40. Find the circulation of the field F = yi + (x + 2y)j around each of the following closed paths. a. (-1, 1) (1, 1) (-1, –1) (1, – 1) b. x² + y? = 4 c. Use any closed path different from parts (a) and (b).

Q: 1/2 5) Consider vector field F(x, y,z) = ( 2/yz – 2xz', - 2x²z) for 1/2 x>0, y > 0 and z>0 a)…

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Q: 4. If div F >0 in a region enclosed by a small sphere, the outward flux of the field across the…

A: Given: Divergence F>0 is enclosed by a small sphere in a closed surface . The objective is to…

Q: 8. Compute the flux of the vector field. F(x,y,z) = (2x -z,-2y, 1-2x) through S= {(x,y,z) € R³: z =…

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Q: 3 e flow in the field F = 4i + 9j + 2k along the path C, where C is the curve r(t) = 7 cos 4ti + 7…

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Q: Find the wark done by the Foree Field FoK x>ye^> along the parabola X=y+1 from (bo) to (2,1).

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Q: 2. Let f(x, y, z) = In(x² + 2) + y²e#* - cos(2yz). Find fzyz-

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Q: Let z = cosh t. Find the differential dz when t = In 3 and dt = 0.2 O 4/3 O 4/15 O 8/15 O 1/3

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Q: Prove 8xyz < (x + y)(y + z)(x + z) for every x, y, z E [0, +∞[.

A: We have to prove given identity: Concept:

Q: 8. The component of the field F= (1,2,3) normal to the plane z = -3x + 3y -1 is (a) 0. (b) –2//19.…

A: If we are given a plane equation in the form Ax+By+Cz=D, then the normal vector to the plane will…

Q: The work done in carrying a -1C charge from point A(0,0,0) to the point B(6,0,0) in the field E =…

A: Given,     E=2x.ax^+ 2yay^  VmQ=-1CPoints are A(0, 0, 0)  and  B(6, 0, 0)

Q: Find the work done by the three-dimensional inverse-square field 1 F(r) = r on a particle that moves…

A: r(t)=&lt;6,2,3&gt;+t&lt;2-6, 6-2, 9-3&gt; r(t)=&lt;6-4t, 2+4t, 3+6t&gt; r'(t)=&lt;-4, 4, 6&gt;…

Q: For Problems 2–3, you are given a vector field F and a curve C. Use the Fundamental Theorem of Line…

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Q: 10. Evaluate the divergence of the following vector field at the point (0,1,2): F = (xyz, x? + y? +…

A: Given: F=xyz,x2+y2+z2, xy+yz+xz       The point 0,1,2 To find the divergence

Q: Sow that 2 x² + y² + z² - c²t² is Lorentz invariant.

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Q: 14 4. Consider the vector field F and the curve c below. 5 5 F(x, y) = x²y³ ₁ + x³y² Ĵ. C:r(t)…

A: The given problem is to evaluate the given path integral of the vector field over given curve.

Q: 2 find the divcrgence and curl of the vectur field f gven by F p) f(xyz) =cas (2x)it X sin ryit z°k…

A: the given vector field F is: F(x,y,z)=cos2xi+2sinπyj+z2k we have to find the divergence and the curl…

Q: 9. Show that F is a conservative vector field and use this fact to evaluate f F dr along the given…

A: Given information: Fx,y=y2i+x2j C is the arc of the parabola y=2x2 from (-1,2) to (2,8). Gradient…

Q: Let f(x, y, z) = 3xyz + y?z. Find the gradient field Vf.

A: Formulae of gradient field is given below

Q: Determine and state the work done by the vector field over the given path in situation. F (x,y) = (x…

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Q: 3. Find a field F(x,y) = (M(x, y), N(x, y)) in the xy-plane with the property that at each point

A: Let f = xi + yj  Then F = ff = unit vector = xi+yjx2+y2

Q: In R°, the Klein-Gordon equation governs the quantum mechanical wavefunction (r) of a relativistic…

A: Suppose, for a given set of boundary conditions  ψ=for we have ∂ψ∂n=g on S we have two solutions to…

Q: 1. Let F(x, y, z) = (2re²*+3», 3e* +3v +2/E, ). 2.x cez²+3y a. Show that F is conservative using its…

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Q: Find the circulation of the field F = yi + (x + 2y)j around each of the following closed paths. a.…

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Q: Express aw/ar and ôw/as as functions of r ands using chain rule w = = In(x² + y² + z²), x = res sinr…

A: Given problem is :

Q: 1. Use isoclines to sketch the direction field for y' = x+ 2y and sketch the solution curve that…

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Q: [1] Show that the functions In x, In(x) are linearly dependent on (0, co)..

