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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.
A derivative of a function of one variable expresses a change in that function relative to its argument. For example Newtons law for rectilinear motion is F = mass times the second derivative of distance with respect to time.
But the world involves functions that depend on several variables. For example the pressure of a gas depends on density and temperature. The speed of sound (squared), it turns out, in a nebula in space (which is very nearly at constant temperature due to radiative transport) is the partial derivative of the pressure with respect to density keeping temperature fixed.
Your happiness H depends on how much money, m, you make and the number of hours, h, you spend with your family. H = H(m, h). But how much money you make also depends on how much on how much time you spend with your family. The more time you spend with them the less money you will make. So m = m(h) and we must write
H = H(m(h), h)
Now, we want to know how many hours "h" to work to maximize happiness so we take the _total_ derivative of H with respect to h and set it equal to zero:
dH/dh = (partial H/partial m)*(partial m/partial h) + (partial H/partial h) = 0.
(1) Maxwell's equations of electromagnetism
(2) Einstein's general relativity equation for the curvature of space-time given mass-energy-momentum.
(3) The equation for heat conduction (Fourier)
(4) The equation for the gravitational potential of a blob of mass (Newton-Laplace)
(5) The equations of motion of a fluid (gas or liquid) (Euler-Navier-Stokes)
(6) The Schrodinger equation of quantum mechanics
(7) The Dirac equation of quantum mechanics
(8) The Yang-Mills equation
(9) The Liouville equation of statistical mechanics