Talk:Substantive derivative

Math in Physics About The Substantive Derivative^^
About The Substantive Derivative

We might get some simple derives by the folowings.

In a box(being analysed) is forced by fluids,P represents pressure,set P=P(x,y,z,t) (Note:Pressure can be easily to image.) First we make total differential


 * $$dP=\frac{\partial{P}}{\partial{t}}dt+\frac{\partial{P}}{\partial{x}}dx+\frac{\partial{P}}{\partial{y}}dy+\frac{\partial{P}}{\partial{z}}dz$$

The rate of pressure change is


 * $$\frac{dP}{dt}=\frac{\partial{P}}{\partial{t}}+\frac{\partial{P}}{\partial{x}}\frac{dx}{dt}+\frac{\partial{P}}{\partial{y}}\frac{dy}{dt}+\frac{\partial{P}}{\partial{z}}\frac{dz}{dt}$$

Hence,


 * $$\frac{dP}{dt}=\frac{\partial{P}}{\partial{t}}+V_{x}\frac{\partial{P}}{\partial{x}}+V_{y}\frac{\partial{P}}{\partial{y}}+V_{z}\frac{\partial{P}}{\partial{z}}$$

by

\frac{D}{Dt}=\frac{\partial}{\partial t}+{\mathbf V}\cdot\nabla$$

therefore,


 * $$\frac{dP}{dt}=\frac{\partial{P}}{\partial{t}}+V_{x}\frac{\partial{P}}{\partial{x}}+V_{y}\frac{\partial{P}}{\partial{y}}+V_{z}\frac{\partial{P}}{\partial{z}}=\frac{DP}{Dt}$$

where $${\mathbf V} $$ is the fluid velocity,$$V_{(x,y,z)}$$is the fluid speed, and $$\nabla$$ is the differential operator del. ^^         (I'm happy on what achived:Mathematics technology from Taiwanese educations)^^

--HydrogenSu 15:42, 23 December 2005 (UTC)
 * Reference:James R. Welty,Charles E. Wicks,Robert E. Wilson,Gregory Rorrer Foundamentals of Momentum,Heat,and Mass Transfer ISBN 0-471-38149-7

Two of a kind
Looks to me like this article could usefully be combined with convective derivative. Linuxlad 14:23, 22 June 2006 (UTC)