5. Let a, b > 0. Verify Stokes' Theorem f F· dĩ = [f, curl(F) - n dS for 2xz, i+xyj the vector field F(x, y, z) = a? 12 k, where the surface o is y?, the portion of the elliptical paraboloid given by z =1- and the boundary curve C is the corresponding ellipse cut out by the ry-plane. a2

Q: In Exercises use the surface integral in Stokes' Theorem to calculate the circulation of the field F…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Consider the solid E determined by the inequalities x² + y² + z≤ 4, and let S denote the surface…

A: Here we will use Stokes theorem  to calculate  flux of curl (F) on surface S bounded by curve C…

Q: Evaluate the surface integral ∬SF⋅dS for the given vector field F and the oriented surface S. In…

A: Compare the given equation with general equation and find the value of P, Q and R.

Q: (a) Given the vector field F = xy + 2yzý + 3zx 2, calculate the line integral around the closed path…

A:

Q: Suppose we have part of a hyperboloid of two sheets -4x? -4y? + z? = 9 where z 20 (so the upper…

A: Equation of hyperboloid of two sheets is: −4x2−4y2+z2=9 Where z≥0 and z≤15

Q: The following figure shows a half-cylindrical closed surface, defined by the cylinder x² + y² = 4…

A: As per the question we have to find the total flux of the vector field V through the flat top…

Q: Let C be the ellipse in which the plane 2x + 3y - z = 0 meets the cylinder x2 + y2 = 12. Show,…

A:

Q: Consider the surface S given by the graph of z = x sin y on the domain D = {(x,y) |0 < x < 3,0 < y…

A: There are two parameters for surface parametrization

Q: 4. For the following vector fields and surfaces, calculate the flux integral E F. dS. (a) F(r, y, z)…

A:

Q: Consider the surface integral of the vector field F [xy2, 3x2, x2z]over a surface S, where S…

A: From Gauss divergence theorem, we have ∬SF→⋅dS=∭VdivF→dV   The vector field is F→=xy2,3x2,x2z.…

Q: 11. Suppose that E is the solid upper hemisphere of radius a > 0, whose base is a circular re- gion…

A:

Q: 3. Consider the vector field F = (y, -x, x² + y?). Find the flux of F through S if: 9– x2 – y? above…

A:

Q: 1. Use the divergence theorem to evaluate the surface integral F · dS, S where the vector field is F…

A:

Q: 3. Verify Stokes' Theorem for the vector field F= 2zi +3xj+5yk, taking S to be a portion of the…

A: Given F=2zi+3xj+5yk

Q: Consider the surface integral of the vector field F = [xy?, 3x², x²z] over a surface S, where S…

A: We will use of gauss divergence theorem

Q: Consider the two-dimensional vector field sin(Arctan(y/x)) F(x,y) = cos(Arctan(y/x)), defined on…

A:

Q: Let f(x, y, z) = (x² + y² + z²)¬/2. Show that the clockwise circulation of the field F = Vf around…

A: Calculate: F=∇f   =∂∂xx2+y2+z2-1/2,∂∂yx2+y2+z2-1/2,∂∂zx2+y2+z2-1/2   =-x2+y2+z2-3/2x,y,z

Q: 12. Evaluate the surface integral JJ, F ·dS for the vector field F(x,y,z)=(-x,-y,z'), where S is the…

A: This is the problem of multivariable calculus, surface iuntegral.

Q: Let a, b > 0. Verify Stokes' Theorem f F - dĩ = SS, curl(F) - ndS for 2xz the vector field F(x, Y,…

A: Let a, b&gt;0. Verify Stokes' Theorem ∮F→· dr→ =∫∫σ curl F→. n dS for the vector field F→x, y,…

Q: Use the Stokes' Theorem to caleulate the circulation of the field F = -6y*7 + 6x*J° + 7z*R° around…

A: Given field, F→=-6y3i^+6x3j^+7z3k^ where C is the curve of the paraboloid x2+y2=z cut by the…

Q: 4. Verify that Stokes' Theorem is true for the given vector field F and surface S. F(x, y, z) = xyi…

A:

Q: ) Verify Stokes' Theorem for the hemisphere S : x²+y²+z² = 9, z > 0, %3D its bounding circle C: 1?…

A: Stokes theorem  If a smooth surface S bounded by closed ,smooth boundary curve C oriented in…

Q: Consider the force fields F, (x, y) = (y2 + 2xe",x(3y + xe + 1)) and F2 (x, y) = (M,N). Let C %3D be…