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Q: Find the work done by the three-dimensional inverse-square field F(r) : 1 r on a particle that moves…

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Q: Find the work done by the three-dimensional inverse-square field 1 r on a particle that moves along…

A: Solution

Q: Is the following field path-independent/conservative? Cos(xy)i+ cos(ry) i F(x, y) 5xy 2 - No Yes

A: The given field is of the form Fx,y=Pi+Qj. Here P=5xy-cosxyx,Q=5x22-cosxyx.

Q: 1. Show that the field F = (x + cos a)i+ (K +4 – y sin y)j + (2³ + K – 9)k is conservative and find…

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Q: compute CF d? for the veder field. KI+x3 xy> where Cis the line segment from CO.0) to Clo3)

A: The objective here is to calculate the integral: ∫CF→⋅dr→; Where F→=1+x2,xy2 And C is the line…

Q: Find the divergence of the field. F = (- 6x + 8y – 4z)i + (- 8x – 5y + 3z)j+ (x + 5y + 2z)k div F =

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Q: 1).Evaluate ∮CF•dr, whereF=⟨e^x−y^3,cosy+x^3⟩, and C is a circle of radius 2, centered at the…

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Q: If. E-Pi+Q+Rk is.a.veeter field.and. Po.Q.and R. have.centineons. 2nd order.partial.derivatives.then…

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Q: Which of the following graphs is the slope field for 15/1 1~/1ST ~\~ in 13t-ii-i ~~~ -1 1 2 3 4 5…

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Q: 1 Justify the Cauchy-Riemann equations for f(2) = e² where f(z) is holomorphic.

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Q: д. L. = a(x, y)+b(x, y) +c(x, y). – d(x, y) -

A: Proof: Consider V be any vector space of all functions of two variable x  and y whose partial…

Q: 3.27 Evaluate the divergence of the following vector fields (a) A хуа, + у'а, — Xza, (b) В — р2'а,…

A: Topic:- vector algebra Note:- as per our guideline if more than one question uploaded than only…

Q: Find the work done by the three-dimensional inverse-square field 1 F(r) = r on a particle that moves…

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Q: 7. Let S be the surface of a solid cube with domain D = {1 < z < 2, 1 < y < 2,1 < z< 2}. For field F…

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Q: show that f(z)=(z-1) cos(1/z+2) has an essential singularity at z=-2

A: Introduction: Laurent series has an important role in determining the singularities. Suppose, a…

Q: Find the divergence of the field F. yj - xk F= 1/2 +: 2 OA. 3/2 2 -X O B. 3/2 2 X -y OC. 3/2 O D. 0

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Q: 7. a) Show that F(x,y)= is a conservative field. b) Find a potential function ø(x, y) fa [show your…

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Q: 12. Compute the work done moving an object through the field F(x, y) = (y, x) one revolution around…

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Q: The curl ofvector field V = 2xy î + (x² + z²) ĵ+2zy k is O and irrotational (xy 1 – 2xyz k ) and…

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Q: 2 2 Find the gradient field of the function f(x.y.z) = In /x ?. + 4y +z The gradient field is Vf =…

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Q: Use Green's Theorem to find the counterclockwise circulation and outward flux for the field F and…

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Q: 5. Consider the DE: dy(t) + ty(t) = 1, dt a. Sketch the slope field of the DE in fig. 4. b. Draw…

A: The slope field of given differential equation with solution curves is given as follows :

Q: 25. Evaluate |F•T ds for the vector field F = x²i – yj along the curve x = y? from (4, 2) to (1,–1).

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