Differential Notation in Physics

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  • 0:06 Our Changing World
  • 3:11 Differential Equations
  • 3:21 Position
  • 4:31 Temperature
  • 6:08 Lesson Summary
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Lesson Transcript
Instructor: Robert Egan
Stop. Look around. Everything is changing. In this lesson, you'll learn what a differential equation is and how these equations can describe the world around you.

Our Changing World

Take a look around you. Everything is changing. For example, if you're standing on Earth it looks like the positions of the stars are changing. That is to say that each star has a vector; it has an x position, a y position and a z position in space. Over time, it looks like it's moving. Its position along x is changing, its position along y is changing and its position along z is changing. You see this because it's either going up in the sky or closer to the horizon. So, at least relative to our point of view, the star's position is changing.

As you walk along, your altitude can change as a function of your position on earth.
our changing world

You also see changes in landmass. For instance when there is an earthquake, on a short-term time scale, the position of your house might move. In the long-term time scale, the position of your house - let's call it x - might move a great deal. It might move so your position is closer to Oregon than to Arizona. However, your independent variable isn't always time. Your position doesn't always depend on time.

If you're on Earth, you have a certain amount of atmospheric pressure of air pushing down on you, right? When you fly up in a plane, that atmospheric pressure decreases - that's why your ears pop. So the pressure exerted on you depends on your altitude. It's a different kind of change with respect to altitude. If you're just walking along Earth, your altitude can change as a function of your position on Earth. Your altitude can change as you walk along the direction x. You could be going straight south, and you may walk down a valley or up a hill. At the end of the day, your altitude is changing as a function of position.

According to the movies, the force between two people gets stronger the closer they are to one another. If you see someone from across the room, you might be drawn to them. When you're closer to them, maybe you want to give them a big kiss on the lips. This is the same as saying the force between two people, the force of attraction, changes as the distance between them changes. Here force would be your dependent variable, and distance would be your independent variable.

If you're jumping out of a plane to go sky-diving, then before you pull your parachute, the force of the air on you is going to depend on your velocity. The higher your velocity, the more the force of this air is going to push back on you. This means that the air resistance depends on your velocity.

The force of air on a sky-diver depends on the velocity.
parachuting

Differential Equations

All of these physical things can be described by differential equations. A differential equation is an equation that relates a variable and its rate of change. So let's take another look at position.

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