Use Green's theorem to find the counterclockwise circulation and outward flux for the field F = (y'-e' sin x)i +(y +e' cosx) j around and along the curve C, where C is the upper hand loop of the lemniscate r= sin20 %3D

Q: Sketch the Part of tthe slope Field of y'=L for X and y be tween -5 and 5. Then add to your sketch…

A: Slope field of y' =1/y

Q: Use the vector field F(x, y) = x 2e yi + cos x sin yj and Green’s Theorem to write the line integral…

A: Counterclockwise is the direction opposite to that in which the hands of a clock travel; movement in…

Q: 8.) Compute circulation of the field F = (-y,x) around the unit circle in two ways: (a) Parameterize…

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Q: A quarter cylinder of radius 2 and height 5 exists as shown in the figure below. Surface integral of…

A: Given that the quarter cylinder is of radius 2 and height 5. This means that the radical distance…

Q: 9) Check whether the vector field F = (-cos x cos y, sin x sin y, - sec² z) is a gradient vector…

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Q: Find the total work done in moving particle in a free field given by F = 3xyi – 5zj + 10xk along the…

A: The total work done by a vector field, F→ moving a particle along the curve r→(t) is calculated…

Q: Consider the rotational field F = (-y, x). Compute the work done, |F•T ds, in the presence of the…

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Q: A pasticle depicts ts motion in Such a ulay at time ť is P=xîty where, 2= acoskt and y=bsinkt. that…

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Q: Why does a two-dimensional vector field with zero divergence ona region have zero outward flux…

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Q: Use Green's Theorem to calculate the circulation of F = 2xyi around the rectangle 0 < x < 5, 0 < y <…

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Q: By using two methods, find the work done by the force field F(x, y) = ysin xỉ + xe³ ĵ to move a…

A: Given: To compute:  ∫C F.dr=?F→(x,y ) = y sin x i^…

Q: Let S be a simple closed surface in R' that encloses a solid region with volume equal to 5. Find the…

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Q: Determine the total work done by the force field F(x, y) =. over a particle that follows the path…

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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: Find the flux of the field F = x2 j - xz k outward (normal away from the yz-plane) through the…

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Q: 23. Find the outward flux of the field i + (e + tan'y)j 1 + y², F 3xy %3D across the cardioid r =…

A: Given: The following formula is used to find the outward flux, Comparing with the general form,…

Q: Find the flow of the velocity field F =(x + y)i - (x2 + y2)j along each of the following paths…

A: The velocity field F=(x+y) i -x2+y2j along each of the following paths from 1,0 to -1,0 in the xy…

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

A: By using green theorem, we calculate the counterclockwise circulation and outward flux of the given…

Q: A particle starts at the point (-2, 0), moves along the x-axis to (2, 0), and then along the…

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Q: The motion of a liquid in a cylindrical container of radius 2 is described by the velocity field…

A: Given,F(x,y,z) = -19y3i + 19x3j + 7k⇒curl F = ijk∂∂x∂∂y∂∂z-19y319x37          = 0i + 0j +…

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

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Q: | Use Green's Theorem to calculate the circulation of F = 4xyi around the rectangle 0 < x < 8, 0 <…

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Q: .) Find the work done by field F = (-y,-x) along curve C: counter clockwise along semicircle y = v4…

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Q: Calculate the divergence of the scalar field F(x,y)=3x2+siny.

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Q: Compute the flux of the vector field F = xi + yj + zk through the surface S, which is a closed…

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Q: Find the line integral F.dr of the vector field F=ax] over a curve C defined by a C rectangle in the…

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Q: 4. Verify that the two integrals in the circulation form of Green's Theorem are equal along a circle…

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Q: 15. Verify Stokes' Theorem for the vector field F = 2xyi + xj + (y+z)k and the surface z = 4x² - y²,…

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Q: Verify the claim made in the given problem below section d by showing that the net outward flux of F…

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Q: Evaluate line integral of vector field F(x, y, z) = y sin(x²) i+ xyz j+ (x +y + z) k along line…

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Q: A particle starts at the point (-1, 0), moves along the x-axis to (1, 0), and then along the…

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Q: Therefore, div(curl G) = sin(y) – sin(y) + 8 = Since div(curl G) is different from 0, we conclude…

A: Given that:div(curl G)=siny-siny+8=

Q: F =(-y,x,z) across the solid sphere E %3D

A: we have to find the outward flux

Q: A particle starts at the point (-2,0), moves along the x-axis to (2,0) and then along the semicircle…

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Q: Express the field D = (x2 + y)-(xax + yay) in cylindrical components

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Q: A particle moves in the positive direction once around the square with vertices (0, 0), (0, 2),(2,…

A: Given, the force field Fx,y=x-sinxi+y2+xj. We have to find the work done.

Q: Consider the conservative vector field 4 8 F (x, y, z) 3y cos (372), 37y sin (372) – a² > 2xz, 2x +…

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Q: The vector field F is shown in the xy-plane and looks the same in all other horizontal planes. (In…

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Q: Find the net outward flux of the field F = (z,3z-xx) across the sphere of radius 9 centered at the…

A: Given: The F=&lt;z, 3 z-x, x&gt; across the sphere of radius 9 centered at the origin. To find: The…

Q: Find the outward flux Of The field F across the boundary surface of the solid.

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Q: Use the Divergence Theorem to compute the flux of the field 1 cos (y)) out of the region E that lies…

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Q: The outward flux of the vector field F=(x,0, z) across the cylinder x + y = 1, for 0<z<1, is %3D…

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Q: The motion of a liquid in a cylindrical container of radius 2 is described by the velocity field…

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Q: Is the vector field shown in the figure conservative? O Yes O No Explain. Let C be a circle centered…

A: Graphically, a vector field is conservative if it has no tendency to swirl around. For a…

Q: A hot air balloon is shaped like a truncated sphere like the figure. The gases escape through the…

A: Given a hot air balloon has a truncated spherical shape. The velocity vector field  Vx,y,z=∇×Fx,y,z…

Q: Compute the flux of the vector field F = 2xi + 2yj through the surface S, which is the part of the…

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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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Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as: