Jai Shree Krishna,
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The aim of every Avatara is to save the world from some great danger, to destroy the wicked and protect the virtuous. Says Lord Krishna: "Whenever there is decay of righteousness, then I myself come forth. For the protection of the good, for the destruction of the evil-doers, for the sake of firmly establishing righteousness, I am born from age to age".
The Avataras are one with the Supreme. They are not parts like the individual souls. Avataras or Incarnations are rays of the Lord. When the work of Loka-sangraha is over, they disappear from the world
Avataras are of various kinds. There are Purna-avataras, with full Kalas or rays. There are Amsa-avataras or partial incarnations. There are Lila-avataras.
Lord Krishna was a Purna-avatara. Sri Sankaracharya was an Amsa-avatara. Matsya, Kurma, Varaha, Narasimha, Vamana, Rama and others were Lila-avataras.
Krishna and Rama were the Avataras of Lord Vishnu. Dakshinamurthy was an incarnation of Lord Siva. Dattatreya was the Avatara of the Trimurtis—Brahma, Vishnu and Siva are three aspects of God. Brahma is the creative aspect; Vishnu is the preservative aspect; and Siva is the destructive aspect.
There are ten Avataras of Lord Vishnu. There are: Matsya (the Fish), Kurma (Tortoise), Varaha (the Boar), Narasimha (the Man-lion), Vamana (the Dwarf), Parasurama (Rama with the Axe), Ramachandra (the hero of the Ramayana), Sri Krishna, Buddha (the prince-ascetic, Founder of Buddhism) and Kalki (the Hero on a white Horse who is to come at the end of the Kali Yuga).
Lord Rama and Lord Krishna had no physical bodies. Their bodies were not made of five elements. They had divine forms. They had Chinmaya bodies, though to all appearance, it looked like flesh. They had no real birth and death like human beings. They appeared and disappeared, just as a Yogi does. Their bodies were not left in this world. There is no destruction for their bodies.
Just as a tailor who makes coats for others can make a coat for himself also, God who has created bodies for others, can create a body for Himself as well. There is no difficulty. He is omnipotent and omniscient. As He has control over Maya, He is fully conscious of His divine nature though He assumes a form. Still He is infinite and unconditioned.
Sometimes, the king visits the jail and enters the cell of a prisoner to see how matters are getting on in the prison. He does this for the good of the prisoners. He is quite independent, and yet, out of his own free will, he himself enters the cell. Even so, an Avatara puts on a body out of His own free will for the ascent of man. He is quite independent and has absolute control over Maya, like the king; while the Jiva is a slave of Avidya, so long as he has no Self-realisation.
Jai Shree Krishna