(а) Given the force field F = (z³ cos x + 2xy?)i+ (2x²y – 2)j + (3z² sin x – 4)k (i) Prove that F is conservative (ii) By using formula VØ = F, find a scalar potential Ø for F. (iii) Hence, compute the amount of work done against the force field F in moving an object from the point (0, -1, 1) to (;7, 2, 2).

Q: 1. Determine the velocity field v so that the surface F(r, y, t) (y+t)+ (r +t)2 - a 0 to be a…

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Q: 1. Find the work done by the force field, a. F(x, y) = x²i – xyj in moving a particle along the…

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Q: 10) Find the potential function for the vector field F=-i-4j - k. z² Then compute the path integral…

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Q: 9. Compute the flux of the vector field F(x, y) = (-x sin(1+xy) – 4 cos(TY), y sin(1+ xy) + 6x)…

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Q: | 37. The flow of a gas with a density of & = 0.001 kg/m² over the closed curve r(t) = (-sin t)i +…

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Q: Without using a potential function, find the total work performed by the force field F(x, y) = = (2x…

A: Given the vector field F→x,y=2xsiny+y2ex,x2cosy+2yex. The particle moves along the curve x=sinπy2…

Q: 2. Given that F(x, y, z) = (e", xe" + e, ye²) is a conservative vector field. a. Find a potential…

A: Here we find the potential function using the partial derivatives and integral as appropriate. Then…

Q: 3. Find the work done by the force field F(1, y) = y² i+ ryj on a particle moving from (0,0) to…

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Q: (b) Given the force field F(x, y, z) = (z³ – 2xy)i – x²j + 3xz² k (i) Show that F is a conservative…

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Q: Q2: Find the work done S. F.dr in the force field F = yzî + xzĵ + xyk in moving from (1,1,1) to…

A: The equation of a line joining by the points and is given as the following.

Q: 4. Find the work done by the force field F = (x? sin(y),y) on a particle that moves along the cubic…

A: Given: The force field, F→=x2siny,y on a particle that moves along the cubic y=x3 from 1,1 to 2,8.

Q: b) Show the force field (x, y, z) = (x²y + In(yz)) i+ + y cos(2z) ) j+ y? sin(2z) 3 F k is a…

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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: Given a force field F(x,y)= (3.x² + 6.xy² )i + (6.x²y+ 4y')j (a) Prove that F(.x, y) is…

A: If a force is conservative then it is independent of paths.

Q: 16. Find the work done by the force field F(x, y, z) = (x – y²)i+ (y – 2²)j+(z – x²)k on a particle…

A: For solving this problem we need to know the parametric representation of line segment joining two…

Q: 2: Consider the vector field F = (y cos z – z sin y, x cos z – xz cos y, –xy sin z x sin y + 2e22).…

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Q: Find the work done by the field F(x,y, z) = (6z, 12x, y?) along the curve r(t) (cos(t), sin(t),…

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Q: Calculate the flux of the field F across the closed plane curve C where F=xi+ vj and the curve is…

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Q: 11. Find the flux of the field F =ri – yj across the curve C: r(t) = (cos t)i+ (2 sin t)j, 0 <t< 2n

A: Given vector field is F=xi-yj across the curve C:rt=costi+2sintj , 0≤t≤2π Flux of the vector field F…

Q: Find the work done by the force field F→(x,y)=54x^2y^2i→+(36x^3y−2)j→in moving an object from the…

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Q: 5.Find the circulation of the field F(r, y, 2) = (x – y)i+2yj around the circle r(t) = cos ti+ sin…

A: To find the circulation of the field Fx,y,z=x-yi+2yj around the circle rt=cos t i+sin t j, 0≤t≤2π

Q: The flow of a gas with a density of d = 0.3 kg/m2 987 over the closed curve r(t) = (cos t)i + (sin…

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Q: Q1 (a) Given the force field F = (z3 cos x + 2xy²)i + (2x²y – 2)j + (3z² sin x - 4)k (i) Prove that…

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Q: (iv) Find the work done by the force field F(x, y) = x²i+ yeªj on a particle that moves along the…

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Q: 2. For the vector field given by: F(x, y) = (2x + e¬* cos y) i + (y² + e=* sin y) j a. Show that the…

