From the movie Manhattan (Woody Allen):
Isaac Davis : I got a kid, he's being raised by two women at the moment.
Mary Wilke : Oh, y'know, I mean I think that works. Uh, they made some studies, I read in one of the psychoanalytic quarterlies. You don't need a male, I mean. Two mothers are absolutely fine.
Isaac Davis : Really? Because I always feel very few people survive one mother.
I believe it's simply the case that the verb "לתמוך" naturally always takes the preposition "-ב" with whatever is being looked after. So while the verb "support" in English takes no preposition, it does take a preposition in Hebrew. The "ba" sound it makes is a combination of "be" (the normal sound of the preposition" and "ha" (the definite article, "the"). Feel free to correct me if I'm wrong!
Yes. It can function as a preposition that indicates both place and time (when it indicates time, you can consider it as indicating "place in time"). And it also can serve as a indirect object preposition that doesn't imply place or time, like in this sentence. I know it sounds weird in English, but in Hebrew, the object after לתמוך is indirect.
It's not only the most common, it's the only option in this particular sentence. And the preposition ב is actually the only option in this case. You can't write היא תומכת הילדים שלה because after the verb "to support" in hebrew an indirect object must follow. When the ה (definite article) stands alone - without any preposition (like ב), then we're talking about a direct object, which is not the case here (again, in Hebrew the verb to support requiers an indirect object). I guess you wouldn't translate it into She is supportive to her children because this is not an equivalent translation. The Hebrew sentence is about an actual verb (to support) and not just a description of the mother's attitude.
It's just אמא, never האמא. If you ever come across האמא it's a mistake. DL has a mistake here. [Update: I wrote this a year ago, but now I realize that it's more complicated. People on the street say things that are not necessarily "correct" and ultimately those things become how things are said. It's good that DL helps us with the way people actually talk. July, 2020]
yes, it bears the same meaning in this case: supporting someone physically, emotionally, politically, supporting weight or structure etc. And it's very common on political or ideological issues as well: אני תומך במפלגה (i support the party) or: הם תומכים בקומוניזם (they are communist, or: they believe in communism; literally: "they support communism").
To me, it's a total mystery why some Hebrew verbs take direct objects and others take prepositions. Some are completely counter-intuitive, like touching "in" something or supporting "in" someone. (ב is used in Hebrew way more than "in" is used in English.) But in truth some of the same absurdities occur in English, where we watch objects but look AT them. You just have to memorize which verbs go with which prepositions in which situations.
Some people know that אמא already has a definite article, after all, the Bavli and Yerushalmi are central texts to the Jewish faith, and so those folks don't add a superfluous definite article, while others add a definite article because regular folks don't know Aramaic and have not yet become acquainted with the Bavli, and if they are secular, they may never become acquainted. Every language has examples of forms of words that might not be technically correct but are nonetheless colloquial. In English, "ain't" and "is not." The former is so common that even grammarians no longer say it's an error but rather an alternate form. DL focuses on how people actually speak and so they would be remiss if they did not introduce learners to forms of common words that people actually will run into and hear on the street. In a comment above from a year ago, I had originally written that האמא is a mistake, but I updated that to acknowledge that it's more complicated.
Trying to understand the use of -ב here. I feel like an attempt at a literal translation would best illustrate that. Would it be, "The mother supports of the children of hers?" Grammatically wrong, but I know -ב is used for touch, because touching is thought of something done in the object being touched.