the lord of the rings - Why was Gandalf so emotional?

This is relating to the movie. Frodo finally relinquishes his desire for the ring, and it melts in the fires of Mount Doom. The next scene shows the eruption of Mount Doom and the destruction of Barad Dur. Gandalf looks up with tears in his eyes.

I have read the books, and can't seem to remember this being explained or described. I'm just wondering why they depicted Gandalf as being so emotional about the event, when throughout the story it depicts Gandalf as having unparalleled wisdom, faith and knowledge. Out of all the Ainur he listened to Eru's song the longest and contemplated it's meaning.

It strikes me that Tolkien portrayed Gandalf as all knowing, and has ultimate faith that Frodo would carry out the task at hand. So why was he so emotionally surprised that it happened? Especially when he had seen far more surprising things happening throughout the War of the Ring and in the movies he was depicted as emotionally immovable, an almost cold hearted awareness of the evil that lurks within the world.

I understand this is probably going to be down-voted as it has alot of opinion based statements. The TL:DR is: Was Gandalf described in the book as being so emotional at Sauron's fall? Or was this simply for effect in the screenplay?

Answer

Gandalf is certainly very wise, but he is definitely not "all knowing". No one in Tolkien's work, aside from Eru Ilúvatar, is all knowing. Gandalf didn't know what the Palantír of Orthanc was when he first saw it, nor did he know what the Ring was the first couple of times he saw it. He knows a lot, but certainly not everything.

Gandalf hoped Frodo would be successful- and would survive his quest- but he absolutely didn't know for sure that this would be the case. In the first cut, Gandalf realizes that, beyond all hope, Frodo has succeeded, and he is humbled and astonished by it; his first tears are tears of happiness. Keep in mind that this is the reason that the Wizards were sent into Middle-earth in the first place - to defeat Sauron. Gandalf has just accomplished the thing he had been striving towards for thousands of years. He has fulfilled his purpose.

"The realm of Sauron is ended!" said Gandalf. "The Ringbearer has fulfilled his quest."
- The Return of the King, The Field of Cormallen

_{There is even a hint of a smile on his face, showing his joyful astonishment}

When he sees the eruption of Orodruin, he knows that there is a very good chance that Frodo has just been killed. Naturally, he is greatly saddened by this, and these tears are tears of sorrow.

And as the Captains gazed south to the Land of Mordor, it seemed to them that, black against the pall of cloud, there rose a huge shape of shadow, impenetrable, lightning-crowned, filling all the sky.
- ibid

Why is Gandalf emotional in general? We find out in The Silmarillion:

Wisest of the Maiar was Olórin [i.e., Gandalf]. He too dwelt in Lórien [the garden in Aman, not the place where Frodo meets Galadriel], but his ways took him often to the house of Nienna, and of her he learned pity and patience.
- The Silmarillion, Valaquenta; Of the Maiar

Nienna, whose name means "She who weeps", is essentially a goddess of pity, mercy, mourning, and compassion:

Mightier than Estë is Nienna, sister of the Fëanturi; she dwells alone. She is acquainted with grief, and mourns for every wound that Arda has suffered in the marring of Melkor. So great was her sorrow, as the music unfolded, that her song turned to lamentation long before its end, and the sound of mourning was woven into the themes of the World before it began. But she does not weep for herself; and those who hearken to her learn pity, and endurance in hope... she brings strength to the spirit and turns sorrow to wisdom.
- The Silmarillion, Valaquenta; Of the Valar