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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".

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    You've already got the one-and-only preposition required, in to which. Which you could validly have relocated after the main verb applied, even though this would then be immediately followed by the very same word acting as an infinitive marker for the verb to form - that's to say it's not necessarily a problem to have multiple instances of homophonous two, 2, to performing different roles in the noun phrase The two vectors which the 2-form is applied to to form an area element. If it's not a noun phrase, just drop the infinitive marker. Oct 1 '18 at 13:41
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This is the short version of the main phrase

The two vectors form an area element (the parallelogram).

When you add the descriptive phrase, the passive verb, 'is applied,' has its object, 'which,' and the preposition, 'to,' just as you expected.

The two vectors [to which the 2-form is applied] form an area element.

An extra 'to' would be a hypercorrection. This is discussed here, in Language Log, under the title "A phenomenon in which I'm starting to believe in."

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