a) Find the values of a,b,c which make f(x,y,z) = sin (xªy®z©) a potential function for the vector field F(x,y.z)= 4 x°y?z? cos (x*y"z7) i+ 7x*y®z7 cos (x*y'z?) j + 7 x*y?z® cos (x*y?z?) k a= b= c= b) Find the work of F along the curve r(t)= (2t, e t(t - 1), e t(t – 1)) for 0sts1. Do not approximate your answer. For example, leave sin (e) as that, not 0.41. Work = F• dr

Q: Calculate the flux of the vector field F(x, y, z) = 5i + 2j+ zk through the closed circular cylinder…

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Q: 1- Prove that the force field F known as : Prove that the force field defined by F = (y*z' -6xz? )i…

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Q: b) Show the force field M UTM UTM F(r, y, z) = (r²y + In(yz)) i+ UTM 3 + y cos(2z) ) j+ tial…

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Q: F(1, y, 2) = (2c" cos(3y), –3e²" sin(3y) + VV:). a. Find divF(x, Y, 2) and curlF(x, y, z). b. Based…

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Q: 6. Use Green's Theorem to find the work done by the force field F = ( 2xy, x² + x ) on a particle…

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Q: 1. Consider the vector field F(x, y, z) = (2e2" cos(3y), –3e²" sin(3y) + Y a. Find divF(x, y, z) and…

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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: 3 , find a potential function for the vector field F by inspection or show that one does not exist.…

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Q: Find the outward flux of the field F = 8xyi + 6yzj + 6xzk across the surface of the cube cut from…

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Q: Find the circulation and flux for the field F(x, y) = (y – 6x² , x+y²) around and across the…

A: Given data field Fx,y=y-6x2,x+y2 around and across the triangle given by y=0, y=x and x=1.

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: (a) Is the vector field F(x, y) = (xe³, ye²) conservative? Justify your answer. If it is…

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Q: b) Show the force field F(x, y, z) = (x²y + In(yz)) i+ 3 y cos(2:) ) j+( - y° sin( sin(2z) ) k %3D…

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Q: VII. - ) Let C be the curve consisting of the line segment from (0,0) to (2,0), the the portion of…

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Q: Find the work W done by the vector field ⟨2x+yx,x2+4⟩ on a particle moving along the boundary of the…

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Q: Consider the conservative vector field 8. 4 F(x, y, 2) = (- 2xa, - 3y*cos (32)3ny³sin (3nz) – x²). -…

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Q: Find a potential function for the vector field F(x, y) = (6x sin y, 3 cos x ), if it exists. A. No…

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Q: Consider the vector field F (x, y, z) = (2e2" cos(3y), –3e2" sin(3y) +VV?) Y y' 2 COS a. Find…

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Q: a) Show that the vector field F = xî + 2yj + zk is imotational, find the potential function b) Find…

A: (a) If the vector field is irrotational then  ∇×F=ijk∂∂x∂∂y∂∂zFxFyFz=0 Therefore,…

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Q: Find a potential function for the vector field F(x, y) = (6x sin y,3 cos x), if it exists %3D A. No…

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Q: (a) Show that the vector field F(x, y, z) = (1 + 3x²y)i + x³j is conservative and find the potential…

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Q: Find the outward flux of the field F = 3xz2 i + y j - z3 k across the surface of the solid in the…

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Q: Use Stoke's Theorem to find the work done by the force field F(x, y, 2) = (ein + 2y?, 6x + tan- y,…

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

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Q: Find the potential function for the following vector field F(x, y, z) = (2x cos y – 2z3) î + (29+…

A: F = <2xcosy-2z3, 29+2ye2-x2siny, y2e2-6xz> To check F¯ is conservative  not we have t check…

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Q: 6) Given the vector field F = (-x,-y,-z) '(x²+y2+z²)3/2! write the integral to calculate the outward…

A: First we parameterize field F then we calculate the Flux across the sphere using following formula:

Q: 3x cmine whether F(x, y) = e³ª cos 3y i – e³ª sin 3yj is ervative vector field. If so, find a…

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Q: If 0(x,y) is a potential function for a constant vector field F = (a, b), then the graph of the…

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Q: Find the counterclockwise circulation and outward flux of the field F = 7xyi + -5y²j C enclosed by…

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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: - Here is a graph of the vector field F(r, y) = r²i+ y°j. 2 (a) Calculate div F. (b) For which…

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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: 2: Compute the work done by the force field F = (y, -x) to move an object from point A = (1,0) to B…

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Q: Determine whether the vector field…

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Q: a) What is the flux of the constant vector field F = (a, b, c) through any closed surface? b) Is the…

A: Given That : A constant  vector field F=a,b,c and F=x3,y3,z3 To Find : i) The Flux through of…