# 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. Commented 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"? Commented 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". Commented Jun 20, 2022 at 3:47
• I'd translate "ist also..." in this context with "thus is..." Commented 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. Commented Jun 20, 2022 at 12:53