Hello I will help you with your question,
Place the pulley of the alternator facing down and jack the engine up a little by the oil pan and the alternator should come out the bottom behind the axle but if you remove the axle use a bar and a large hammer to strike the housing of the center joint sharply and it will separate. I usually remove the three bolts from the center support to the engine block (top bolt is tough though) and slide the while axle out of the transmission then you have plenty of room.
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Thank you for the help so far. I have tried to remove the alternator from the bottom, with the axle still in, but I just can't see an exit point.
Also, what part should I be striking on the center shaft? I've attached a couple pics if that will help. Thanks again!
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The shell of the joint is where you strike. Use a bar about 1 foot long against the shell and strike the bar with a 3 lb hammer. Two or three blows should free the axle. The diagram of the shaft is below
Click here for diagram
Then brass looking ring on the center bearing is a dust cap and will bend if you pry or hit so keep your pressure away from those.
You picture shows a lot of oil on the alternator and that is likely why it is failed.
Let me know how I can help