This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 18.02 Multivariable Calculus (Spring 2009): Lecture 31 Surface Area and flux across cylinders and spheres. April 24 Reading Material: From Simmons : 21.4. From Lecture Notes : V8, V9. Last time: Surface Area. Vector fields in 3D. Surface Integrals and flux. Today: Surface Area and flux across cylinders and spheres. In the last class we learned that the flux across an oriented surface S is flux of ~ F across S = ZZ S ( ~ F n ) dS Shorthand : We often use the notation ndS = d ~ S , so you could also write flux of ~ F across S = ZZ S ~ F d ~ S. This is a typical surface integral . We already know how to compute one of then, namely the area of a surface: Area ( S ) = ZZ S 1 dS so it will not be difficult to generalize this formula. 2 Evaluating flux Integrals Assume S is a surface given by the graph of a function f ( x,y ). Assume S is oriented by using the upward normal. To convert the flux integral ZZ S ~ F d ~ S into a double iterated integral we use (3.2) above and we observe that d ~ S = ndS = ~ N  ~ N   ~ N  dA = ( fx i fy j + k ) dA 1 and as a consequence we simply have ZZ S ~ F d ~ S = ZZ R ~ F ( x,y,f ( x,y )) ( fx i fy j + k ) dA, where R is the shadow cast by S on the xyplane....
View Full
Document
 Spring '06
 HartleyRogers
 Integrals, Multivariable Calculus

Click to edit the document details