Evaluating Surface Integral with Stokes Theorem

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SUMMARY

The discussion focuses on using Stokes' Theorem to evaluate the surface integral of the vector field F = (x² + y², x, 2xyz) over a hemisphere defined by the equation x² + y² + z² = a² for z ≥ 0. The boundary curve, which is crucial for applying Stokes' Theorem, is identified as the circle x² + y² = a² in the xy-plane. Participants emphasize the need to express the vector field in spherical polar coordinates and clarify the differentiation process to obtain the differential element dr. Visual aids, such as diagrams, are recommended to enhance understanding of the problem.

PREREQUISITES
  • Understanding of Stokes' Theorem
  • Familiarity with vector fields and surface integrals
  • Knowledge of spherical polar coordinates
  • Ability to perform differentiation in multiple dimensions
NEXT STEPS
  • Study the application of Stokes' Theorem in various contexts
  • Learn how to convert Cartesian coordinates to spherical polar coordinates
  • Practice evaluating surface integrals with different vector fields
  • Explore visualizing surfaces and boundaries in three-dimensional space
USEFUL FOR

Students and professionals in mathematics, physics, and engineering who are working with vector calculus, particularly those interested in surface integrals and Stokes' Theorem applications.

trelek2
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use the stokes theorem to evaluate the surface integral [curl F dot dS] where
F=(x^2+y^2; x; 2xyz)
and S is an open surface x^2+y^2+z^2=a^2 for z>=0. So i guess its a hemisphere of radius a lying on x-y plane.
I don't see however how to take F dot dr. What is this closed curve dr bounding this hemisphere? I guess we can take spherical polar coordinates, but still once i have x,y,z in terms of r,phi,theta i still don't know "dr" (just differentiate ?) and what will the limits of integration be? Can someone show me?
 
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The boundary of the surface is just the circle x² + y² = a² in the xy plane. Did you try drawing a picture?
 

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