I'm hoping somebody could help me understand the difference between the following:
$$∂_tc(x,t)$$ $$∂_xc(x,t)$$
My understanding is the the top derivative would be something like velocity but what would that make the bottom derivative?
Thanks.
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1What is $c$ modeling? The interpretation depends on what $c$ represents. – GFauxPas Apr 20 '15 at 21:44
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c is just the concentration of a molecule. So these describe the movement of the molecule as a function of position x and time t – The Nightman Apr 20 '15 at 21:47
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$∂_tc(x,t)$ represents the rate of change in $c$ with respect to $t$ (temperature) at a fixed $x$ (position), whereas $∂_xc(x,t)$ represents the rate of change in n $c$ with respect to $x$ (position) at a fixed $t$ (time).
Here is an example:
Consider a thin metal pipe filled with an aqueous solution being heated at only one end. Let $C(x, \, y)$ denote the concentration of a reactive solute $x$ meters from the beginning of the pipe at time $t$.
Here, $∂_tC(x,t)$ represents the rate of change of concentration of the solute at a specific fixed position from the beginning of the pipe, at any time $t$.
On the other hand, $∂_xC(x,t)$ represents the rate of change of concentration of the solute at a specific fixed time at any position $x$.
Does this help?
MathMajor
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Sorry should have made that clearer. t is time and x is position. Could you give me an example that helps me understand what a change in time at a fixed position would be like? I can understand how velocity would be described by change in position over a fixed time. But what about the former? – The Nightman Apr 20 '15 at 21:41
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This is just what I'm looking for, but what about time not temperature? – The Nightman Apr 20 '15 at 22:14
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