## Finals Week Office Hours + TA Evaluations

Since the final assignment is due during finals week, each instructor/TA will hold office hours next week, but they will be different from the normal schedule.

• Josh will have office hours from 4:30 PM – 6:00 PM on Monday, December 11, outside Smith 232.
• Keenan will have office hours from 5:00 PM – 6:30 PM on Tuesday, December 12, in Smith 217.
• Rohan will have office hours from 4:30 PM – 6:00 PM on Thursday, December 14, outside Smith 232.

If you would like to come to office hours but can’t due to finals, please let us know and we’ll try to work out some other way to help.

The website for students to evaluate TAs is now live. If you’ve had substantial interaction with Rohan and/or me, please take the time to fill it out. 🙂

## Assignment 5 Notes

Later this week, assignment 5 will be released. This will be the last assignment of the course, and it will be due during finals week (Thursday, December 14). Whether this assignment is required depends on how many assignments you have already done:

1) If you have done all of assignments 1-4, assignment 5 is *not* required. You can submit assignment 5 for extra credit.
2) If you skipped one of the earlier assignments, then assignment 5 is required.

Please let us know if you have any questions.

## Complex-valued differential forms

In Assignment 4, you get to work with complex-valued differential forms. These work mostly the same as real-valued differential forms, but there are a couple additional features.

• Recall that the wedge product for real-valued two 1-forms $\alpha$, $\beta$ is defined as
$\alpha \wedge \beta (u, v) = \alpha(u)\cdot \beta(v) – \alpha(v)\cdot \beta(u),$
where “$\cdot$” in the usual product for real numbers. The wedge product for complex-valued 1-forms is identical, except that $\cdot$ is replaced with the complex product
$(a + bi) \cdot (c + di) = ac – bd + (ad + bc)i.$
• A special operator on the complex numbers is the conjugation map $z \mapsto \bar{z}$, where $\overline{a + bi} = a – bi$. This operator can be applied to complex-valued forms, too. For a $k$-form $\alpha$ the conjugate will be
$\overline{\alpha}(X_1, …, X_k) = \overline{\alpha(X_1, …, X_k)}$
Note that conjugations commutes with the exterior calculus operators $d$, $\star$ and $\wedge$. That is, $\overline{d\alpha} = d\overline{\alpha}$, $\overline{\star \alpha} = \star \overline{\alpha}$ and $\overline{\alpha \wedge \beta} = \overline{\alpha} \wedge \overline{\beta}$.

Please ask in the comments if there is anything else you need to be clarified about complex-valued differential forms.