To what extent does Prospero use Ariel to carry out revenge in The Tempest?
I think this statement has a great deal of truth to it. Throughout the play, it is Prospero that uses (or abuses?) his mastery of Ariel and causes him to act as his agent to bring about his revenge. However, when we first meet Ariel, it he seems to present himself as perfectly happy to be Prospero's servant. Note his words in Act I scene 2:
All hail, great master! Grave sir, hail! I come
To answer thy best pleasure; be't to fly,
To swim, to dive into the fire, to ride
On the curled clouds. To thy strong bidding task
Ariel and all his quality.
Of course, it is Ariel that has created the tempest at the beginning of the play that allows the enemies of Prospero to fall directly into his clutches, and at every stage, Ariel is instrumental in bringing Prospero's plans to bear, and also warning Prospero of any threat or concern, such as when Caliban groups together with Stephano and Trinculo.
Thus it seems hard to find any arguments to suggest that Ariel is not used for the express purpose of carrying out the revenge of Prospero. Some critics have likened Prospero to a director of a play that he is in charge of, with the island being the stage, and Ariel being the stagemanager, that makes sure everybody is in the right place and the right time. Certainly, throughout the play, Prospero's power and autocracy is not something that can be ignored, and Ariel is used to support his power.