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Algebra & Trigonometry with Analytic Geometry

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter8: Applications Of Trigonometry

Section8.3: Vectors

Problem 60E

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Problem Set-2
Vector (Unit-1)
1. State divergence theorem and verify it using the function y' i+(2xy + z*) j+ 2yzk for a unit
cube having one corner coincident with the origin.
2. State Stokes' Theorem and verify it with (2xz+3y²) j+ 2yz² k for a square surface of unit
length in the Y-Z plane and having the left-bottom corner coincident with the origin.
3. Show that the divergence of a curl is zero.
4. Show that the curl of a gradient is zero.
5. Show that F = (2xy + z°)i + x²j + 3xz²k is a conservative force field. Find the scalar
potential and the work done in moving from (1,-2,1) to (3,1,4).
6. Evaluate ff Ä.î dSwhere Ä = 18zt – 12j + 3yk, S is that part of the plane 2x + 3y + 6z =
12 located in the 1" octant.
7. Use surface integral to find the area of a sphere of radius R.
RASICS
8. Let o = 45 x²y and let V denote the closed region bounded by the following planes
4x + 2y + z = 8,x = 0. y = 0, z = 0. Evaluate f o dV.
9. If Å = (3x? + 6y)l – 14yzj + 20xz?k, evaluate Ā. dr along the paths
x = t.y = t², z = t³ ,
r(1,1,1)
(i)
(ii)
the straight line from (0,0,0) to (1,0,0) and then to (1,1,0) and then to (1,1,1).
10. Show that f. ř. Ads = 3V where Vis the volume enclosed the closed surface S.
11. Prove that
a) , v x Å dV = f, î x Å ds
b) $ dr xÃ = , (î x ỹ ) × Ả ds

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