Toyota Repair: my battery keeps dieing, alternator output, alternater

Question
iv checked the battery it was replaced the alternter is fine the starter is fine after i get a jump the car wont stay running more than 30 seconds after i take th cables off if i go for some reson th alternater isnt recharging the battery even if i have a short wuldnt the battery still be geting charged or is it just somthing iv missed?

Answer
The alternator should keep the engine running after the battery is disconnected, it's not charging the battery, the alternator output should be tested on the car under actual load conditions with a machine that apply an electrical load to the charging system.