Department of Mathematical and Computational Sciences
National Institute of Technology Karnataka, Surathkal
Engineering Mathematics II - MA111
Even Semester (2016 - 2017)
Assignment Sheet 10
Stokes’ Theorem & Divergence Theorem
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1. Find the circulation of F = y2 î − yĵ + 3z2 k̂, around the ellipse in which the plane 2x +
6y − 3z = 6 meets the cylinder x2 + y2 = 1, counter-clockwise as viewed from above.
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2. Evaluate S (O × F ).n̂ dσ where F = 2yzî − ( x + 3y − 2)ĵ + ( x2 + z)k̂ and S is the
surface of intersection of the cylinders x2 + y2 = a2 and x2 + z2 = a2 that lies in the first
octant.
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3. Evaluate C ydx + zdy + xdz where C is the curve of intersection of the sphere x2 + y2 +
z2 = a2 and the plane x + z = a.
4. Verify Stokes’ theorem for the following fields.
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(i) F = (2x − y)î − yz2 ĵ − y2 zk̂ where S is the upper half surface of the sphere x2 +
y2 + z2 = 1.
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(ii) F = x2 î + xyĵ, where S is the square 0 ≤ x ≤ a, 0 ≤ y ≤ a in the xy-plane.
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5. Without actually computing the integral, show that the circulation of F = O( xy2 z3 )
around the boundary of any smooth Orientable surface in space is zero.
6. Let S be the cylinder x2 + y2 = a2 , 0 ≤ z ≤ h, together with its top x2 + y2 ≤ a2 , z = h.
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Let F = −yî + x ĵ + x2 k̂. Use Stokes’ theorem to find the flux of O × F outward through
S.
7. Let S be the upper hemi-sphere x2 + y2 + z2 = 1, z ≥ 0.
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(i) Find F such that curl F = xey î − ey ĵ.
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(ii) Evaluate S ( x2 ey − yey )) dσ.
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[Hint: By observation, F = xey k̂ and x2 ey − yey = curl F .( x î + yĵ + zk̂)]
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8. Let C be the curve defined by r(t) = cos tî + sin tĵ + (cos t + 4)k̂, 0 ≤ t ≤ 2π and
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F = (z2 + ez )î + 4x ĵ + (ez cos2 z)k̂. Evaluate C F .d−
r.
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9. Let F = yî − x ĵ + (2z2 + x2 )k̂ and S be the part of the sphere x2 + y2 + z2 = 25 that lies
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below the plane z = 4. Evaluate S curl F .n̂ dσ, where n̂ is the outward drawn normal
of S.
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10. Show that the outward flux of a constant vector field F = C across any closed surface
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to which the Divergence Theorem applies is zero.
11. Among all the rectangular solids defined by the inequalities 0 ≤ x ≤ a, 0 ≤ y ≤ b,
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0 ≤ z ≤ 1, find the one for which the total flux of F = (− x2 − 4xy)î − 6yzĵ + 12zk̂
outward through the six sides is greatest. What is the greatest flux?
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12. Use the Divergence theorem to find the outward flux of F = 2xzî − xyĵ − z2 k̂ across the
boundary of the wedge cut from the first octant by the plane y + z = 4 and the elliptical
cylinder 4x2 + y2 = 16.
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13. Use the Divergence theorem to find the outward flux of F = x2 + y2 + z2 ( x î + yĵ + zk̂)
across the boundary of the region 1 ≤ x2 + y2 + z2 ≤ 2.
14. Let D be the region enclosed by the surfaces x2 + y2 = 4, z = 0 and z = x2 + y2 . Let
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S be the boundary of D and n̂ denote the unit outward normal to S. Suppose F is a
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vector field whose components have continuous first order partial derivatives. If div F =
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α( x − 1), for some α ∈ R and S F .n̂ dσ = π, evaluate α.
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15. Let S be the sphere x2 + y2 + z2 = 1. Suppose for some α ∈ R, S [zx + αy2 + xz] dσ = 4π
3 ,
find α.
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16. Evaluate S [ x (2x + 3ez ) + y(−y − e x ) + z(2z + cos2 y)] dσ, where S is the unit sphere.
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