GATE EC 2014 Set 4 — Question 64
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Engineering Mathematics → Vector Calculus → Gauss, Stokes & Green's Theorems
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Question
Given . If represents the portion of the sphere for , then is ______________.
Correct answer
3.13 to 3.15
Solution
By Stokes' Theorem, the surface integral of the curl of a vector field over a surface is equal to the line integral of along the boundary curve of :Given the vector field:The surface is the upper hemisphere for . The boundary curve is the circle in the -plane where .On the boundary curve :
1.
2.
The line integral is calculated as:Substituting the conditions for ( and ):Using Green's Theorem in the plane (or the formula for the area of a region enclosed by a curve):where is the region enclosed by the unit circle .The area of a unit circle is .Value of .The official range for the answer is 3.13 to 3.15.Continue learning with Success Tracker
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