"What I dont understand is how a larger force would be unable to stretch their lines more effectively than a smaller opponent, or how the persuing persian were unable to keep up with the wheeling and charging maneuvers executed by the Macedonian forces. "

Your first fundamental mistake here is not realizing that larger groups are harder to move while maintaining cohesion than smaller groups (a mistake the Persians also made). Imagine taking a kindergarten class on a field trip. Now imagine taking an elementary school on the same field trip, with the same number of chaperones.

The second error you are making is not knowing what Alexander knew about the Persians. He knew their order of battle (the types and numbers of units). He knew how they liked to fight. And he knew that the Persians by this point were equally familiar with him. This meant Alexander knew how to show the Persians something they would expect, and he also had a very good idea how they would respond to it. Alexander, in other words, knows everything about the Persians at Gaugamela a bullfighter knows about a bull before walking into the ring. He knows where their horns are, how they will respond to seeing a red cape, and where the weak spot he can attack and stab them in the heart is.

Alexander's standard battle tactic, developed by his father, is to use his infantry units, the Phalanx troops, to make contact with the enemy in the center or left side, and pin the enemies best infantry down there while using his cavalry to sweep wide around the flank and then sweep in and trap the enemy between the two forces. Alexander had used this very tactic against the Persians successfully before, and he knew showing this tactic would be very familiar to the Persians. He also knew they had amassed a large elite cavalry formation to counter him. Like a bullfighter showing a red cape he aligned his infantry to the center left, and started pushing his cavalry off to the right to begin his standard flanking maneuver. Like a bull responding to red the Persians sent their cavalry out wide to their left to shadow Alexander, and started stretching their lines to the left to keep Alexander from flanking them.

There's a reason why bullfighters use a large brightly colored cape to get a bull's attention. A bull is a very nearsighted animal, and it takes a lot to get their attention. This is to the bull's disadvantage, as it can see the red cape, and not the sword in the bullfighters other hand. The bullfighters standard tactic is to get the bull to not only chase the cape, but to do so on a specific line and drop his head to gore the offending cape at a precise point. If the bullfighter can train the bull to taking this line and dropping his horns to strike at the cape right next to him a small 2-3 inch gap between the bull's shoulder blades will be exposed to him, with the horns to low to protect when he stabs in with his sword. If everything works right the bull will be dropped once his horns are too low. Of course, if the bullfighter misses he is unprotected, and the bull wins.

While Alexander was riding to his far right, pulling the Persian cavalry along with him, he had a unit of peltasts trailing him. The peltasts are light infantry that can move quickly, and they are armed with slings and javelins to throw at the enemy, and swords for self protection. The peltasts were behind Alexander's cavalry, opposite the Persians and it was a dry dusty day so the Persians didn't see them there. As Alexander keeps pushing further and further to his right, and the Persians keep edging out (to their left) to keep him from turning the corner on him the Persians start getting stretched out.

As you pointed out the Persians are a larger group, but they lack cohesion, or an ability to stay in neatly ordered ranks when moving. As they stretch their line gaps start opening between different units. Given time they could have corrected this easily, and used their superior numbers to fill in, but time is something Alexander won't give them. When Alexander sees a gap open up wide enough to run through he cuts back. Instead of moving out to he right slowly he cuts back and charges at full speed back to his left, back to the center of the battlefield. The Persians elite cavalry swing around to follow and block him.

This is where Alexanders peltasts come in to play. When Alexander cuts back the opposite direction his peltasts are now in the open right in front of the Persian cavalry. Instead of charging back towards Alexander across open space the Persians charge directly into a hail of projectiles from a peltasts unit that to all intents and purposes had just materialized in front of them out of thin air. It wasn't enough to destroy them completely, but it broke their momentum just long enough to keep them from blocking Alexander, who is now charging a gap in the Persian lines at full speed with 7,000 elite cavalry at his back.

If you watch American football you have seen this move hundreds of times. The ball is snapped, the running back moves right with his blockers, the linebackers and safeties move to intercept him. At the last second he cuts back, running against the grain through a hole that has opened up in the defense. Running back hits the opening he saw at full speed, runs through it, and scores a touchdown. This is a common but effective American Football move, pioneered by a man named Red Grange who left a young West Point cadet (and captain of the Army football team) named Dwight David Eisenhower with a permanent limp using it.

The Persians had a huge army, but the numbers were fluffed with contingents from all of their conquered territories. Their own units, their heavy infantry and their elite Bactrian cavalry (commanded by Darius kinsman) were what really counted here. Alexander had pinned the Persians best infantry units in contact with his infantry in the center. He had stretched the Persians best cavalry off the edge of the playing field and pinned them there with peltasts. He had found a gap in the Persian front line and was charging through it at full speed. At this point there wasn't a single effective combat unit standing between Alexander and Darius. Darius fled the battlefield. It was that or commit suicide. Once Darius was gone the non Persian units all quit. They weren't thrilled about fighting for their slave-masters in the first place, they certainly were not willing to continue the fight after the Persian king had run away. The battle turned into a rout, and Alexander was able to invade Persia proper virtually unopposed.