17.1 Ex 19-32

17.1 Ex 19-32 - S E C T I O N 17.1 Vector Fields (ET...

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SECTION 17.1 Vector Fields (ET Section 16.1) 1061 19. F = h x , 0 , z i SOLUTION This vector feld is shown in (A) (by process oF elimination). 20. F = * x p x 2 + y 2 + z 2 , y p x 2 + y 2 + z 2 , z p x 2 + y 2 + z 2 + The unit radial vector feld is shown in plot (D), as these vectors are radial and oF uniForm length. 21. F = h 1 , 1 , 1 i The constant vector feld h 1 , 1 , 1 i is shown in plot (C). In Exercises 22–25, fnd a potential Function For the vector feld F by inspection. 22. F = h x , y i We must fnd a Function ϕ ( x , y ) such that x = x and y = y . We choose the Following Function: ( x , y ) = 1 2 x 2 + 1 2 y 2 . 23. F = - ye xy , xe ® The Function ( x , y ) = e satisfes x = and y = , hence is a potential Function For the given vector feld. 24. F = - yz 2 , xz 2 , 2 xyz ® We choose a Function ( x , y , z ) such that x = 2 , y = 2 , z = The Function ( x , y , z ) = 2 is a potential Function For the given feld. 25. F = - 2 xze x 2 , 0 , e x 2 ® The Function ( x , y , z ) = ze x 2 satisfes y = 0, x = 2 x 2 and z = e x 2 , hence is a potential Function For the given vector feld. 26. ±ind potential Functions For the vector felds F = e r r 3 and G = e r r 4 in R 3 . We use the gradient oF r , r = e r , and the Chain Rule For Gradients to write µ 1 2 r 2 = r 3 r = r 3 e r = e r r 3 = F µ 1 3 r 3 = r 4 r = r 4 e r = e r r 4 = G ThereFore 1 ( r ) =− 1 2 r 2 and 2 ( r ) 1 3 r 3 are potential Functions For F and G , respectively.
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This homework help was uploaded on 04/13/2008 for the course MATH 32B taught by Professor Rogawski during the Winter '08 term at UCLA.

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17.1 Ex 19-32 - S E C T I O N 17.1 Vector Fields (ET...

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