9. Given vector field F(7)= ||7||² 7 - 57, where T = x i +y5 + z k. Find th flux of the vector field through the closed surface S oreiented outward, that consists of the uppe hemisphere S1 : x2 + y² + z² = 1, z > 0, that is located above the ry-plane, and the unit disk S2 a2 + y? < 1, z = 0 that is located in the ry-plane. 3 (a) Find the flux by evaluating the flux of F through S1 and S2 and using fs FdA = [s, FdA - %3D Ss, FdĂ. (b) Find the flux by applying the Divergence Theorem.

Q: Find the flux of the vector field V(x, y, z) = 3xy²i + 5x²yj + 2z³k out of the unit sphere. Flux =

A:

Q: 4. Consider the vector field given in Cartesian coordinates by A(r) = (x² + y³, 3xy², z²). (a)…

A: Given : Ar=x2+y2,3xy2,z2

Q: 4. Evaluate the flux of vector field F = zi + 2yj - 2xk across part of the paraboloid z = 4-x² - y²…

A: A vector field F=zi^+2yj^-2xk^ across part of the paraboloid z=4-x2-y2 that lies above the circle…

Q: 1) Find the work done by the vector field F(x, y, z) = (x², 4xy³, xy²) on a particle that %3D…

A: Given force is F→x,y,z=x2,4xy3,xy2 Let r→=rt→ be the boundary of the rectangle with vertices  0,0,0,…

Q: Find the curl for the vector field at the given point. F(x, y, z)= 6xyzi+6yj+6zk (6, 7, 6) O…

A: To find the curl for the vector field  at the given point.Fx, y, z=6xyzi+6yj+6zkPoint=6, 7, 6

Q: 2. Use Green's Theorem to find the flux of F(x, y) = (-y², xy) across the square with vertices…

A: Green's Theorem:The circulation of vector field F=&lt;M,N&gt;in a closed path and the flux of the…

Q: Consider the function f(r, y, z) = r'y + yz+ (y – x)z² and the vector field F = xyi + (-x + z*)t yzk…

A:

Q: Compute the curl of the following vector field. 4r (4x, 4y, 4z) 7/2 (4x +4y +4z) " 2r The curl of F…

A:

Q: (b) Given a potential functionf = Ax-2y³z² + Bx³z² – Cxy, where the coefficients A, B,C, are the…

A: Here A = 0, B = 2 and C = 8

Q: Determine the vector field of F. F(x, y) 3D yi — хј

A: Let us make a table to draw the vector field as shown in the image below.

Q: Let C₁ be the directed line segment that starts at (2,0) and ends at (-3,-5) and C₂ be the parabola…

A:

Q: Find the Curl of the vector field. F(x,y, z) = 7e" sin(yi+ 8e* cos(y j+ 14z k a. &e" cos(y) +7e"…

A: solution of the given problem is below...

Q: point -2 has parameterization F(t) = (4(1-t), 3t) for 0 s 2

A: This is a problem of vector calculus.

Q: +20x-22y-14-0 become when referred to the rectangul axes through the point (-2, -3), the new axes…

A:

Q: 3) Let S be a portion of the surface 3z = 12 – 6x – 3y in the first octant. Suppose that S is…

A: 3z=12-6x-3y divide by 3 z=4-2x-y ⇒2x+y+z=4 clearly it's plane The plane cuts the x-axis at (2,0,0),…

Q: Given the 2-D vector field : G(x, y) = (2y)î + (x)j and the triangular curve T : line segments from…

A: Given: G→x,y=2yi^+xj^ Triangular curve T, line segments from point A0,0 to point B4,0 to point D4,2…

Q: Compute the flux of the vector field F =2xi +3yj through the cylindrical surface shown below, if a=3…

A: Given that the vector field is F(x,y)=2xi+3yj Let us parametrize the cylinder as x=cosθ, y=sinθ, z=r…

Q: Consider the vector field F(x,y,z) = (3x²y)i + (x³ + 1) + (92²) and the curve C made up of the…

A:

Q: d) An open surface F is given by F = {(x,y, z) |z = 4 – x² – y² , z > 0} . Compute the flux of V…

A: V(x,y,z) = &lt;2x+3y,2y+3x,-4z&gt;F={ (x,y,z) : z=4-x2-y2 , z≥0}Consider, g(x,y,z) = z+x2+y2-4…

Q: Suppose the vector field v(x, y, z) = (4xy2, -4xz², 5x) represents the velocity field of a fluid.…

A:

Q: Q2) Find the flux of the vector field F(x,y,z) = (x,y, z) , where the surface S is the portion of…

A:

Q: Consider the vector field F(x, Y, z) = 3x?yi+ (x³ + 1) j+9z²k. Let C be the circle cut from the…

A: Given vector field is  F(x,y,z)=3x2yi+(x3+1)j+9z2k C be circle cut from sphere x2+y2+z2=8 by the…

Q: 2. Given vector field F= (-2y, z-2x, y + z), find the work done by ♬ on a particle that moves along…

A:

Q: Let C₁ be the line segment from the point (-4, 8) to the point (2, -4), C₂ be the arc on the…

A:

Q: 3) Find the curl of the vector field F(r, y, z) = (ry") i + (ya)j+ (8ry) k %3D

A: We know,If, F(x,y,z)=Fx i +Fy j+Fz kCurl of  F(x,y,z)=∇× F(x,y,z)=ijk∂∂x∂∂y∂∂zFxFyFz

Q: Consider the vector field F = (2, -3). Let C be the straight line segment oriented from point (3,0)…

A: Given vector field, So,

Q: 1. For each of the following vector fields F(1, y), decide if there is function f(z, y) such that F…

A:

Q: Let F be the vector field (-2 sin(2x - y), sin(2x - y)). Find two non-closed curves C₁ and C₂ such…

A:

Q: What is the curl of the vector field F(z, y, z) = ryi+ zyzj + yzk O a. (-ry + z)i + (yz – 2)k O b.…

A: We have to find

Q: Compute the flux of the vector field F 3yi + 4j – 3xzk through the surface S, which is the surface y…

A: We apply formula for finding this

Q: Find the potential function for the vector field given as 2 F(x.y.?) = Oa. $(x, y, z) =x*y + x²2?…

A: Answer:   The correct option is B.

Q: h) Given vector field F (x, y, z) = (2x + 2y + 4z) i + (2y + 2x – 32) j + (2z – 3y + 4x) k. (a)…

A:

Q: Ex: Find the work done by the vector field F = (y-x²)i + (z - y²)j + (x-z²2)k Over the curve r(t) =…

A: Given: F=y-x2i+z-y2j+x-z2krt=ti+t2j+t3k

Q: A conservative vector field is given as F = 6xe'i + (3x²e" –10yz')j-15y²z°k. Find the potential…

A:

Q: 2x 2y Consider the vector field F = |x² + y2’ x² + y² (a) If C1 is the unit circle x² + y² = 1,…

A: Compute the circulation and flux using parameterization:  ∮C1Fdr=∮C12xx2+y2,2yx2+y2·dx,dy…