(a). A two-dimensional fluid motion is given by the complex potential w(z) = iz* (cos(nt) – i sin(nt)), where n is a constant. O Find the lines of flow and equipotentials. [ii] Show that they are orthogonal to each other.

Q: Let v = (x²z, 1 – 2xyz – 3y + x²y, 3z - x²z) be the velocity field of a fluid. Compute the flux of v…

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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…

A: Solution:

Q: Let i = (3 – 2æyz - - xe cos y, yʻz, e² cos y) be the velocity field of a fluid. Compute the flux of…

A: Given that V=3-2xyz-xezcosy,y2z,ezcosy And the surface x+y2+z2=1 It is bounded. Then the flux is,…

Q: Q#3. Find the work done by the force field 4 1 F(x, y)=- x² +y x'+y? on a particle that moves along…

A: Given- Fx,y=1x2+y2i+4x2+y2j To Find- The work done by the above force field on a particular that…

Q: panticle of mass m. moves in the one-dimersianal potential U(x) = U. %3D 92 Sketch Ulx).

A: The function is a rule that associates or fixes a number to the particular point x. If , then the…

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: 7. Find the circulation of the field F(x, y) = xi+yj around the circle r(t) = (cos t)i + (sin t)j.

A: Since you've asked multiple questions, as per our answering guidelines we are supposed to answer…

Q: Exercise 10.6 Evaluate the line integral using a method of your choice. e¯*(cos y dx + sin y dy),…

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Q: A fluid motion is given by v=(y+z)i+(z+x)j+(x+y)k show that the motion is irrotational and hence…

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Q: 6. Evaluate the line integral (x? + y² )dx – x dy along the curve C: x? + y² =' counterclockwise…

A: Given: ∫c x2+y2 dx- x dy  We have to Evaluate the given line integral along the curve C: x2+y2=1…

Q: 3. A flow is represented by the velocity field (y² + z²)i + xzj + 2x²y²k, check if: a) A possible…

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Q: Find the differential of w = x ev sin z at the point (x, y, z) = (1, 1, 0). %3D O dy + dz O dy O dx…

A: Obtain the differential of the given function as follows. w=xeysin zdw=ddxxeysin zdx+ddyxeysin…

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: The motion of a liquid in a cylindrical container of radius 3 is described by the velocity field…

A:

Q: Use Gauss's Law to find the charge contained in the solid hemisphere  x2 + y2 + z2 ≤ a2, z ≥ 0,  if…

A: Given that

Q: Use separation of variables to solve the following heat equation in a disk 1 ðu ди = k dt 1 ди r dr…

A: Given: To solve the following heat equation in a disk, use variable separation, Let us consider the…

Q: Find the circulation and flux of the field F = - xi – yj around and across the closed semicircular…

A: The circulation of a vector F=M i+ N j along the curve C is defined by ∮CF·T ds=∮CMdx+Ndy. Here…

Q: Q3: A- Use Greens theorem to evaluate the circulation or flux for the velocity field : 1 f(x, y) =-…

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Q: (4 xe cos y, y cos z, e cos y) be the velocity field of a fluid. Compute the flux of v 9 where x > 0…

A: Let v=4-xcosz-xezcosy,ycosz,ezcosybe the velocityfield of a fluid.Compute the flux of v a cross the…

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: (b) Let E = {(x, y, z) : –1< # < 0, 0 < y < VI– a², 0< z< 2}. Write the integral II| cos (2² + y°)dV…

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Q: Fluid flowing has a velocity field given by V =u, sin[a(t – ý/v.F +v,j on=4 %3D where uo Vo, and @…

A: Given: V→=u0sinωt - y/v0i→+v0j→, where u0, v0 and ω are constant.

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: If the velocity of a steady flow is given by V=(1x ,-1y,0) where 2 is a constant and F = -gzk ,…

A: According to the given, Velocity of a steady flow is given by:

Q: (a) Suppose that a black hole has magnetic field F = (2a* + e" sin(2), y³ + xe*, 2z³ + e" cos(z) +…

A: TO FIND: Magnetic flux across the black hole

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: 36. Convert the integral •Vī-y² (x² + y²) dz dx dy Jo to an equivalent integral in cylindrical…

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Q: Find the circulation of the Fİeld F(x; y) = 3xi+yj around the circle r(t) = (cos6t)i + ( sin6t)j;…

A: We need to find the circulation of field F→=3xi^+yj^  around the circle r→t=cos6ti^+sin6tj^ where…

Q: 38. The flow of a gas with a density of 8 = 0.3kg/m² over the closed curve r(t) = (cos t)i + (sin…

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Q: 2. Calculate the circulation of the flow v=-8xy i+ 4 yj around the triangle with vertices (0, 0),…

A: Given that: v=-8xyi+4y2j The triangle is with the vertices (0,0), (3,0), (3,2) To calculate the…

Q: Find the outward flux of the field F = sin(y) i +xzj +3zk across the boundary of the region D: the…

A: We need to find the outward flux ∬DF→.dD We can use divergence theorem which equates ∬DF→.dD with…

Q: 3. Use the surface integral is Stokes' Theorem to calculate the flux of the curl of the field F…

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Q: Show that polar form of equation of continuity for a two dimensional incompressible fluid is av…

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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 gradient vector field F(x, y, z) = V (e² + y² + z²) in %3| mov ing a…

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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: 3. Solve the Laplace equation inside a 60° wedge of radius 1 subjected to the boundary conditions…

A: The Laplace equation is given by: r2∂2u∂r2+r∂u∂r+∂2u∂θ2=0 The boundary conditions are: ∂u∂θr,0=0,…

Q: 1. Use Green's Theorem to find the counterclockwise circulation and outward flux for the given filed…

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Q: Find the flux of F = (3z + 1)k upward across the hemisphere x2 + y2 + z2 = a2, z&gt;= 0 (a) with the…

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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: 15. Let R be the plate in the shape of the region under y = sin x for 0 < x < in Figure 13(A). If R…

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Q: The electric potential at each point (x,y) in the region R: x 2 0, y s1 is given by v - 48xy – 32x -…

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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: (5 2ryz the surface x + y° + z? = 1 where z > 0 and the surface is oriented away from the origin.…

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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: Q3: A- Use Greens theorem to evaluate the circulation or flux for the velocity field : 1 f(x, y)=-…

A: GIVEN: f(x, y) = 1x + yi - 1x + yj

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

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Estimation

A new framework of data analysis where the new statistical methods are used for interpreting the results and analyzing the data is known as estimation in statistics.