
(a) Given the vector field F = (0,22 + 2xy) = ui + (x2 + 2xy)j Find u for 7 to be conservative and find the potential, if it exists (b) Given u= (e? – zły, xy + y) = (e– r’y)i + (xy + y); Evaluate I= dos u dr where is the circle with radius r = 1 and center at the origin.
f(x, y) = x2 + y2 + 2xy + 6. 1- Find all the local extremas and 2) does the function f have an absolute max or min on R2
(1 point) (a) Show that each of the vector fields F-4yi + 4x j, G-i ЗУ x2+y2 x?+yi J, and j are gradient vector fields on some domain (not necessarily the whole plane) x2+y2 by finding a potential function for each. For F, a potential function is f(x, y) - For G, a potential function is g(x, y) - For H, a potential function is h(x, y) (b) Find the line integrals of F, G, H around the curve C...
j [3x2 +5y + i(x2-y2)] dz along (0,0) =x. |Evaluate
5. (2 Points) Let -2 -y F (x2+y2+z2)2/3' (2+y2)2/3' (r2+y2+22)2/3 Find the work done by this force field on an F(t) = (1+3t, 1 + 4t2, 1 +5t3) object that moves from (1,1, 1) to (4,5,6) along the curve
Let F(x,y,z) = ztan-1(y2) i + z3ln(x2 + 2) j + z k. Find the flux of F across the part of the paraboloid x2 + y2 + z = 8 that lies above the plane z = 4 and is oriented upward.
Q6 [10+1+3=14 Marks] Let F be a force field given by F(x, y) = y2 sin(xy?) i + 2xy sin(xy?)j. (a) Show that F. dr is exact by finding a potential function f. (b) Is I = S, y2 sin(xy2) dx + 2xy sin(xy?) dy independent of path C? Justify your answer. (c) Use I to find the work done by the force field F that moves a body along any curve from (0,0) to (5,1).
Use Green's theorem to evaluate line integral F.dr, where F(x, y) = (y2 – x2)i + (x2 + y2)j, and C is a triangle bounded by y = 0, x = 6, and y = x, oriented counterclockwise.
QUESTION 2 Calculate and simplify det if f(x,y) = O x2 - y2 + 2xy (x2 + y2)2 O 4x(+- x² + 3y2) (x² + y²)3 o 4x(x² – 3y?) (x² + y²)3 OO O x2 - y2 + 2xy (x² + y²)?
1 Let F = (22+y2) -y i + x j). Find a nontrivial curve Cį such that F .dm = 0 and a nontrivial curve C, such that fc, F.dm #0. Justify Sc, F your answers.