Transcription of Stokes’ Theorem
{{id}} {{{paragraph}}}
Example: tourism industry
Stokes’ Theorem
Stokes ~F(x,y,z) = y,x,xyz and~G= curl~F. LetSbe the part of the spherex2+y2+z2= 25that liesbelow the planez= 4, oriented so that the unit normal vector at(0,0, 5)is 0,0, 1 . Use Stokes Theorem to find S~G d~ s a picture of the use Stokes Theorem , we need to first find the boundaryCofSand figure out how it should beoriented. The boundary is wherex2+y2+z2= 25 andz= 4. Substitutingz= 4 into the firstequation, we can also describe the boundary as wherex2+y2= 9 andz= figure out howCshould be oriented, we first need to understand the orientation ofS. We are toldthatSis oriented so that the unit normal vector at (0,0, 5) (which is the lowest point of the sphere)is 0,0, 1 (which points down). This tells us that the blue side must be the positive want to orient the boundary so that, if a penguin walks near the boundary ofSon the positive side (which we ve already decided is the blue side), he keeps the surface on his left.
clockwise (from our vantage point). So, using Stokes’ Theorem, we have changed the original problem into a new one: Evaluate the line integral Z C F~d~r, where C is the curve described by x2 + y2 = 9 and z= 4, oriented clockwise when viewed from above. Now, we just need to evaluate the line integral, using the de nition of the line integral ...
Tags:
Information
Domain:
Source:
Link to this page:
Please notify us if you found a problem with this document: