This preview shows page 1. Sign up to view the full content.
Unformatted text preview: div( F G ) = G curl FF curl G . Letting F = f and G = g , this gives us that div( f g ) = 0, since the curl of any gradient eld is . 2. Find the ux of the vector eld F ( x, y, z ) = x ( x 2 + y 2 + z 2 ) 3 / 2 i + y ( x 2 + y 2 + z 2 ) 3 / 2 j + z ( x 2 + y 2 + z 2 ) 3 / 2 k through the surface x 2 + y 2 + z 2 = 1, oriented outward. Note that F always points in the same direction as the normal to this surface. So, at any point on the surface, F ( x, y, z ) n =  F ( x, y, z )   n  = s x 2 ( x 2 + y 2 + z 2 ) 3 + y 2 ( x 2 + y 2 + z 2 ) 3 + z 2 ( x 2 + y 2 + z 2 ) 3 = s 1 ( x 2 + y 2 + z 2 ) 2 = 1 . Therefore, if we call the given surface D , the ux integral is given by Z D F d S = Z D F n dS = Z D dS, which is the surface area of D . This is 4 . 1...
View Full
Document
 Spring '07
 Hutchings
 Math, Derivative, Scalar

Click to edit the document details