My car ran fine then stoped and jumped started 3 times and then dies after 15 min is it the alternator?

Hey guys I have a 1988 Nissan sentra and I was driving home on the highway everything was fine then at every light the car seemed like it wanted to stall it did once fired it back up was fine again still seeming like it wants to stall. then i drive up the hill come back down my wipers and raido stoped working then it died i got a jump then it was working ok i noticed the headlights starting to dim so i sped up didnt make a diffrence then it died again had to get another jump start then i drove to my place and it died thankfully right there. the car wont start now only if i jump start it from another car I was told by various people its the alternator thats gone i tested the battery its reading 12-13 Volts Please need your help!