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Looking at usage frequencies in Google:

"in polar form" - 162000 occurrences

"in the polar form" - 85100

"in a polar form" - 26200

The context I am interested is math and it seems that most occurrences I see in Google belong to it. Same for "Cartesian":

"in Cartesian form" - 33700 occurrences

"in the Cartesian form" - 26900

"in a Cartesian form" - 162

I am not asking which one is correct, since it seems that all of them are correct. I am asking which would be the subtle difference in meaning between the three.

I am familiar with the theory of articles in English and have improved my confidence considerably, but still cannot figure it out in this particular case.

  • CONTEXT IS NEEDED!!! – Hot Licks Mar 28 at 2:27
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"in a adjective form" will be used when different forms are possible in a certain situation.

  • Some fuels can be stored in a solid or in a liquid form.

"in the adjective form" will be used when that form has been specified.

  • The gas is the most stable when stored in the liquid form (of all the possible forms in which it can be stored).

"in adjective form" will be used when that form is the only possible one.

  • The gas is typically stored in liquid form.
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Here are three sentences:

I can write [some imaginary number] *in polar form*.
Trigonometric functions are used *in the polar form* of imaginary numbers.
You can also represent a point on a graph in a polar form.

The third is a bit subtle. "In polar form" would work and would probably be used with an audience familiar with the idea of polar coordinates. But for an audience to whom polar forms are a new concept, at least as applied to the problem at hand, the "a" acknowledges that novelty. It's like the difference between "You can use GPS for that" and "You can use a GPS for that." The first tells you that the problem can be solved using a technology with which you are familiar, the second that the problem can be solved using tech with which you may not be familiar.

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