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How can I construct a relative clause of the following sentence containing the complex preposition "with respect to"?

Original sentence: The expression is differentiated with respect to variable x.

Construction (a): Variable x with respect to which the expression is differentiated

Sounds unnatural.

Construction (b): Variable x to which the expression is differentiated with respect

Sounds even more unnatural.

Construction (c): Variable x which the expression is differentiated with respect to

Better, but it's not pied-piping.

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    It would help if you gave a full sample sentence, and explained what "pied-piping" means in this context.
    – Max Williams
    Commented Mar 22, 2016 at 15:46
  • @MaxWilliams Pied-piping referes to the syntactical phenomenon that when a movement occurs (in here the argument 'variable x' is moved to the SpecCP position of the clause), its encompassing phrase, rather than itself, is moved. Consider the sentence "Alice spoke with Bob". When I ask "Who did Alice speak with?" I'm not using pied-piping; however, if I ask "With whom did Alice speak?" I'm using pied-piping because the whole prepositional phrase "with whom" is moved to SpecCP.
    – Tongfei Chen
    Commented Mar 22, 2016 at 15:53
  • Are you intent on using with respect to? If not, consider variable of differentiation x.
    – bradimus
    Commented Mar 22, 2016 at 17:09
  • @bradimus This is standard mathematical usage.
    – Tongfei Chen
    Commented Mar 22, 2016 at 23:59

3 Answers 3

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(a) is fine, though rather stilted. (c) is okay. (b) is ungrammatical, because it violates a constraint on extraction, though I'm not sure just what that constraint is. Considering a few other examples suggests that you can't extract the complement of a noun -- e.g. "Who is he the uncle of?" / *"Of whom is he the uncle?". So maybe that is what goes wrong with (b), if the PP "to __" is complement of "respect".

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Construction (A) is correct, if a comma is added to make it a nonrestrictive relative clause:

  • Variable x, with respect to which the expression is differentiated, depends on variable y.

Pied-piping is most common in non-restrictive clauses.

As noted, with respect to is a fixed phrase, therefore reified, and syntactic rules may or may not apply to its parts. In case (2), which is (also as noted) ungrammatical, it doesn't apply to the to in with respect to.

So solution (3), stranding the preposition at the end, is by far the most common strategy,
and involves no acrobatic syntax, nor piping of any kind, pied, laid, or meerschaum.
Ordinary language is generally clear language.

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I think (a) is correct. Firstly, as a native speaker, although it's not an everyday form it sounds acceptable to me. Secondly, I can find examples of other complex prepositions that use this structure.
"The renewal of a note may not be issued after the sale of bonds in anticipation of which the original note was issued" from a California statute limiting school bonds. "That will be an exceptional occasion, in view of which she asks the Lord to increase the unity of all Christians until they reach full communion" from a Papal Encyclical "Ut Unum Sint".

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