Question
Is Lord Bhairava same as Lord Shiva, or are they a little different? Is Bhairava a part incarnation or complete Shiva?
Answer
Lord Shiva took his fifth incarnation as Bhairav and his Shakti was called 'Bhairavi'. So he is incarnation of shiva .
An incarnation may not be same as the one who is taking avatar like narsimha is not like how lord Vishnu is shown.
Shiva Purana says in
1.1.7 Shiva Obliges Brahma Nandikeshwar continuing with the tale said: "After according same status to Vishnuji as that of his own, Lord Shiva opened his third eye from it manifested 'Bhairav'. He ordered Bhairav to kill Brahmaji. Bhairav severed the fifth head of Lord Brahma with his sword. Brahmaji became very terrified. He was trembling in fear. Lord Vishnu felt pity on his condition and requested Lord Shiva to forgive him.
and here
1.3.16 The first incarnation of Lord Shiva was as Mahakal and his corresponding Shakti was called Mahakali. His second incarnation is Tar and his Shakti was called ‘Tara’. The third incarnation is Bhuvaneshwar and his Shakti was called ‘Bhuvaneshwari’. Fourth incarnation as ‘Shodash’ and his Shakti was called ‘Shodashi’ or ‘Shri’. Fifth incarnation as Bhairav and his Shakti was called ‘Bhairavi’. Sixth incarnation of Lord Shiva is famous as ‘Chhinamastak’ and his Shakti by the name of ‘Chhinamasta. Seventh incarnation as ‘Dhoomvan’ and his Shakti was known as ‘Dhoomvati. Eighth incarnation was as Baglamukh and his Shakti as Baglamukhi. Nineth incarnation of lord Shiva became famous as Matang and his Shakti as ‘Matangi’. Tenth incarnation as ‘Kamal’ and his Shakti as Kamala.
by the way while searching for Partial incarnation I could only find Ashwathama as partial Incarnation which is mentioned here
1.3.29 Indra went to Arjuna in the guise of a celibate and asked about the purpose for which he was doing penance. Arjuna told him that he wanted to defeat the Kauravas. Indra then told Arjuna that it was not in his capacity to help him achieve victory over the Kauravas, because of Aswatthama who was a partial incarnation of lord Shiva.
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