Let F be the vector field (x², 2r, 2²). Let C be the boundary of Lv **---- the surface S given by 4x? + y? + 2² = 5, z > 1. Give C the counter-clockwise orientation when viewed from (above) the positive z-axis. Evaluate the integral // (7 × F) - dš by using a different surface bounded by C.

Q: Let CC R² be the circle with radius 8 centered at the origin. Let F: R2 → R² be the vector field…

A: Use the relation, T^=drdtdrdt     Where rt=R cost i^+R sint j^ And N=T^×k^

Q: 4. Find the positively oriented flux over the surface created by the paraboloid z = 16 – x² – y2…

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Q: Compute the curl of the following vector field. F = = (3z2 sin y, 3xz2 cos y, 6xz sin y)

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Q: Compute the flux of the vector field F 3.ri + 3yj through the cylindrice surface shown below, if a=3…

A: To calculate the flux of the vector field.

Q: Use the Divergence Theorem to compute the flux across the sphere x2+y2+z2 = 1 within the vector…

A: The answer is as follows:

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Q: Let F be the vector field F = (14x – ", - ", – }=*) Compute the normal at the point (1, 1, 1) to the…

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Q: let F be the vector field Cx, 2x,3). let c be the of the Surface S gien by 4xt y1855 it. 871.Given c…

A: We will use reverse of Stokes theorem for vector  field F to convert given integral into line…

Q: Use the divergence theorem to calculate the flux of the vector field F = x3i - 3x2yj + 6zk through…

A: The divergence theorem is defined by the formula, ϕ=∮∂VF·ndS=∮V∇·FdV Consider the given vector…

Q: Is there a vector field G on R³ such that curl G = (x sin y, cos y, z - xy)? Explain

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Q: Is there a vector field Ğ on R³ such that curl Ġ = (xyz,–y*z³,y*z*)?

A: the given function is, G→=xyzi^-y4z3j^+y3z4k^ If there exists a vector field G then the divergence…

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Q: What is the flux of the vector field (x2,y2,z1 ) through a cuboid with x ranging from 0 to 8.3, y…

A: use divergence theorem

Q: Find a potential functioon for the conservative vector field given by y3/2 3 xy + z, 2 sin z 2.x…

A: According to question,Given vector field,F→x, y, z = 2xx2 + cos z-y322x, -32xy+z, -sin zx2 + cos z+y

Q: Show that F of F. = (2xy + z³)i+x²+3xz²k is conservative force field. Find the scalar potential

A: *By Bartleby policy I have to solve only first one as these are all unrelated problems.These are all…

Q: Calculate the integral line of vector field F(x,y) around the circle x²+y²=1 oriented…

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Q: Let C be any smooth simple closed curve with positive orientation that lies entirely in a plane with…

A: Step:-1 Stock's Theorem:- ∫CF.dr=∬Scurl(F). ds=∬Scurl(F). n dA Part (a):- Given that F→=9y, 3x, 6z…

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Q: Why is the upward flux of a vertical vector field F = ⟨0, 0, 1⟩ across a surface equal to the area…

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Q: Find the curl of the vector field F. F(x, Y, 2) = arcsin yi+ V1 - xzj + yêu y²k

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Q: Suppose a particle travels clockwise on a triangular path connecting the points (0,0), (3,1), and…

A: greens theorem

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Q: Verify Stoke's theorem for the vector field F = (2x - y) I- yz*J-yzK over the upper half surface of…

A: Stokes theorem is used

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Q: The vector field F is shown in the xy-plane and looks the same in all other horizontal planes. (In…

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Q: Find the curl of the vectòr field F(x, y, z) = 2xyzi + 8yzj + 5zk

A: According to our guidelines we are supposed to do only one question.  Kindly repost other question…

Q: If F = xy i+ (x² + 4yz)j+ 2y²z k. Evaluate the line integral between A (0, 2, 0) and B (3, 6, 1).…

A: Given, F= xy i +(x²+4yz)j +2y²zk r0= (0,2,0) r1= (3,6,1)

Q: Use the Divergence Theorem to find the flux of the vector field F = (3x2 – 3yz + 423) i + (4.xz –…

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Q: The outward flux of the vector field F=(x,0, z) across the cylinder x + y = 1, for 0<z<1, is %3D…

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Q: Verify Stoke's theorem for the vector field F = (2x - y) I– yz"J - y z K over the upper half surface…

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Q: Find a vector potential field for Ġ = [6xz+x³, –3r²y – y², 4.x + 2zy – 32²], i.e. find F such that V…

A: The solution is below.

Q: 4- Evaluate the surface integral of the vector field F = = [ax, by, cz] over the sphere a2 + y² + z²…

A: From the given problem:   F→=axi+byj+czk   Sphere:  x2+y2+z2=36.   From the divergence theorem:…

Q: 4. Verify that the two integrals in the circulation form of Green's Theorem are equal along a circle…

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Q: Calculate F=, around the triangle with vertices: (0,0,2), (2,0,0) and (0,2,0). Can you use Stoke's…

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Q: the vector field F is shown in the xy plane. y 21 2 Is div F positive, negative, or zero? Positive…

A: Given : The vector field F is shown in the xy plane :   We have to determine that : Is div F…

Q: 4- Evaluate the surface integral of the vector field F = [ax, by, cz] over the sphere r² + y² + 2² =…

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Q: Consider a conservative vector field: F(x, y) = (y cosx + y2, sinx + 2xy – 2y) Find a potential…

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Q: : Verify the divergence theorem for A=4xi +(2Y+z Vector field ) | +(52-2x)k in the regim beunded by…

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Q: Let C be the helix that winds around the cylinder x2+y? = 1 (counterclockwise viewed from the…

A: here i used parametric equation of helix and done simple calculation like dot product and after that…

Q: Give an example of a vector field F (x, y, z) that has value 0 at only one point and such that curl…

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Q: Compute Sc F.d3 for the vedor field in the Cylinder with out ward facing and the surface which…

A: It is given that F→=xi+yj+ezk and the surface is x2+y2=4,1≤z≤5