A smooth vector field F has div F(1,4, 4) = 7. Estimate the flux of F out of a small sphere of radius 0.005 centered at the point (1, 4, 4).

Q: Compute the curl and divergence of the vector field F(x, y, z) = (T 2², 2yz) . Is F conservative? If…

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Q: Find the divergence of the vector field F <7", 22°, zy* div F -

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

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Q: A small surface S is divided into three pieces of area 0.2. Estimate j L F · dS if F is a unit…

A: Lets consider the given surface S  is of the form with area 0.2, where F is a unit vector field…

Q: Show that the outward flux of a constant vector field across any closed surface to which the…

A: We have to Show that the outward flux of a constant vector field across any closed surface to which…

Q: Find the work done by vector field F(x, y, z) = xi + yj + (x + z)k on a particle moving along a line…

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Q: If a vector field F (x, y, z) = (evz-, In (x² z), sin z), which of the following values is equal to…

A: We have to find div F(-1,0,1).

Q: Let divF(x,y, z) = 4x³ + 4y³ + 42³ , where F represents the flow of a vector field. Estimate the…

A: Concept Used:- Divergence Theorem Flux=∫∫∫∇.F dVwhere ∇.F is the divergence of FVolume of the…

Q: Find the gradient vector field of f(r, y) = In(x + 3y)

A: Solve for the gradient vector

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: Compute the flux of the integral of the vector field F(z,y, 2) = (2,y, 2) through the half cylinder…

A: Given vector field is, Fx,y,z=x,y,z. It can be written as,  Fx,y,z=xi^+yj^+zk^. Flux of vector field…

Q: A smooth vector field F has div F(2, 5, 5) = 2. Estimate the flux of F out of a small sphere of…

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Q: Find the flux through through the boundary of the rectangle 0 < x < 6,0 < y < 3| for fluid flowing…

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Q: Find the conservative vector field for the potential function by finding its gradient. (x, у, 2) %3…

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Q: Determine if the vector field F(x,y)= (x+y,x−y) is conservative? If it is, find its potential.

A: Given Data The vector field is: F(x,y)=(x+y,x-y).   Write the vector field in standard form, The…

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

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Q: Determine the line integral if the vector field is F(x, y) = x²yî – 2xyf On a rectilinear path that…

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Q: Show that the vector field F = 2xyi + (x - 2 y)j is conservative and evaluate the line integral %3D…

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Q: Find the divergence of the vector field F→(x,y,z)=xyzi→−yzj→+xzk→at the point (8,3,2) .   Enter…

A: The given vector field is : Fx,y,z=xyzi-yzj+xzk To find the divergence at 8,3,2

Q: Find the flux of the vector field F = (x, z, - y) through the helicoid with parameterization r(u, v)…

A: Given :-  The flux of the vector field  F→ =  x3 , z , -y through the helicoid with the…

Q: Find the flux through through the boundary of the rectangle 0 < x < 6,0 < y < 5 for fluid flowing…

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Q: Find the gradient vector field (F(x, y, z)) of f(x, y, z) = In(4x + y + 62) . F(1, y, 2) =

A: we have to find the gradient vector field of f

Q: Verify the Divergence Theorem in the calculation of the flux of the vector field: through the…

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Q: Compute the flux of the vector field (x, (æ°, – ay*), out of the rectangle with vertices (0,0),…

A: To compute the flux of the vector field x3,-xy4, out of the rectanglewith vertices 0,0,2,0,2,4…

Q: Show that the vector field F=2xyi +(x -2y)j is conservative and evaluate the line integral |F•ds…

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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 the divergence of the vector field At the point (2,4,1). F(x, y, z)=x²2√√yi + -j + zk

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Q: Find the flux of the vector field F = (y, - a, z) through the helicoid with parameterization r(u, v)…

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Q: Estimate the circulation of a vector field F around a circle of radius R = 0.1, assuming that curl2…

A: It is known that ∮c2F·ds-∮c1F·ds=∬DcurlFdxdy where c1 and c2 are circles of radius 0 and 0.1,…

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

A: Given that Fx,y=-2yx2+y24+2x,2xx2+y2/4+xy+1. The the integral line of vector field F(x,y) around the…

Q: Compute the flux of the vector field F = 5xi + yj through the cylindrical surface shown below, if a…

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Q: Compute the flux of the vector field F = xi + yj through the surface S, which is the part of the…

A: Given: F→=xi→+yj→ z=36-x2+y2 ds=1+∂z∂x2+∂z∂y2dxdy       =1+4x2+4y2dxdy   and…

Q: disk

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Q: Let divF(x, y, z) = 4x³ + 4y3 +4z³, where F represents the flow of a vector field. Estimate the flux…

A: Given divF→x,y,z=4x3+4y3+4z3radius r=0.1 Centered at point 1,1,1

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

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

Q: Compute the flux of the vector field F(x,y, z) = yi– 2xyk over the triangle T with vertices (1,0,0),…

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Q: A smooth vector field F has div F(5, 2, 5) = 2. Estimate the flux of F out of a small sphere of…

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Q: Find the flux of the vector field F = across the part of the plane | z = 2 + x + 3y above the…

A: Given: F¯=y,-z,x z=2+x+3y The formula to determine the flux is: ∫S∫F·nds=∫D∫(-agx-bgy+c)dA Here…

Q: Consider the vector field F = ⟨ 1 , y ⟩ .Compute the path integral of this field along the path:…

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Q: Find the conservative vector field for the potential function by finding its gradient. f(x, y, z) =…

A: Given : f(x,y,z)=9x2y+5z2   To Find : Conservative Vector field by finding it's gradient

Q: Compute the flux of the vector field F(x, y, z) = 3xj + 3yk through the surface S, where S is the…

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Q: Show that the vector field F = 2xyi +(x² - 2y)j is conservative and evaluate the line integral (F•dš…

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Q: Let divF(x, y, z) 4x3 + 4y3 + 4z³ , where F represents the flow of a vector field. Estimate the flux…

A: Given that a small sphere of radius 0.1, centered at the point  (1, 1, 1).And                  div F…

Q: he line integral of the vector field F= over the curve C which is the quarter of the rcle from…

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Q: If the vector field A = âx3x² + âyy – â̟5z³ is given, express à in cylindrical and spherical…

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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: Compute the flux of the vector field F = 2xi + 2yj through the surface S, which is the part of the…

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Q: Compute the flux of the vector field F = zi + 8xj through the parameterized surface S oriented…

A: We have find the flux of the given vector field through the parameterized surface S oriented upward.

Q: Q3 Show that the vector field F= 2xyi +(x -2y)j is conservative and evaluate the line integral |F•ds…

A: Solution

Q: Sketch the vector field F(x, y) = 3i + 5j and calculate the flux then the circulation of F| along…

A: (1) The equation of line segment is: x+3-2+3=y-1-5-1=tx+3=t⇒x=t-3y-1=-6t⇒y=-6t+1 The curve…