How partial derivatives arise in physical applications?

Question

Please, Please, Support your opinion with examples.

Answer 1

The famous partial differential equation is the wave or Harmonic equation in physics. Then there is the Navier-Stokes equation in fluid mechanics. Just about any situation with multiple variables can be modeled with some partial differential equations. The problem is not so much as how the partial derivatives arise, but how to solve it.

What you learn in a partial differential equation class is not how it is actually solved in real-life. They only give you the simple cases. In real-life engineering, you have software programs using numerical techniques like finite differences to solve the partial derivative problems.

For example, the space shuttle foam problem during lift-off can be modeled with the Navier-Stokes partial differential equation. I believe it has 7 variables. If you leave all 7 variables unconstrained fluid flow over the shuttle structure, then it will probably require a few parallel supercomputing servers to crunch the numbers to look at all the different possibilities (or solutions) of how a foam may fall during lift-off. The results give the engineers an idea of what to expect with a free flowing foam near the Shuttle during lift-off and how they can design things differently to avoid any free flowing objects.

Answer 2

Applications Of Partial Derivatives

Answer 3

The derivatives are the rate of change or the slope.
If any property varies with respect to two or more variables then you define the derivatives partially one being constant

The temperature distribution on the top surface of a coffee cup. the partial derivative of temperature with respect to 2 coordinates. This means the slope or the temperature variation along x is the partial derivative of temp vs x y being constant.

I wish I have some diagrams. if you still don't get it let me know I will draw some diagrams

Answer 4

In quantum mechanics, Schroedinger's equation is actually formed of 2 partial derivatives. Any problem where you are studying the evolution of 1 parameter while not caring about the other. Also taking the gradient is extremely common. MAxwell's equations have alot of those.

Answer 5

When factors that are dynamic act independently.
A car is in motion: the fuel enters the Engine, buy the car itself is travelling alone, so the fuel is in motion because the car is in motion, and the fuel is in motion because it's moving into the engine.

Answer 6

The distribution of heat in an unevenly heated object is represented by the heat equation:

http://mathworld.wolfram.com/HeatConductionEquation.html

The vibration of a two dimensional object (drum head) is represented by the wave:

http://mathworld.wolfram.com/WaveEquation.html