Compute the flux SS F.nds through the piece of the cylinder of radius 2, centered on the z-axis, with 0≤x, 0≤ y and 0 ≤ ≤ 4 for the vecto S field F = 3zi – 5z³k where the normal of the surface pointing outwards as shown in the figure below. y 16.
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- 8. Find the work done by the gradient vector field F(x, y, z) = V (e" + y? + z?) in moving a particle along the circular helix r(0) = (cos 0, sin 0, 0) for 0 < 0 < 2n.In last week's lab you calculated the line integral of the vector field F = (2x, 3yz, zyz) around the boundary of the surface z = 1-², −1≤ x ≤ 1,0 < y<1, counterclockwise from above. The correct value was 8/15. Verify that the line integral you calculated last week is the same as the upward flux of V X F.6. Compute fF dr where f(x, y) = (y² cos x, 2y sinx) and C is the straight line path starting at (1,1) and ending at (1,3). Hint: is the vector field conservative? (Answer: 8 sin 1)
- 7. You must explain your reasoning. Consider the following radial vector field F: (a) Is Fa conservative vector field? (b) Suppose that for the two curves C (left) and C2 (right), we have F. dr = 5 What is fe F dr? (c) Which is greater, fe (F n) ds or fo (F-n) ds? Or are they equal?9. The outward flux of the vector field F=(x,0, z) across the cylinder x + y² = 1, for 015. Find the total upward flux of the field F = across the surface o: z=1 - 4x² - 4y² above the plane z = 0.Recommended textbooks for youAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, IncorporatedNumerical Methods for EngineersAdvanced MathISBN:9780073397924Author:Steven C. Chapra Dr., Raymond P. CanalePublisher:McGraw-Hill EducationIntroductory Mathematics for Engineering Applicat…Advanced MathISBN:9781118141809Author:Nathan KlingbeilPublisher:WILEYMathematics For Machine TechnologyAdvanced MathISBN:9781337798310Author:Peterson, John.Publisher:Cengage Learning,Advanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, IncorporatedNumerical Methods for EngineersAdvanced MathISBN:9780073397924Author:Steven C. Chapra Dr., Raymond P. CanalePublisher:McGraw-Hill EducationIntroductory Mathematics for Engineering Applicat…Advanced MathISBN:9781118141809Author:Nathan KlingbeilPublisher:WILEYMathematics For Machine TechnologyAdvanced MathISBN:9781337798310Author:Peterson, John.Publisher:Cengage Learning,