Q.8-) Given the vector field; (a) Is B irrotational(or conservative)? (b) Find the net flux of...
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) =
Consider the vector field F2(x, y)-(-y,z) and the closed curve C which is the square with corners (-1,-1), (1,-1), (1,1), and (-1,1) and is traversed counter-clockwise starting at (-1,-1) (a) Compute the outward flux across the curve C by calculating a line integral. (b) Use an appropriate version of Green's Theorem to compute the above flux as a (c) Compute the circulation of the vector field around the curve by computing a line (d) Use an appropriate version of Green's...
(25 %) Q4. A vector field is given as v=e"’i+e+*+j+evk a) Determine the curl of this vector field b) Determine the divergence of this vector field c) If this vector field shows a flow field, explain if the flow is rotational or irrotational. Also, explain if the flow is compressible or incompressible. d) Compute the rate of change of Q(x, y, z) at Po in the direction of r, where P(x, y,z)=2xy + xe”; Po = (-2,1, 6) and r=-2i+j+6k
(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 +...
5. The electric field in a certain region of space is given by the vector field Vector E(Vector r)= Vector E(x,y,z)= (x-z)hatx+(z-y)haty V/m Find any two points P(x1,y1,z1) and Q(x2,y2,z2) such that the electric field at P is perpendicular to the electric field at Q. Evaluate the electric field at each of these two points. (Hint: Use the dot product.).
10. Use the Divergence Theorem to compute the net outward flux of the vector field F= <x^2, -y^2, z^2> across the boundary of the region D, where D is the region in the first octant between the planes z= 9-x-y and z= 6-x-y. The net outward flux is __. 11. Decide which integral of the Divergence Theorem to use and compute the outward flux of the vector field F= <-7yz,2,-9xy> across the surface S, where S is the boundary of...
Q4 please and thank you
(3) You are given that the vector field f in Q2 is conservative. Find the corresponding potential function and use this to check the line integral evaluated in Q2. (4) Consider the vector field F(x, y) -ryi - 2j (-Fii F2j) and let C be the closed curve consisting of three segments: the straight line from (0, 0) to (1,0) followed by the circular arc from (1,0) to (0,1) followed by the straight line from...
Find Flux of given Vector Field across given Surface. F 5x j-zk; S is the portion of the parabolic cylinder y 9x for which 0 S z S 3 and-1sxS1; direction is outward (away from the y-z plane) Select one: 190.275 O -30 O .000125 O 30 O 10 O -10 O 1.2T
Find Flux of given Vector Field across given Surface. F 5x j-zk; S is the portion of the parabolic cylinder y 9x for which 0 S z...
In Exercises 25-28, a net is dipped in a river. Determine the flow rate of water across the net if the velocity vector field for the river is given by v and the net is described by the given equations. 26. v = (x _ y, z + y 4, z~ ), net given by y = I-x2-z2, y 0, oriented in the positive y-direction
In Exercises 25-28, a net is dipped in a river. Determine the flow rate of...
(a) Use surface integral(s) to calculate the flux of the vector field or through the given surface. (b) Use the divergence theorem to calculate the flux of the vector field through the given surface. 4. F(x, y, z) =x2yi - 2yzj + x2y2k; S is the surface of the rectangular solid in the first octant bounded by the planes x= 1,y=2, and z=3. Show your work and give an exact answer.