Let F =i+x^2j, and let C be the boundary of the region {(x, y):x^2 + y^2 ≤ 1 and x ≥ 0}: the right half of the unit disk. Find the outward flux of F across C.

Q: Find the flux of the vector field F = (y, z, x) across the - part of the plane z = 4+3x+2y above the…

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Q: Find the flow of F=xzi-yk through the upper part of plane z-1 in the x +y² +z = 4sphere. %3D

A: The general form of a vector-valued function in three dimensions is given by F=F1i+F2j+F3k, where…

Q: Compute the flux of F = xi+yj+zk through the curved surface of the cylinder x² + y² = 1 bounded…

A: The given vector field is F→=xi+yj+zk. The goal is to evaluate the flux or surface integral of the…

Q: Sketch the level curves f(r,y) = c of the function f(x,y) = ln(r² + y²) for c = -1,c = 0,c = 1,c =…

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Q: Calculate the flow of F(x, y, z) = (y², x, x²) across the surface z = 4-x²-y² above the z = 1 plane

A: The field is F→(x, y, z) = y2, x, x2. The surface is . Then fx, y, z=z+x2+y2-4 Obtain the normal:…

Q: 8. Find the flux of F = (2x – 5y², 3y, x) out of the sphcre of radius 2 centered at the origin.

A: The flux of the vector space is given by the formula, ∫S∇.F dS where S is the sphere of radius 2…

Q: Determine the area of the surface of z = x² + y² that lies below the plane z = 4. 2

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Q: Consider the function F(x,y,z) = x2 – yz + z2 and the point P(−3,5,−1). Find the direction (this…

A: Given consider the functionf(x,y,z)=x2-yz+z2 and thepoint p(-3,5,-1)

Q: Let S be the surface defined by the vector function R(u, v) = (v + 4, 4u - v, uv²). Set up an…

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Q: 3. Find a vector function that represents the curve of intersection of the paraboloid z =. and the…

A: To find the vector function that represents the curve of intersection of the paraboloid z=x2+y2, and…

Q: Use Divergence theorem to find the outward flux of F = (7x², et², cos³ (y² + x)) of the cube in R³…

A: Given that  and the cube  bounded by the planes The Divergence Theorem

Q: he upward flux of

A: Introduction: Let F be a function and let the region is S, then the flux of the function is given…

Q: Let F = = (x, y, z) (x² + y² + z²)3/2 Use the fact that V F = 0 when (x, y, z) following surfaces.…

A: Step 1:- To find the flux of the vector field F=(x2+y2+z2)3/2(x,y,z)​ through the given square…

Q: Let F (8xy, 3y, 8z). = The curl of F = (000). Is there a function f such that F = V f? ☐ (y/ (y/n)

A: Step 1: Step 2: Step 3: Step 4:

Q: The gradient vector for the surface xy + xz + yz = 1 at the point (1, 1,0) is (1, 1,0). Select one:…

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Q: The gradient vector for the surface xy + xz + yz = 1 at the point (1, 1,0) is (1, 1,0). Select one:…

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Q: F(1, y.:)= (x², 2x²y*, 5yz*) Given: Find (a) div F (b) curl F

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Q: Graph the solid that lies between the surface z= 2xy/( x2 + 1) and the plane z =x + 2y and is…

A: Given surfaces are z=2xyx2+1, z=x+2y  and are bounded by planes x=0 , x=2 , x=-1 , y=0 , y=4. To…

Q: Find the outward flux of the vector field F (x, y, z) = x³i + y³j+(z³ + xz)k across the surface of…

A: The vector field given is as follows: Fx, y, z=x3i^+y3j^+z3+xzk^ We are asked to find the outward…

Q: Let F(x, y, z) = (x², 2xy + e³², 3ye³z). Find div F(1,2,0) and curl F(0, 1, 1).

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Q: calculate div(F) and curl(F). F = (xy, yz, y² – x³)

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Q: Find a unit normal vector to the surface 6x2 + 2y2 + z2 = 225 at the point P : (5, 5, 5), and…

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Q: Use the Divergence Theorem to find the flux of F = (xy², x²y, y) outward through

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Q: Given: Find (a) div F (b) curl F

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Q: Find the flux of F = z²k upward through the part of the sphere æ² + y² + z² = a² in the first octant…

A: Given vector field is  F=z2k to find the flux through sphere x2+y2+z2=a2 in the first octant.

Q: Find the outward flux of F = (4x +25y*,0,) 10z through the surface 25 4 9

A: Solution:-

Q: Determine the centroid in the y

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Q: Let f(x,y,z) =x+y+z, let B be a ball of radius 1 around the origin,and let S be its surface a).…

A: GIVEN :-  Let f  x, y , z = x + y + z , let B be a ball of radius 1 around the origin , and let S be…

Q: Integrate g(x, y, z) = x + y + z over the portion of the plan 2x + 2y + z = 2 that lies in the first…

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Q: Let F = –9zi+ (xe"² – 2xe³) }+ 12 . Find ſg F ·dÃ, and let S be the portion of the plane 2x + 3z = 6…

A: The vector field is F→=-9zi→+xexz-2xex2j→+12k→ The equation of the plane is 2x+3z=6⇒ gx, y=6-2x3.…

Q: Find the flux of F = yi+2xj-3zk through the unit disk centered at (0,3,0) in the plane y=3, oriented…

A: Given F→=yi^+2xj^-3zk^ through the unit disk centered at 0,3,0 in the plane y=3 oriented towards the…

Q: Calculate F - dS where F = (e" sin y, e" cos y, yz²) S is the surface of the box bounded by the…

A: Given F¯={exsiny,excosy,yz2)

Q: Find the flux of F= (y, z-y, z). across the tetrahedron with vertices (0,0,0), (1,0,0), (0, 1,0),…

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