major / main compartment
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major
The most important subject of study that students focus on at a university or college. It is often supplemented by one or more minor subjects.
Sie studiert Germanistik im Hauptfach und Geschichte im Nebenfach.
She is studying German studies as her major and history as her minor.
The word is transparently composed of 'Haupt-' (meaning 'most important/main') and 'Fach' (subject or compartment). So the literal meaning is 'the most important subject/compartment'.
Das Hauptfach des Rucksacks ist groß genug für einen Ordner.
The main compartment of the backpack is large enough for a binder.
You can easily tell which meaning is intended from the context. Words like 'studieren' (to study), 'Universität' (university), or 'Nebenfach' (minor) point to the academic subject. Words like 'Rucksack' (backpack), 'Koffer' (suitcase), or 'Tasche' (bag) refer to the compartment in a container.
Mein Hauptfach an der Uni war sehr anspruchsvoll.
My major at university was very demanding.
In a university context, 'Hauptfach' (major) is often used with the preposition 'in' or 'als' (as). Typical formulations are 'etwas im Hauptfach studieren' (to study something as one's major) or 'etwas als Hauptfach haben/wählen' (to have/choose something as one's major).
Ich habe Physik als Hauptfach gewählt.
I chose physics as my major.
focus / center of gravity
'der Schwerpunkt' and 'das Hauptfach' can both describe what someone focuses on especially in their studies. However, 'Hauptfach' is a fixed term from school, university, or college: it is the main subject of study alongside possible minor subjects. 'Schwerpunkt' is more general and can appear in many areas, for example in a paper, a course, or a project. The words can be confusing because people often have a 'Schwerpunkt' within their 'Hauptfach'. To be precise: the 'Hauptfach' is the subject itself; the 'Schwerpunkt' is the area that you focus on within it or within a task.
Biologie ist mein Hauptfach, und Chemie ist mein Nebenfach.
Biology is my major, and chemistry is my minor.
Im Zug hat sie noch einmal gelernt, weil der Schwerpunkt der Prüfung auf Statistik liegt.
She studied again on the train because the focus of the exam is on statistics.