There fore integraldisplay γ 2 xyz e x 2 dx z e x 2

Info icon This preview shows pages 3–4. Sign up to view the full content.

View Full Document Right Arrow Icon
dant of the choice of path and we need only evaluate at the endpoints. There- fore, integraldisplay γ 2 xyz e x 2 dx + z e x 2 dy + y e x 2 dz = f (0 , 1 , 2) f (2 , 0 , 1) = (1)(2)( e 0 ) (0)(1)( e 4 ) = 2. 6. (a) (i) We parametrize γ by γ ( t ) = ( 1 , 1) + t (2 , 2) = ( 1 + 2 t, 1 + 2 t ), 0 t 1. Now integraldisplay γ ( x 2 + x ) dx + xy dy = integraldisplay 1 0 ( (( 1 + 2 t ) 2 + ( 1 + 2 t ))(2) + ( 1 + 2 t )( 1 + 2 t )(2) ) dt = integraldisplay 1 0 ( 16 t 2 12 t + 2 ) dt = bracketleftbigg 16 3 t 3 6 t 2 + 2 t bracketrightbigg 1 0 = 4 3 . (ii) We parametrize γ by γ ( t ) = ( t, t 3 ), 1 t 1. Now integraldisplay γ ( x 2 + x ) dx + xy dy = integraldisplay 1 1 ( ( t 2 + t )(1)+( t )( t 3 )(3 t 2 ) ) dt = integraldisplay 1 1 ( 3 t 6 + t 2 + t ) dt = bracketleftbigg 3 7 t 7 + 1 3 t 3 + 1 2 t 2 bracketrightbigg 1 1 = 32 21 .
Image of page 3

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
MATB42H Solutions # 4 page 4 (b) If the vector field F ( x, y ) = ( x 2 + x, xy ) were conservative, the answers in part (a) would have been the same. Since they are different, F ( x, y ) can not be con- servative. 7. (a) We wish to parametrize the curve of intersection of x 2 + y 2 + z 2 = 3 and z = 1 y . Substituting for z in the first we have x 2 + y 2 + 1 2 y + y 2 = 3. Completing the square we have x 2 + 2 parenleftbigg y 1 2 parenrightbigg 2 = 5 2 , an ellipse in the xy –plane, which can be parametrized by x = radicalbigg 5 2 cos t , y = 1 2 + 5 2 sin t . Now z = 1 y = 1 2 5 2 sin t , so we can parametrize the curve of intersection, in the counterclockwise direction, when viewed from above, by γ ( t ) = parenleftbigg radicalbigg 5 2 cos t, 1 2 + 5 2 sin t, 1 2 5 2 sin t parenrightbigg , 0 t 2 π . (b) By inspection we note that F ( x, y, z ) = (2 x, 2 yz, y 2 ) is conservative and the general potential function is g ( x, y, z ) = x 2 + y 2 z + C . Since c , the curve of intersection, is a closed curve, we have integraldisplay γ F · d s = 0.
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern