Hello! Please help with this advanced math question. Let F = 22 +y2 +(-ył +21). Find a nontrivial curve C, such that fc, F.dñ = 0 and a nontrivial curve C, such that fc, F.dh +0. Justify your answers.
Hello! Please help with this advanced math question. Let F = 22 +y2 +(-ył +21). Find a nontrivial curve C, such that fc, F.dñ = 0 and a nontrivial curve C, such that fc, F.dh +0. Justify your answers.
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).
Calculus 4
Let f(x,y) = A)-i-j E) i+j 1. Find the gradient vector Vf (1, 1) at the point (x,y) = (1,1). B) - 1 - 1 D)-i-j 10. . Find the largest value of the directional derivative of the function f(x,y) = ry + 2ya at the point (3,y) = (1,2). A) 53 ' B) V58 C) V63 D) 74 E) 85 y + The function (,y) = 2 + y2 + A) (-3,5), saddle point C) (-1,3), maximum...
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.
Let F(r, y, z)(z4+ 5y3)i + (y2 surface of the solid octant of the sphere x2+yj2 + 22 = 9 for x> 0, y> 0 and z> 0 )j+ (3z + 7)k be the velocity field of a fluid. Let B be the Determine the flux of F through B in the direction of the outward unit normal
Let F(r, y, z)(z4+ 5y3)i + (y2 surface of the solid octant of the sphere x2+yj2 + 22 = 9 for x>...
Consider the vector field F(x, y, z) = (z arctan(y2), 22 In(22 +1), 32) Let the surface S be the part of the sphere x2 + y2 + x2 = 4 that lies above the plane 2=1 and be oriented downwards. (a) Find the divergence of F. (b) Compute the flux integral SS. F . ñ ds.
Find extremals
b) J(y) oe2(y - y2)d, y(0)=1, y(3) free c) J(y) Jo2+'y )d, y(0) = , y(1) free. d) J(y) 22y2 - y2)d, y(1) 1, y(e) free. (1) X= B (ii) y is a geodesic; (iii) Xp is not zero and is parallel along 6.5 Let y be a curve with Y(0) = P, T(1) = Q. If Xe T,M, let X be the parallel translate ofX along y. Define y: T,M --- TqM by y*= X(1) (a) Prove...
5. Let F = (3x2 + 2xy + y2)i + (x2 + 2xy + 4y3)j, and let C be any curve in R2 which begins at the origin and ends at (1, 2). Find ( F.dr.
Let f(x,y) = x2 - xy + y2 - y. Find the directions u and the values of Duf(1, -1) for which the following is true. a. Duf(1, - 1) is largest c. Duf(1, - 1) = 0 e. Duf(1, -1)= -3 b. Duf(1, -1) is smallest d. Duf(1, -1) = 4 a. Find the direction u and the value of Duf(1, - 1) for which Duf(1 - 1) is largest u=[i+Oj and D,f(1, - 1) = 0 b. Find...