This preview shows page 1. Sign up to view the full content.
Unformatted text preview: Leibniz Theorem and the Reynolds Transport Theorem for Control Volumes Author: John M. Cimbala, Penn State University Latest revision: 20 September 2007 1-D Leibniz Theorem The one-dimensional form of the Leibniz theorem allows us to differentiate an integral in which both the integrand and the limits of integration are functions of the variable with which the integral is being differentiated: ( ) ( ) ( ) ( ) ( , ) ( , ) ( , ) x b t x b t x a t x a t d F db da F x t dx dx F b t F a t dt t dt dt = = = = = + Example : Find 2 x ct x x d e dx dt = = . This integral cannot be solved in closed form and then differentiated. However, with Leibniz rule, the solution is easily found. The above expression reduces to 2 2 c t ce (to be done in class). 3-D Leibniz Theorem The one-dimensional Leibniz theorem can be extended to three dimensions (volume and area integrals) as follows: ( ) ( ) ( ) ( , ) ( , ) ( , ) A t t A t d F x t F x t d d F x t u dA dt t = + G G G G v V V V V G where...
View Full Document
This note was uploaded on 07/23/2008 for the course ME 521 taught by Professor Cimbala during the Fall '07 term at Pennsylvania State University, University Park.
- Fall '07