Your way of thinking of it is essentially correct. Part of the confusion is that when you ask "what T really is", the question is ambiguous. There are two functions of interest here: one is T(x,y,t), where we put in whatever x, y, and t values we want, to measure the temperature of the plate at a position and time we choose. The other is T(x(t),y(t),t), where (x(t),y(t)) is the position of the bug (or moving temperature sensor, if you like).
I found your example some how difficult to understand, since it's not clear to me what T really is. Either is T the temperature of the bug or the temperature of the plane? To better understand your idea, I think of T as the outcome of a temperature sensor moving across the plane (measuring the spacial change in temperature) while at the same time the plane itself keeps changing its temperature e.g. due to external heating, so that T also measures the temporal change of the plane's temperature.
Thanks for the video. Here's a question: In steady-state where F(X,Y) the total derivative ( DF/DT) is not zero right? It's the partial derivative that is zero?
It really depends on what you mean by "steady-state". If the parameters of a system are not changing---for example, no external forces are changing, no knobs are being twiddled---then you have the partial derivative of F(x,y,t) with respect to t being 0. But even with the partial being zero, a particle moving within the system will usually experience a nonzero total derivative of F with respect to t, since its x and y are changing as t changes.
Your way of thinking of it is essentially correct. Part of the confusion is that when you ask "what T really is", the question is ambiguous. There are two functions of interest here: one is T(x,y,t), where we put in whatever x, y, and t values we want, to measure the temperature of the plate at a position and time we choose. The other is T(x(t),y(t),t), where (x(t),y(t)) is the position of the bug (or moving temperature sensor, if you like).
davidmetzler 1 year ago
I found your example some how difficult to understand, since it's not clear to me what T really is. Either is T the temperature of the bug or the temperature of the plane? To better understand your idea, I think of T as the outcome of a temperature sensor moving across the plane (measuring the spacial change in temperature) while at the same time the plane itself keeps changing its temperature e.g. due to external heating, so that T also measures the temporal change of the plane's temperature.
mdinka 1 year ago
Thanks. That's a very easy to understand explanation. I wish I had seen something like this when I first learned about PDE's back in the 60's.
jimmymars 1 year ago
Awesome video, makes perfect sense, thanks for posting!
sjb167 1 year ago
Tanks for this contribution. Nice example.
Holographys 1 year ago
Thanks for the video. Here's a question: In steady-state where F(X,Y) the total derivative ( DF/DT) is not zero right? It's the partial derivative that is zero?
nikan4now 2 years ago
It really depends on what you mean by "steady-state". If the parameters of a system are not changing---for example, no external forces are changing, no knobs are being twiddled---then you have the partial derivative of F(x,y,t) with respect to t being 0. But even with the partial being zero, a particle moving within the system will usually experience a nonzero total derivative of F with respect to t, since its x and y are changing as t changes.
davidmetzler 2 years ago