I am reading A Theorem on Force-Free Magnetic Fields (1958) by L. Woltjer. He used “integrating by parts” twice and I am confused about his results.
The first one is in the equation (7). I understand the first step because he just used the relation (6) he got earlier, but for the second step, if I integrate it by parts, I guess it will be like $$\int_V A\cdot \nabla\times \frac{\partial A}{\partial t} =-\int_V \dfrac{\partial A}{\partial t}\cdot(\nabla\times A)dV+\int_SA\cdot\dfrac{\partial A}{\partial t}dS.$$
I also found a similar question Integration by parts of $\varphi \cdot \operatorname{curl}(u)$, but even if I use the Divergence theorem, I got $$\int_V \dfrac{\partial A}{\partial t}\cdot(\nabla\times A)dV+\int_V\nabla\cdot\left(\dfrac{\partial A}{\partial t}\times A\right)dV=\int_V \dfrac{\partial A}{\partial t}\cdot(\nabla\times A)dV+\int_S\dfrac{\partial A}{\partial t}\times AdS,$$ which still has a different sign.
Even though it does not affect the final result “zero”, I hope someone could help me do integration by parts correctly for this case.
Similar thing occurred from equation (9) to equation (10). Equation (9) is $$ \int_{V}[2 \operatorname{curl} A \cdot \operatorname{curl} \delta A-\alpha(\delta A \cdot \operatorname{curl} A+A \cdot \operatorname{curl} \delta A] d V=0 $$ Equation (10) is $ \int_{V}[\operatorname{curl} \operatorname{curl} \boldsymbol{A}-\alpha \operatorname{curl} \boldsymbol{A}] \cdot \delta A d V=0 $, while my result is $$\int_V\text{curl curl}A\cdot\delta A-\alpha(A\cdot\text{curl}\delta A+\delta A\cdot\text{curl}A)dV=0.$$ I have no idea how to deal with the extra part.
I guess I can understand both if either one is clarified. Thank you!!!