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Q: Consider the force field F = (2rze"*: – 4, cos' z, r²e*** – 3y cos? z sin z - Find the work done by…

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Q: Q6: Find the outward flux of the vector field F=(x²y+e'sin(y))i+ (x²+ e*cos(y))j through the right…

A: Solution

Q: 4. EvaluateLy ds where C is the upper half of circle x2 + y2 = 9, traversed in the counterclockwise…

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Q: 1.) Given the conservative vector field F(x,y,z) = ( y In z,x In z, xy find the work required to…

A: The conservative vector field Fx,y,z=y ln z, x ln z, xyz. The parametric equation of line segment…

Q: 2. Find the gradient field F = Vy for the potential function p. Sketch a few level curves of o and a…

A: Given φ(x,y)=2x2+4y2 The gradient field  F=∇·φ=∂φ∂x,∂φ∂y=4x,8y Some vectors of F are  At x=1 and y=1…

Q: Find the work done by the force field F(x, y) = x sin y1 + y J on a particle that moves along the…

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Q: Q2: Find the work done S. F.dr in the force field F = yzî + xzj + xyk in moving from || (1,1,1) to…

A: The given question is related with work done. We have to find the work done ∫CF→ . dr  in the force…

Q: Find the potential function for the flow with velocity field i- (2ry + 2ze2*)j- [2ye 2y + z sin (2)…

A: Given:  u=y-2+2xi-2xy-3+2ze-2yzj-2ye-2yz+zsinz-coszk Let u=∇f…

Q: Find the work done by the force field F(x, y) = x 2 i yex  on a particle that moves along the…

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Q: (iv) Find the work done by the force field F(x, y) = x²i+ye¹j on a particle that moves along the…

A: We have to find the work done by the force using line integral

Q: yz Find the flux of the field F up through the part of the 1+ xy, paraboloid z = 1 - x² – y² that…

A: Given field F=yzxz1+xy  The part of the paraboloid z=1-x2-y2 cut off by the xy-plane with upward.…

Q: Find the work done by the force field F on a particle moving along the given path. F(x, y) = xi +…

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Q: d the work done by the force field F in moving a particle along the curve C. F = xyi + 3j + 9xk; C:x…

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Q: 2. Find the complex potential function f(z) = u + iv, given that 2u +3v = e*[cos(y) - sin (y)]. Also…

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Q: 2. Find the circulation and flux of the field F(r.y) = (-y,r) along and across the curve 7(1) =…

A: We have to find the circulation and flux of the field F(x,y)= (-y,r) along and across the curve…

Q: 12. Find the flux of the field F(x, y, z) = 4xi + 4yj + 2k outward (away from z-axis) through the…

A: Flux of the given field is given by

Q: Find the potential function of the vector field F(x,y)=-9y²e²'¡- 18ye'j whe Provide your answer…

A: Given vector field Fx,y=-9y2exy2i-18xyexy2j. The objective is to find the potential function for the…

Q: (a) Given the force (3z2 sin x - 2)k field F = (z³cos x + 2xy?)i + (2x²y - 1)j + X (i) Prove that F…

A: Given the force field F=z3cosx+2xy2i+2x2y-1j+3z2sinx-2k We need to prove that F is conservative. We…

Q: 12. Find the flux of the field F(x, y, z) = 4xi + 4yj + 2k outward (away from z-axis) through the…

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Q: Find a potential function f for the field F. 7y j+ y² +z? 7z 21 F = (In x+ sec(10x + 10y)) i+ sec…

A: Given that F = (ln(x) + sec^2 (10x + 10y)) * i + (sec^2 (10x + 10y) + (7y)/(y ^ 2 + z ^ 2)) * j +…

Q: 1. Find the work done by the gradient of f (r. y) = (r+ y)? counterclockwise around the circle 2+ y?…

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Q: Calculate the flux of the field F across the closed plane curve C where F= xi+ vj and the curve is…

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Q: Find the flow of the velocity field F = b. a. Na g (2,8) : 4x (0,0) X (5y² + 1)i + (10xy)j along…

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Q: Consider the gravitational field: xî + yỹ + zk F = -GmM · (x² + y2 + z² )ž where G, m, M are…

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Q: Q4: A- Find the flux of the field F = i+j+k √√x² + y² +2² bounded below by the plane z-0 and bounded…

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Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.