# Meaning of "ab" and "ist also" in mathematical texts

Read the following German text and its official translation:

Die Gauss-Krümmung K eines Flächenstücks f:U --> R3 der Klasse C3 hängt nur von der ersten Fundamentalform ab (ist also eine Größe der inneren Geometrie). (Read in Google book)

The Gaussian curvature K of a two-dimensional surface element f : U --> R3 of class C3 depends only on the first fundamental form ("and" is consequently an intrinsic quantity of the surface). (Read in Google book)

Q1: What is the meaning and role of "ab" here? As I checked by Google translate, the translation of the sentence without "ab" is same as with "ab".

Q2: I know that it is better to add "and" in the beginning of parenthesis, but why there is no "und" in German text? Is the German sentence a complete and meaningful sentence? (without "und")

I know that the translation of "ist also" is "is thus, is consequently, ..".

• Q1 is basically the same issue as your earlier question. If you're a beginner you have to check the end of every clause so make sure there isn't a misplaced prefix. In many cases the difference is significant: hängen = "hang", abhängen = "depend". In technical literature there can be quite a gap between the verb and the prefix. Jun 19, 2022 at 20:02
• Why is the "and" okay in English? Doesn't "and" mean additionally, and isn't that in contradiction to "consequently"? Jun 19, 2022 at 20:48
• @CarstenS: I think it is correct with and without "and". "AND" used to join two words, phrases etc referring to things that are related in some way but "consequently" mean "as a result". Jun 20, 2022 at 3:47
• I'd translate "ist also..." in this context with "thus is..." Jun 20, 2022 at 12:00
• @Carsten S: To me, "(is consequently ...)" sounds incorrect. I would probably create a new sentence "(K is consequently ... .)" Without "K" there is no subject, which is okay when there is a conjunction, but odd sounding without one. "John plays guitar and sings," is preferred over "John plays guitar, sings." Maybe the parentheses replace a conjunction in German, but not so in English. Jun 20, 2022 at 12:53

## 1 Answer

Re Q1: The verb here is abhängen. This is where ab comes from. Side remark: Incidentally, the English verb depends is constructed in an analogous way -- de-pend.

Re Q2: To add und is fine; one could also say ... und ist daher ... The way it is written in the book sounds more like speaking informally, but it is not wrong.

• Of course the difference between abhängen and depend is that abhängen is separable, but to depend is not. Separable verbs like in German do not exist in English; the closest thing to German separable verbs in English are so-called phrasal verbs.
– RHa
Jun 19, 2022 at 15:49
• @RHa Yes, this is completely correct. The side remark is just meant to illustrate that the construction of the words is similar as is often the case in many Indoeuropean languages. Jun 19, 2022 at 15:54
• @RHa: Phrasal verbs in English are divided into particle verbs, prepositional verbs and particle-prepositional verbs. German also has prepositional verbs, for example freuen auf = "to look forward to", with "look forward to" being a particle-prepositional verb. An example of a particle verbs is "give in"; the difference being that the "in" does not precede a noun. Separable verbs in German correspond to particle verbs in English. I think abhängen von is a prepositional verb where the verb is separable, while "depend on" is a prepositional verb. Jun 19, 2022 at 20:35