It is a question vector calculus and Maxwell's laws. I put it this way. Let's say, we are working in a 3-Dimensional space ( e.g x⋅y⋅z=4⋅3⋅2, a certain room/class of that size ) .
Within this room, the heat obey a certain equation ( for e.g. T=25+5z ) .We know that heat flows from higher temperature regions to lower temperature regions. With this information in mind
How could I be able to determine the amplitude and the direction of travel of the thermal energy with the Del operator?
I'm not looking for a definitive response, but an equation that could give me potentially the result for the amplitude and the direction. I also want to know if thermal energy follows a loop pattern within my room (and be able to explain it mathematically by using the del operator once again of course)?
You can find the lecture source here.
From those, we are able to get the Amplitude:
For one dimensional heat flow, we have q=kT2−T1L , where T1 and T2 are terminal temperatures and L is the length of the material. Switching to 3-D spaces, we need to look at the fourier system of the heat transfer before proceeding further. qxA=−KdTdx. So integrating we obtain qxA∫L0 dx=−k∫T2T1 dT.
So given that the heat flow ( rate of conduction ) in the 3-D space (x,y,z) is given by the following equation q=−k∇T=−k(ˆi∂T∂x+ˆj∂T∂y+ˆk∂T∂y).
What about the direction?
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