For the following vector field, compute (a) the circulation on and (b) the outward flux across the boundary of the given region. Assume the boundary curve is oriented counterclockwise. F= (x.y); R is the half annulus {(r,0): 1srs4, 0s0sn}

Q: For the following vector field, compute (a) the circulation on and (b) the outward flux across the…

A: Given, F=x+y,x2-yℝ=x,y:y2≤x≤18-y2 By the general theorem circulation, ∮Pdx+Qdy=∬Qx-PydAWhere,…

Q: For the following vector field, compute (a) the circulation on and (b) the outward flux across the…

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

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Q: Given the following vector field and oriented curve C, evaluate SF.T ds. C F= (-3y,3x) on the…

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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: Find the counterclockwise circulation and outward flux of the field F = xyi + 4yʻj around and over…

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Q: Given the force field F, find the work required to move an object on the given orientated curve. F =…

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Q: For the force field given by F(x, y, z) = e cos yi – e sin yj + 2k show that f. F. dr is independent…

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Q: The flow of a gas with a density of δ = 0.001 kg/m2 over the closed curve r(t) = (-sin t)i + (cos…

A: Given that

Q: For the following vector field, compute (a) the circulation on and (b) the outward flux across the…

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

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Q: For the following vector field, compute (a) the circulation on and (b) the outward flux across the…

A: We need to calculate circulation & outward flux across the boundary which can be calculated as

Q: For the following vector field, compute (a) the circulation on and (b) the outward flux across the…

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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: The flow of a gas with a density of d = 0.001 kg/m2 over the closed curve r(t) = (-sin t)i + (cos…

A: Given information:

Q: Consider the force field F = (y, –2x² y) in R². Suppose that an object moves along the parametrized…

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Q: Consider the following vector field F (x, y, z) = yz cos (xy) i+ xz cos (xy) j+ sin (xy)k (a) Use…

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Q: 1. Determine the velocity field v so that the surface F(r, y.t) = (y+t)+ (z+t)? - a² = 0 to be a…

A: The given surface is - F(x,y,t) = (y+t)2 + (x+t)2-a2.......(1) The velocity field (V→) components…

Q: Consider the semicircle r(t) = a cos(t)i + a sin(t)j with 0 0. For a given vector field F, the flux…

A: Given  rt = acosti+asintj, 0≤t≤π & a>0 for a given vector f the flux across r(t) = ∫CF.Nds…

Q: Consider the vector of electric field E(x, y, z) = (sinh x)i – (cosh y)j – xyz k a. Compute the…

A: The given vector field is, E(x,y,z) = (sinh x) i - (cosh y) j - (xyz) k   the divergence and curl…

Q: For the following vector field, compute (a) the circulation on and (b) the outward flux across the…

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Q: Find the flux of the field F =-x i - y j + z2 k outward (normal away from the z-axis) through the…

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Q: Find the flux of the vector field F(x, y, z) =(x)i+(y)j+(z)k through the surface S of the…

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Q: For the following vector field, compute (a) the circulation on and (b) the outward flux across the…

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Q: Integrate the vector field h(x,y) = y^2i + (2xy-e^2y)j over the circular arc C parametrized by r(t)…

A: The answer is given by using the formula of line integral

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: 12. Use a parameterization to find the flux across the surface cut from the cylinder x? + y? = 1 by…

A: Image is attached with detailed explanation.

Q: Given the following vector field and oriented curve C, evaluate |F•T ds. F= (x,y) on the parabola…

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Q: Given the vector field F (x, y, z) = 2xyi + (x² + z?) j +2yzk a. Verify that the vector field…

A: Ans :A)

Q: For the following vector field, compute (a) the circulation on and (b) the outward flux across the…

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

A: Flux of a vector field F→ along surface S =∫∫SF→.dS Divergence theorem  ∫∫SF→.dS→=∫∫∫E divF→dV Where…

Q: Use the Divergence Theorem to compute the net outward flux of the field F = (x,yz) across the…

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Q: Consider a force field F = yi+zj+k that acts along a path from foothill A to the top B as in the…

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Q: Use the Divergence Theorem to compute the net outward flux of the field F = (x,y,z) across the…

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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: The field F = -y³i + (x³ + e=y)j + 0k is the velocity field of a flow in space. Determine the…

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Q: Find the outward flux of the field F = xz i + yz j + k across the surface of the upper cap cut from…

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Q: Determine whether the vector field is conservative, and, if so, find a potential function. (If the…

A: If curl of vector field is zero then conservative otherwise not conservative

Q: Let D be the upper half-plane (y > 0) and consider the vector field F : D+R² given by F(x, y) = (…

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Q: Show that the vector field ? = ? + sin(z)? + ycos(z)? is conservative and then find the…

A: We have to show that the vector field F=i+sinzj+ycoszk is conservative and then find the…

Q: Find the outward flux of the field F = 2xy i + 2yz j + 2xz k across the surface of the cube cut from…

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

A: Given vector field is F(x,y,z)=xy i+yz j+zx k. To find the flux F across the surface S: Sis the part…

Q: Given the radial field (x, y, z) r (x2 + y? + z2)3/2 r|3/2 Find the outward flux across the sphere…

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Q: Consider a force field F = yi+zj+k that acts along a path from foothill A to the top B as in the…

A: Given - F=yi+xj+k  Surface of mountain is the graph of the function- z=1-x2-y2 To find - Work done…

Q: Given the force field F, find the work required to move an object on the given orientated curve. F=…

A: Let F=Pi^+Qj^ ∂Q∂x=∂P∂y=1 The force field is conservative.

Q: Find the work done by the force field F(x,y) = x²yi + y²j around the circle with radius 2.

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Q: Consider the following vector field. F(x, y, z) = (2e* sin(y), 7e sin(z), 4e² sin(x) (a) Find the…

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Q: For the following vector field, compute (a) the circulation on and (b) the outward flux across the…

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Q: For the following vector field, compute (a) the circulation on and (b) the outward flux across the…

A: Counter clockwise  circulation on the region R  for the vector field F=M,Nis given by…