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Unformatted text preview: LESSON 25 Divergence and the Divergence Theorem READ: Section 18.3 NOTES: The divergence of the vector field- F = F 1- + F 2- + F 3- k is defined to be div- F = F 1 x + F 2 y + F 3 z Example : If- F = xe y- i + xyz- j + sin yz- k then div- F = e y + xz + y cos yz . Notice that the divergence is a scalar field; that is, vectors are plugged in and numbers are produced. The divergence shows some of the features of ordinary derivatives. For example, it is linear . That is, div a- F + b- G = a div- F + b div- G, It also obeys a law reminiscent of the product rule for derivatives. Let- F = - V be the flux density of a flowing fluid. The vector- F ( x,y,z ) points in the direction of the velocity, and its magnitude has units such as grams per square meter per second. In other words,- F ( x,y,z ) tells us how much mass of fluid (3 grams say) is flowing in the direction of- V ( x,y,z ) at the point ( x,y,z ) per square meter per second. Now imagine a little rectangular box of the fluid, with one corner of the box at ( x,y,z ), and with side lengths given by x , y , and z . We want to compute the mass of the fluid flowing through one face of the box....
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This note was uploaded on 02/24/2011 for the course MATH 265 taught by Professor Jerrymetzger during the Winter '11 term at North Dakota.
- Winter '11