MTH301 Calculus II GDB Solution Fall 2013

Discuss the applications of  partial derivatives in daily life with at least 2 examples.

Solution: 

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.

 

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