Alexander Jack was born in Scotland in 1782, son of William Jack. He was Licentiate of the Royal College of Surgeons (LRCS) in 1803 and appointed surgeon in the Royal Navy 26 May 1804.
Alexander Jack was surgeon on HMS Shannon when the American Frigate Chesapeake was taken in 1813.....Read an account of the battle, deaths and injuries in The Gentleman's Magazine and his Medical Journal with accounts of treatment of the injured at National Archives .
He was entered in the Navy List of Medical Officers of 1814. He was appointed surgeon to the Tiber in 1814 - Naval Chronicle.
He was awarded Doctor of Medicine (M.D.) in 1815.
Alexander Jack was employed as Surgeon Superintendent on the convict ship Caledonia to Van Diemen's Land in 1820. The Caledonia departed Portsmouth on 10 July 1820 and arrived in Hobart on 17 November 1820. 
His Will was written while he was on the Caledonia - .I Alexander Jack, Surgeon of the Caledonia Convict ship being in bodily health and of sound mind and memory and being aware ofthe perils and dangers of the sea and other uncertainties of this transitory life publish my last Will and Testament.......He bequeathed his estate consisting of landed property and houses to his father William Jack of Cupar, Fife, Scotland. At his father's death the estate was to pass to his sister Jean, wife of Alexander Ramsay, writer of Cupar, Fife and afterwards to Jean's daughter Margaret. He also bequeathed money to the eldest child of his only brother David of Saint John, New Brunswick, Canada.
The Will was witnessed by J. Mustleman, Brevet Major 53rd Regt., Matthew Rutledge, Gentleman and Robert Carns, owner of the ship.
Alexander Jack died in 1822 and was buried at St. Peter and St. Paul's Church, Fareham, Hampshire on 28 May 1822. 
His Will was proved 15 June 1822 (National Archives UK)
Notes and Links
1) Family of David Jack, brother of Alexander Jack at Find a Grave
3). Alexander Jack was surgeon on HMS Shannon when the American Frigate Chesapeake was taken in 1813.....Read an account of the battle, deaths and injuries in The Gentleman's Magazine.
4). 'These are to certify that I, the undersigned David Rowlands, M.D., F.R.S., late surgeon of H.M. Naval Hospital at Halifax, in Nova Scotia, was there when H.M.S. Shannon arrived with her prize, the American frigate Chesapeake, on Sunday, the 6th of June, 1813; the former was commanded by the present Captain Wallis, owing to the dreadful wound which Captain Broke had received in the action with the enemy a few days previous. On the 7th of June I was requested by Mr. Alexander Jack, the surgeon of the Shannon, to visit Captain Broke, confined to bed at the commissioner's house in the dockyard, and found him in a very weak state, with an extensive sabre wound on the side of the head, the brain exposed to view for three inches or more; he was unable to converse, save in monosyllables, and I am sure totally unable to dictate or write an account of the action for some time afterwards, owing to his severe wounds, loss of blood, and the shock his whole frame must have experienced by the blow on the head. 'I continued to attend him twice a day for weeks afterwards, in conjunction with Mr. Jack, to whom every possible credit is justly due for his skilful treatment and care in bringing his brave captain on shore alive. 'I grant this certificate to Captain Wallis, being called upon to do so by the death of Mr. Jack the surgeon. 'Given under my hand, this 8th day of December, 1841.- Admiral Sir P. B. V. Broke ...:
A Memoir By John George Brighton, Sir Philip Bowes Vere Broke
 Ancestry.com. England, Select Deaths and Burials, 1538-1991. Original data: England Deaths and Burials, 1538-1991. Salt Lake City, Utah: FamilySearch, 2013.
 National Archives. ADM 101/120/3B 1812-1813 Medical and surgical journal of HMS Shannon by Alexander Jack for 30 July to 29 November 1813 during which time the said ship was employed in Halifax, passage and home station