I've noticed that in a manual transmission vehicle when going uphill I have two options:
I haven't measured it but typically in the higher gear I end out flooring the throttle, while the lower gear I don't need to go quite that far.
Am I wasting gas if I stay in the higher gear, or is the engine just not able to consume that much so I'm just making a bunch of gas available, but the engine just doesn't suck it in?
Bear with me, I'm going to clear up a myth, then answer your question. You will see that they are related.
Higher throttle doesn't mean higher RPM. Higher throttle just means more fuel entering the combustion chamber. This will tend to accelerate your engine, but if it's under an accelerating load (increasing incline) at a constant gear, the engine will not necessarily accelerate. It will, however, burn more fuel.
That being said, when you downshift, your engine does more Rotations Per Minute. It is now bringing less fuel into the combustion chamber, but it's bringing it in much more often.
All this means that when you are running under a load, you basically want to be running the engine at the balance point between entering tons of fuel into the combustion chamber at very low RPM and entering tons of fuel into the combustion chamber at very high RPM.
For most cars, that point is between 1,500 and 3,000 RPM. In other words: your fuel efficiency doesn't just have to do with higher vs. lower gear: it has more to do with your proximity to your engine's optimal RPM.
Exception case is when you are not running under a load (accelerating, climbing a hill). When you are on flat ground or descending a hill, you want the lowest RPM possible.
My Volvo V70 (model year 2006) has the digital dashboard fuel consumption meter. I also have a nice, fairly long uphill climb on my way to work, so I've had opportunities to try various methods. This is a manual transmission car.
According to the car, the fuel consumption is nearly the same whether I downshift from fifth to fourth gear and let off the gas pedal a little (going 70-80 km/h; the speed limit is 80 km/h), or stay in fifth gear and floor the pedal. Either method nets me roughly the same speed at the top of the climb, and churns through fuel at a rate of approximately 15-18 L/100 km while climbing (compared to something like 6-9 L/100 km during level cruise under power, depending on specifics).
The big difference is that if something happens, and I am in fourth gear, I have a lot more margin for acceleration -- I could probably get the car to at least 100 km/h without breaking a sweat while going uphill if I had to. (No, I'm not going to try it unless I absolutely have to.) If I'm in fifth gear and have the pedal floored just to maintain speed, there is no such margin, and I end up having to shift down while going uphill. Doing so loses a fair chunk of speed (something like 10 km/h) in the short time it takes to shift.
By shifting down the gear you just add more torque to be able to accelerate up hill. If you don't want to accelerate uphill you can reduce throttle and keep climbing. That brings you into the constant speed uphill situation (constant drag).
Generalizing that situation with a stupid 70yrs old motor: The more RPM you have the more fuel power you loose due to internal motor friction increased oil pump activity, just every device coupled to RPM directly with a belt.
But if you have too little RPM your motor will have to burn all the fuel you give it via your pedal. That may be too much at a time. The pressure on the sealings rises and the oil gets heated more than fine at certain points of friction. Not good.
Also you lose fuel power due to drag... as the motor parts are quickly getting faster on ignition but then suddenly get slower due to the high gear you chose and the hill.
If your motor squeals, you hear motor friction and waste fuel. If your little 1.3L motor growls and gurgles like a huge 3L Ford Mustang, you are breaking your cylinder head sealings right now.
Now for modern Motors: Most cars have adaptive compressed injection motors nowadays. It is not widely marketed and explained as most people don't care about technology when buying cars. Most of those technologies are actively tested in motor sports like F1 but mostly Rallies like Paris-Dakar.
The point is: the cylinders of modern motors are not just flat at the top. They are curved at different angles on different places. There are several injection nozzles that aim at those angled surfaces. When the fuel beam collides with those surfaces it is distributed differently and determines if if will be dense (and burn quickly) or scattered and burn slow.
(There are even areas where fuel will burn quick in the cycle or late in combination with dense or scattered. A modern car has more than three nozzles per cylinder that can be combined in output to reach different 'fuelburning' properties per cycle. The software does the calculations several thousands times a second.)
The software of the motor decides how to burn the fuel. As an input the engine software takes your fuel pedal movement.
Modern motors (last 5-10 years) govern your pedal input depending on their firmware (yes, engines are IT devices too by now.)
At too high RPM the motor firmware can't beat internal motor friction. You waste fuel if software does not govern quickly.
At too low RPM your engine sealings get all the stress. But motor firmware tries to ensure RPM.
If it is not smart enough to shut you off and reduce throttle by software, you actively break your engine every time it gurgles in a slow gear. (E.g. DodgeViper still allows to totally break the engine by manual input in 10 minutes. I've seen people exploding the viper-cylinder-sealings in that time.)
Engines are not made to run at very slow RPM (under 2000) for minutes when you are climbing. They are designed for 2000-5000, as RPM directly determines the pressure release per minute via the valves.
Just keep in at a nice snurring sound (2.200-3600 rpm) and drive up that hill without(!) getting faster. The software, programmed by people who have an army of engineers behind, does the rest if you have a 2007 up car.
It is easier to explain back in the carburetor day. When the gas did get sucked. A higher RPM would create more suction. More throttle the valve on the gas delivery would open wider. So even at low rpm when you floor it then more gas will be delivered. At a higher RPM even more gas is delivered due to more suction. Fuel injection is the same - RPM is part of the equation for how much gas to actually deliver. So in the two different gears the actual fuel consumption will be about the same.
