Compute Integral of F over S: Vector Calculus

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SUMMARY

The discussion focuses on computing the surface integral of the vector field F over the surface S defined by the equation z = 4 - x² - y², where z is non-negative. The vector field is given by F(x,y,z) = xcos(z) i - ycos(z) j + (x² + y²) k, with a downward-pointing normal. Participants highlight that the divergence of F is zero across R³, indicating the existence of a vector field G such that F = Curl G, although G is not explicitly defined. The urgency of the request underscores the complexity of the problem, which has stumped the original poster for several hours.

PREREQUISITES
  • Understanding of vector calculus concepts, specifically surface integrals.
  • Familiarity with vector fields and their properties, including divergence and curl.
  • Knowledge of the mathematical representation of surfaces in three-dimensional space.
  • Experience with the application of the Divergence Theorem and Stokes' Theorem.
NEXT STEPS
  • Study the Divergence Theorem and its applications in vector calculus.
  • Learn about Stokes' Theorem and how it relates to surface integrals of vector fields.
  • Research methods to compute curl and divergence of vector fields in three dimensions.
  • Explore examples of computing surface integrals over various surfaces in vector calculus.
USEFUL FOR

This discussion is beneficial for students and professionals in mathematics, particularly those studying vector calculus, as well as educators seeking to understand complex surface integrals and vector field properties.

lembeh
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Urgent help! Vector Calculus question...

Let S be the surface given by the graph z = 4 - x2 - y2 above the xy-plane (that it is, where z [tex]\geq[/tex] 0) with downward pointing normal, and let

F (x,y,z) = xcosz i - ycosz j + (x2 + y2 ) k

Compute [tex]\oint[/tex][tex]\oint[/tex]sF dS. (F has a downward pointing normal)

(Hint: Its easy to see that div F = 0 on all R3. This implies that there exists a vector field G such that F = Curl G, although it doesn't tell you what G is)
 
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Please help me with this. I've been working on it for 6 hours now and still can't figure it out...Your help is really appreciated!
 


I was about to suggest something but apparently, you have to show some attempts before we can help you.
 

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