.