Show that the vector field F = 2.xyi +(x² – 2 y)j is conservative and evaluate the line integral |F•ds along the boundary of the square with vertices (1,0), (2,0), (2,1) and (1,1) taken counter C clockwise.

Q: The potential of the conservative vector field F = (y,x,1> is Of=z+xy O f=x+yz O f=y+xz Of = xyz

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Q: Which of the following functions is a potential function for the conservative vector field F(x, y,…

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Q: Evaluate the integral curves for the vector field F(x,y) : i – 4xy²j . О a. - 4у — х2 = C 4x2y2 Ob.…

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Q: Find the work done by vector field F(x, y, z) = xi + 9xyj - (x + z)k on a particle moving along a…

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Q: Determine whether the vector field F(x,y,z)=(y,x,1) is conservative on R° . If so, determine a…

A: Given the vector field Fx,y,z=y,x,1=yi→+xj→+k→.

Q: Consider the vector field F(x, y,z)= 2.xy i+(x² +z² ) j+2yzk. c) Is F a conservative vector field?…

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Q: Suppose the vector field F = yi + (x + Z)j + (y + 2z)k is conservative. Find a potential function of…

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Q: Find the conservative vector field for the potential function h(x,y,z)=3xyln(x+y) by finding its…

A: Find gradient

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

A: It is required to find the conservative vector field for the potential function given in the…

Q: Find the potential function p, for the conservative vector field F(x, y, z) = (2xe² – y²sin x)î +…

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Q: Evaluate line integral of vector field F(z,9) = i+ 1- x? y2 2x along the curve x = cos(t), y =…

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Q: 3. Consider the vector field F = 2xy i + (x² + 2y) j. Verify the Green's Theorem in the closed…

A: Definition of green's theorem :

Q: Find the conservative vector field for the potential function by finding its gradient. f(x, y) = x²…

A: Here we have to use the concept of gradient

Q: Evaluate the line integral of the vector field (3x2 + 3y + 2xy, 3y2 + 3x + x2) along the curve y = x…

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Q: Consider the line integral cos(x+2y)dx+2cos(x+2y)dy where S is a curve starting at (0,0) and ending…

A: We have to show that F is conservative. When F is given as F=cosx+2yi+2cosx+2yj For the conservative…

Q: Show that the vector field F = x²i + (2yz)j+ y² k is conservative and then find the corresponding…

A: A vector field F is conservative if curl(F)=0 or ∇×F=0 The potential function is given by f such…

Q: Consider the vector field F = 5 sin(x² )ï + (3x² + 2e* )j. Find the line integral F· dr, where C is…

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Q: Compute the line integral of the vector field F=⟨3zy−1,4x,−y⟩ over the path c(t)=(et,et,t)for −4≤t≤4…

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

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Q: etch the vector field F(x, y) = xi + yj and calculate the line integral of F along the line segment…

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Q: Calculate the work done by the vector field F(x, y)= ,2xy) on a particle that moves once around the…

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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: Evaluate the outward flux of the vector field F = (2x² – 5e" + sin(z²), –4ry + In(x*), 5z – 3x²y®)…

A: We can see that the region given here is closed, so…

Q: Determine whether the vector field…

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Q: Show that the vector field F(x, y, z) = (le* cos y + yz)i+(xz - e* sin y)j+(xy + 4z) k is a…

A: Given: F(x,y,z)= (excosy+yz)i+(xz-exsiny)j+(xy+4z)k To show that F is conservative and if it is,…

Q: Evaluate the outward flux of the vector field F = (2x? – 5e" + sin(2²), –4ry + ln(x*), 5z – 3x²y°)…

A: Given

Q: Sketch the vector field F(x, y) = i + 3j and calculate the line integral of F along the line segment…

A: We have to sketch the vector field.

Q: Find the gradient vector field (F(r, Y, 2)) of f(x, y, z) = In(4x + 6y + 22) . F(7, y, 2) = (

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Q: For the vector field in three-dimensional space F = (4x°y – sin(x – 2))i+ e² v#j– yzk Evaluate…

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Q: Given that the vector field F(x, y) = (2x-3y, 2y-3x) is conservative in the xy-plane. If (x, y) is a…

A: We have to find the value

Q: Compute the line integral of the vector field F=⟨6y,−6x⟩ over the circle x2+y2=1 oriented clockwise…

A: Given: F=6y dx - 6x dyand Let  x= cos t,   y= sin t    ⇒   dx = -sin…

Q: Find all potential functions of the conservative vector field F (x, y, z) = (2xz + y², 2xy,x² + %3D…

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Q: If F = Pi+ Qj+Rk is a vector field on R and P, Q. and R have continuous second-order partial…

A: Given:F=Pi+Qj+Rk is a vector field  in ℝ3To show that:∇.curl F=0

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

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Q: 3. Find a potential for the conservative vector field: F(x, y, z) = (2xy + 2x, x² – 2yz³, -6z –…

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Q: Determine if the vector field F(x,y,z)= (yz,xz+2y,xy+1) is conservative. If it is, find its…

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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: 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: 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: F(x, y) = 6xyi + 3y²j is a conservative vector field.

A: Given :F(x,y)=6xyi+3y2j

Q: Show that the vector field F(x, y, 2) = (2x cos y – 3)i – (x² sin y + 2²)j – (2yz – 2)kis a…

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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: The vector field F(r, y) = (2xy +y) dx + (æ + 2xy) dy is conservative. %3D The function f(x, y) = x²…

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Q: Consider the conservative vector field given by: F (x, y) = (x - ycos (x), y - sin (x)) A potential…

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

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Q: Evaluate F Î ds for the vector field F(x, y) = yì – xj counterclockwise along the circle x² + y² = 1…

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Q: Sketch the vector field F(z, y) = i – 2j and calculate the line integral of F along the line segment…

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Q: Find the conservative vector field for the given potential function. f(x, y) = 9x2 + 2xy + 10y2 F(x,…

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