Math Help - Types of Derivatives.

1. Types of Derivatives.

Hi All.

I was wondering as to under which conditions can one use simple du/dt instead of Du/Dt ? Here u is the velocity, t is the time and D/Dt represents the "material or total derivative". So both derivatives represent different types of accelarations. So when does D/Dt becomes d/dt ?

Thanks.

2. Originally Posted by beezee99
Hi All.

I was wondering as to under which conditions can one use simple du/dt instead of Du/Dt ? Here u is the velocity, t is the time and D/Dt represents the "material or total derivative". So both derivatives represent different types of accelarations. So when does D/Dt becomes d/dt ?

Thanks.
The notation "D/Dt" is used in problems dealing with fluid flow and is "rate of change with the flow".

For example, if T(x, y, z, t) is the temperature at each point (x, y, z) and time t of water in some channel and you lean over and stick a thermometer into the water to record the temperature at a specific point, you would be measuring the temperature of different "bits" of water as they flow by the thermometer. The rate of change of your readings would be dT/dt (or, more correctly, $\partial T/\partial t$). If, instead, you attach a thermometer to a small block of wood and let it move with the flow, so that your thermometer records the temperature of a single "bit" of water as it flows, the rate of change of your readings would be DT/Dt.

If I remember corrrectly (it's been a while), If the velocity vector is $\vec{v}(x, y, z, t)$, again at each point (x, y, z) and time t, then $\frac{DT}{Dt}= \frac{\partial T}{\partial t}+ \nabla\vec{T}\cdot\vec{v}$ where $\nabla T= \frac{\partial T}{\partial x}\vec{i}+ \frac{\partial T}{\partial y}\vec{j}+ \frac{\partial T}{\partial z}\vec{k}$
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