I am given the integral

\begin{gather} \int_V \hat{e}_z \times \vec{u} dV \end{gather}

where $\hat{e}_z$ is the unit vector in the $z$ direction and $\vec{u}$ is a vector field. Can I pull $\hat{e}_z \times$ out of the integral and rewrite it as

\begin{gather} \hat{e}_z \times \int_V \vec{u} dV \end{gather}

  • $\begingroup$ No. You have to be integrating a scalar function, not a vector. $\endgroup$ May 6, 2016 at 15:49
  • $\begingroup$ Yes, the operations can be interchanged. However, if the unit vector depended on the integration variable(s), then the answer would be in general no. For example, if the unit vector was, say, the unit vector $\hat \theta$ in spherical coordinates $(r,\theta,\phi)$ and the integration extended over either $\theta$ or $\phi$, then the interchange is not permissible. $\endgroup$
    – Mark Viola
    May 6, 2016 at 16:20


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