Your in sight is something I had not thought of and I'll give it some more thought. I appreciate your time and effect in engaging me on this. I feel that, when studying some phenomenon, I sometimes get lost in the math solving problems and miss some greater truth. Thanks again.
I am asking something slightly different. To illustrate, I'll use the one-dimensional wave equation, say, for a vibrating string.
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where y is the displacement (a function of both x and t), t is time, and x is the independent space variable.
This equation says is that the acceleration of the...
I'm interested in the latter question. I do understand the mathematics (calculus) for finding solutions. Many physical phenomena have the property that time variation is proportional to spatial variations such as its gradient, divergence, curl, or Laplacian. I'm wondering whether there is any...
Many physical laws involve relationships between time derivatives to space derivatives of one or more quantities. For example, thermal conduction relates the thermal energy time rate of change [dQ/dt] to temperature space rate of change [dT/dx]. In fluid flow, the Navier-Stokes Theorem relates...