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3. Show that the divergence of a curl is zero.

3. Show that the divergence of a curl is zero.

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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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 xà = , (î x ỹ ) × Ả ds
Transcribed Image Text: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 xà = , (î x ỹ ) × Ả ds
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