I'm German and I'm reading the English language General Relativity lecture notes by my professor, who is also German. As I read I take notes and corrections to help him improve the notes whenever I find errors. I don't want to correct him where he's actually right and I'm not entirely sure if this is a mistake or not (emphasis mine):
A 2-form in ℝ³ can be interpreted as as a space-filling staggering of parallel bars. The two vectors to which the 2-form is applied form a area element (the green parallelogram in the figure). The number of bars enclosed by this area element is the result value of the 2-form.
Apart from the obvious typos, shouldn't it be
[…] The two vectors to which the 2-form is applied to form an area element […]
I am asking if I need to add a preposition after the verb apply when it's used in a passive voice, where a "to" has already been added to the object in a relative clause. (Forgive me if that's not what I'm asking, High School English grammar has been a while)
As a clarification of terminology for people not familiar with this kind of mathematical lingo, one usually says "We apply 2-form α to vector u".