A: Let R be a region in the xy-plane that is bounded by a closed, piecewise smooth curve C, and let  F…

Q: Question 11 Let S be the surface consisting of the part of the solid cylinder r² + y? = 9 between…

A:

Q: Find the flux of the field F(x, y, z) = 4x i + 4y j + 2 k outward (away from the z-axis) through the…

A:

Q: Q1 Use Stokes theorem to compute the flux of the curl of a vector field F = (2x – y)ĩ – yz? j- y²zk…

A:

Q: Consider the vector field F = a?i+ ?j+ y²k Calculate the divergence: Calculate the curl, Suppose S…

A:

Q: 3. Verify Stokes' Theorem for the vector field F = 2zi +3xj+5yk , taking S to be portion of the…

A: We will use the basic knowledge of vector calculus to solve this question.

Q: Let S be the part of the plane 4x + 3y + z = 4 which lies in the first octant, oriented upward. Use…

A:

Q: Consider an ellipsoid E given by the equation x2, y?, 22 = 1, c2 a > 0, b > 0, c > 0. a? 62 1. Show…

A: NOTE:- By Bartleby rules, only question (1) is answered Let E be an ellipsoid given by the equation…

Q: . Verify that Stokes' theorem is true for the vector field F(r, y, z) = (x, y, 2), where S is the…

A:

Q: (3) Evaluate the surface integral F . ds for the given vector field F and the given oriented surface…

A:

Q: (a) Compute the flux of the field F = r i+yj-zk across the cylindrical surface a + z = 1, 0 <y53…

A: If E be a  solid region bounded by surface of S with positive orientation then for a vector field…

Q: Let E be the solid region bounded by the upper half-sphere x2 + y² + z² = 4 and the plane z = 0. Use…

A:

Q: Let ƒ(x, y, z) = (x2 + y2 + z2)^(-1/2). Show that the clockwise circulation of the field F = ∇ƒ…

A:

Q: Evaluate the surface integral ff, F · dS for the given vector field F and the oriented surface S. In…

A:

Q: i) Evaluate the surface integral where F = (x,y,zx) and S has upward orientation. S is the part of…

A: The solution is below.

Q: 12. Evaluate the surface integral F-dS for the vector field F(x, y,z) =(-x,-y.z), where S is the…

A: The given problem is to evaluate the given surface integral in the part of the cone  between the…

Q: - L ---- Let F be the vector field (r², 2r, z²). Let C be the boundary of the surface S given by 4x²…

A: Given- Let F→ be the vector field x2, 2x, z2. Let C be the boundary of the surface S given by 4x2 +…

Q: 2. Find the surface integral SS. ř · dS for the vector field F = xyî + yzĵ + zxk where S is part of…

A: To find Surface integral ∬SF→·dS→: The Surface S is of the form z=g(x,y) and is oriented upwards.…

Q: 10. Verify Stokes' Theorem, fF.dr=ffcurl(F) ds, for the vector field F= (z, 2x, 2y) and the surface…

A:

Q: Let o be the surface 3r+ 10y+9z = 4 in the first octant, oriented upwards. Let C be the oriented…

A: “Since you have posted a question with multiple sub-parts, we will solve first three sub-parts for…

Q: find the surface integral of the field F over the portion of the given surface in the specified…

A:

Q: 6) Given a hemisphere bounded by S,: z = /25 – x2 – y? and S2:z = 0.Find the outward flux of the…

A: The flux of a vector-valued function F(x,y,z)=F1i+F2j+F3k over a surface S is defined as ∫SFdS.…

Q: Evaluate the surface integral JJS ² F. ds for the given vector field F and the oriented surface S.…

A: Consider the vector function F = x i+y j+8 k Here surface S is the boundary of the region enclosed…

Q: 4. A) Find the work done by the force F (x, y) = yx²i – y²j over the curve r (t) = t’i+t*j, 0<t<1.…

A:

Q: 1 Find the line integral | Fdrof the vector field F :(xi + yj+ zk) where C is the boundary of the…

A: see below the answer

Q: 12. Find the flux of the field F(x, y, z) = 4xi + 4yj + 2k outward (away from z-axis) through the…

A:

Q: Consider the surface S given by y=8−x2−z2, y≥0 oriented in the positive direction of the y axis. be…

A: We will find out the required value.

Q: Let S be the part of the plane 2x + 2y + z = 4 which lies in the first octant, oriented upward. Use…

A: Given  plane  2x+2y+z=4