Thanks B. for the answer. I drove the car to the store I bought that battery. The guy in the store was too busy to check the alternator. He just checked the battery and replace another new one for me. So everything is ok right now but I don't know how long it will last.

I will check the alternator again this weekend.

But, if there is corrosion, then the problem may be caused by the poor electrical contact ??? Then what should I do? ????
Because the answer is yes, there are rust and corrosion where the battery sitting on. The car was parked in the dealer lot for too long before I bought, and the battery was leaking caused that corrosion. I didn't know why I bought the car even that I saw the rusting... hic hic..

I hope it's not the alternator; it's not cheap at all !. ...