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Find te gradient of | r= v youi result. the magnitude of the position vector. Comr

Find te gradient of | r= v youi result. the magnitude of the position vector. Comr

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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16:48 O. O O Assignment-1.pdf 13. a) Show that Vo , the gradient of a scalar function points in the direction of ma change of o. b) Also show that |V@| gives the slope of o along this maximal direction. 14. Show that the gradient of constant surface o (x, y.z)=c at any point on o points t the direction of the normal to the surface at that point. 15. Find a unit normal to the surface x²y + 2xz = 4 at (2,-2,3). . Find te gradient of | r |= Vx² + y° + z² , the magnitude of the position vector. Comr Jour result. 17. Find the gradients of the following functions (i) x² + y' + z* (ii) x²y°z* 18. A surface is given as f(x, y) = x² + y² – 2 = 0. Find Vf at the points (V2,0)(0, /2). comment on the results. 19. Find the directional derivative of Ø = x?yz + 4xz²at (1-2,-1) in the direction of 21 - j -2k. 20. In what direction from the point (1,3,2) the change of Ø = 2xz – y?is maximum? What is the magnitude of the maximum? 21 Find the angle between the surfaces x? + y? +z² = 9 and z = x² + y² – 3 at the (2,-1,2). 22. Prove the following りマ() - = - i) ūr" = nrn-2# 23. Calculate the divergence of the following functions. i) x* i+ 3.xz ii) xyi+ 2yz j+ 3zxk 24. Calculate the divergence of E = and comment on the result you get. 25. A) Calculate the curl of the following functions i) - yitxj ii) x j ii) B) Prove that = 0
Transcribed Image Text:16:48 O. O O Assignment-1.pdf 13. a) Show that Vo , the gradient of a scalar function points in the direction of ma change of o. b) Also show that |V@| gives the slope of o along this maximal direction. 14. Show that the gradient of constant surface o (x, y.z)=c at any point on o points t the direction of the normal to the surface at that point. 15. Find a unit normal to the surface x²y + 2xz = 4 at (2,-2,3). . Find te gradient of | r |= Vx² + y° + z² , the magnitude of the position vector. Comr Jour result. 17. Find the gradients of the following functions (i) x² + y' + z* (ii) x²y°z* 18. A surface is given as f(x, y) = x² + y² – 2 = 0. Find Vf at the points (V2,0)(0, /2). comment on the results. 19. Find the directional derivative of Ø = x?yz + 4xz²at (1-2,-1) in the direction of 21 - j -2k. 20. In what direction from the point (1,3,2) the change of Ø = 2xz – y?is maximum? What is the magnitude of the maximum? 21 Find the angle between the surfaces x? + y? +z² = 9 and z = x² + y² – 3 at the (2,-1,2). 22. Prove the following りマ() - = - i) ūr" = nrn-2# 23. Calculate the divergence of the following functions. i) x* i+ 3.xz ii) xyi+ 2yz j+ 3zxk 24. Calculate the divergence of E = and comment on the result you get. 25. A) Calculate the curl of the following functions i) - yitxj ii) x j ii) B) Prove that = 0
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