The vector field + (x,y) = (ycos (xy) + 2e*, xcos(xy) + 2ycos (y2)) is conservative....
2. a) Find a potential of the vector field f(x, y) = (a2 +2xy - y2, a2 - 2ry - y2) b) Show that the vector field (e" (sin ry + ycos xy) +2x - 2z, xe" cos ry2y, 1 - 2x) is conservative.
+ 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...
Is the below statement True or False? The vector field F(x,y) =<xy?, x?y) is conservative. True False
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)-
3. Verify Stokes' Theorem for the vector field F(x, y, z)= (x2)ĩ+(y2)]+(-xy)k where S is the surface of the cone +y parametrized by (u,v)-(ucos v, u sin v, hu) x2+y2 a at height h above the xy-plane Z = a V 0<vsa, OSvs 2n, and as is the curve parametrized by ē(f) =(acost,asint, h), 0sis27 as x2+ a
3. Verify Stokes' Theorem for the vector field F(x, y, z)= (x2)ĩ+(y2)]+(-xy)k where S is the surface of the cone +y parametrized...
Consider the vector field. F(x, y, z) = (98 sin(y), 4e' sin(z), 2e sin(x)) (a) Find the curl of the vector field. curl F = (b) Find the divergence of the vector field. div F =
1. (20 points) Identify if the following vector fields are conservative. If there exists a vector field that is conservative, you must also find a potential function for that field. (a) F(x,y,z) = (x3 – xy +z)i + 2 (b) F(x,y,z) = (y+z)i + (x+z)j + (x+y)k (& +y +y-22) i + (- y2)k
(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 +...
Give 2 different proofs that F is conservative. F = (ycos(x) + y, sin(x) + x) Provide at least one theorem or equation or formula number from the sheet I provided you in your solution. Make it clear where the theorem/equation is being used.
3. Consider the vector field F(x,y) = (27x D = {(1,y): 0 < rº + y2 <2}. +ya) defined on the region D where a) Directly compute SF. Tds using the definition of the line integral, where C is the unit circle oriented counterclockwise. b). Use Theorem 3.3 (Test for Conservative Vector Fields) from the text to determine if F is conservative. Is your answer consistent with part a)? If not, what is the source of the discrepancy?