Continuity mistake: Chef says, "I gotta remember, never get out of the boat" as he sits in front of Chief Phillips on the starboard side of the boat. In the next shot as they pull away from the tree they've been hiding under he's walking around on the other side of the boat. (00:59:30)
Continuity mistake: During the arrow attack Chief Phillips grabs a machine gun and starts shooting. At first he and Chef's guns are both pointing in the same direction, starboard. In the next shot one gun is facing one way and the other is facing the opposite way. (02:22:15)
Continuity mistake: The gentleman in the tan jacket to the left of Roxanne is eating when Hubert says, "You don't understand our mentality". In the next shot he's reaching over and pouring a glass of wine for her. (02:05:20)
Continuity mistake: The gentleman sitting next to Roxanne in the tan jacket gets up and says, "Look, Captain," and Roxanne reaches over with her right hand as she holds her cigar in her left. In the next shot she's sitting with both hands on the table and the cigar in her right hand. (02:07:20)
Continuity mistake: As Chief Phillips orders Chef to, "Check what's in that vegetable basket," the Vietnamese woman runs past and hands him papers that he takes with his right hand. In the next shot as he pushes the woman the papers are in his left hand. (01:36:25)
Continuity mistake: After he decides to decline the offer to spend some time with the bunnies Chief Phillips picks up some tools off the roof of the boat. In the next shot he's already on the floor of the boat turned around and ready to go down the hatch to work on the engine. (01:27:25)
Continuity mistake: As the PBR backs out of Kurtz' compound the person at the controls is wearing a shirt. Lance is sitting on the front of the boat so the only other person that could be at the helm is Captain Willard who is not wearing a shirt in the previous or next shots. (03:14:00)
Continuity mistake: As he first spots the French plantation from the bow of the boat Captain Willard's watch reads 12:45. Then when Chef puts his rifle down it's 4:00. then two shots later as the captain walks forward towards the Frenchmen it's 5:50. (01:55:10)
Continuity mistake: As they come along side the sampan Chief Phillips says, "Look in that forward hooch, bring the people out of there." Clean slips on something and spins the machine gun "Canned Heat" towards the sampan as he ties the boats together. In the next shot the machine gun is spun to the left pointing away from the sampan. This reversal happens several other times. (01:35:10)
Continuity mistake: Clean tunes in his transistor radio to AMVN and we hear it continuously playing in the background as Captain Willard goes from standing with his shirt on looking at Chief Phillips as he pilots the boat, to sitting shirtless looking at pictures of Colonel Kurtz between shots. (01:13:50)
Continuity mistake: A little old Vietnamese woman wearing a bright red scarf around her neck is feeding bullets into the machine gun. Six seconds later as the machine gun is blown up her stunt double is not wearing the bright red scarf. (00:40:55)
Continuity mistake: After they first meet Captain Willard looks at Dennis Hopper and says, "Take who away?". The red wind spinner to left of the Captain is in tatters, but a few seconds later when Dennis Hopper walks back to the front of the boat it's back together again. (02:30:20)
Continuity mistake: When Colonel Kurtz says, "And they call me an assassin," the statue in front of the TV casts its shadow on the left of the picture tube. The lighting doesn't change throughout this scene but the next time we see it the shadow is one quarter of the way to the center of the picture tube. Later the shadow moves back to the side of the TV again. (00:13:55)
Continuity mistake: As they begin to break out of the fog bank Chief Phillips says, "Lance on the 50s". Lance doesn't move from where he is on the roof of the boat but Chef goes to the machine gun twice between shots and points it in different directions each time. (02:21:00)
Continuity mistake: Clean hammers on the helicopter door and says, "I got my rights, man." He's in direct sunlight when you see him from inside the helicopter and not in the shot of him on the outside. (01:30:00)
Continuity mistake: Clean starts out to the left of Captain Willard, then just before Colonel Kilgore lands he's behind the Captain chasing his helmet. In the next shot as the helicopter lands he's now in front of the Captain. (00:26:45)
Continuity mistake: After leaving the Do Lung Bridge everyone sits around reading their mail from home. Lance takes the canister of purple smoke and proceeds to throw it overboard as he says, "Hot potato. Hot potato." In the next shot he has the canister in his hand again. (01:51:15)
Continuity mistake: When Lazaro's boat initially approaches, the guy on the front has a flare in both hands that he's just about to ignite. When they cut to the next shot he doesn't have anything in his hands. (01:16:15)
Continuity mistake: As Captain Willard and Lance run away from Colonel Kilgore, their boat is coming towards them with no colored smoke behind it. A few seconds later we see it again with green and purple smoke behind it. (00:51:05)
Continuity mistake: There's a shot of two injured soldiers in wheelchairs sitting front row dead center as playmate of the year Carrie Foster is brought out to the audience. There's another soldier crouching between them who disappear between shots. (01:07:50)
Answer: It's not any specific song; it's just the kind of generic piano music you'd hear at dinner at a high-class restaurant of the era. Think of scenes in movies, films, etc. set or made in the '50s and '60s, where the characters go to a nice dinner and there's someone at the piano playing unobtrusive music to accompany the food/conversation.