Show that the gravitational field F(x)--mMG is conservative with the potential function f(x) mMG(...
Decide whether or not the vector field is a gradient field (i.e. is conservative). If it is conservative, find a potential function. (ii) F(x,y)-6ญ่-12xVJ (iv) F(x, y, z)-< ye", e + z,y >
Decide whether or not the vector field is a gradient field (i.e. is conservative). If it is conservative, find a potential function. (ii) F(x,y)-6ญ่-12xVJ (iv) F(x, y, z)-
. Let F(a,y)-(3e +secztan,e -90 (a) Show that F is a conservative. (b) Find a function f (potential function) show that F (c) Use above result to evaluate Jc F. V. dr, where C is a smooth curve that begin at the point (2, 1 ) and ends at (0,3). (cos t, sin t) fromtto t particle that moves along the curve. (Write the value of work done without evaluating (d) Find the work done by the force field F(r,...
QUESTION 10 (a) Verify that the force field F = yi+zj+4k s conservative. By finding its potential function ф, evaluate the work done by F in moving a particle over the path from (6 marks) Apply Green's Theorem to evaluate the line integral фу2dr+x2dv in which C is (b) the boundary of a triangle made by x-0, x+y-1, y-0 (6 marks)
(1 point) Determine whether the vector field is conservative and, if so, find the general potential function. F = (cos z, 2y!}, -x sin z) Q= +c Note: if the vector field is not conservative, write "DNE". (1 point) Show F(x, y) = (8xy + 4)i + (12x+y2 + 2e2y)j is conservative by finding a potential function f for F, and use f to compute SF F. dr, where is the curve given by r(t) = (2 sinº 1)i +...
+ cos(y) is conservative by responding to the 2. Show that the vector field F(x,y) = (ye* + sin(y))i + ( following steps: a.) Determine both P(x,y) and Q(x,y) given F. b.) Demonstrate your answers in a.) satisfy Clairaut's theorem. c.) Partially integrate P with respect to r to obtain the potential S(= y) = P(x,y)da = (1.x) + C) where (a,b) is the anti-derivative of P(x,y) with respect to r and C(y) is a function of y such that...
Let F(x,y,z) = <2y2z, 4xyz, 2xy2> be a vector field. (a) Knowing that F is conservative, find a function f such that F = Vf and f(1,2,1)= 8. (b) Using the result of part(a), evaluate the line integral of F along the following curve C from (0, 0, 0) to (3.9, 1.4, 2.6). y2 + x4z3 + 2xy(x3 + y4 + 24)1/3 = K ; K is a constant Answer: Next page
Consider the vector field F(x, ) (4x3y -6ry3,2rdy - 9x2y +5y*) along the curve C given by r(t)(tsin(rt), 2t +cos(xl)), -2ss 0 To show that F is conservative we need to check a) b) We wish to find a potential for F. Let r,y be that potential, then Use the first component of F to find an expression for ф(x, y)-Po(x,y) + g(y), where ф(x,y) in the form: Differentiate ф(x,y) with respect to y and determine g(y) e Using the...
1. Determine if Ę is conservative. If F is conservative find a potential function for Ē. If Ę is not conservative explain why you know this. (a) F = (ye-1, -2, 22) (b) F (y cos(xy), x cos(sy), – sin z)
Determine whether or not the vector field is conservative. If it is conservative, find a function f such that F = Vf. (If the vector field is not conservative, enter DNE.) F(x, y, z) = 4xyi + (2x2 + 10yz)j + 5y2k f(x, y, z) =
et F(r, v) (3z2e* + sec z tan z,ze - 90y*). (a) Show that F is a conservative. (b) Find a function f (potential function) show that F Vf. (c) Use above result to evaluate JeFdr, where C is a smooth curve that begin at the point (2, 1) and ends at (0, 3). (cost, sint) from -2 to t = 줄 particle that moves along the curve. (Write the value of work done without evaluating d) Find the work...