EXAMPLE 3 Integral Around a Closed Path Let f(x, y, z) = xy sin(yz). Evaluate C V. dr, where C is the closed curve in Figure 5. Solution By Theorem 1, the integral of a gradient vector field around any closed path is zero. In other words, Vf•dr = 0. FIGURE 5 The line integral of a conservative vector field around a closed curve is zero.

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: Given vector field is F=  x2y + ex siny  i + x2 + ex cosy j Curve r2=cos2θ

Q: 5) Consider the conservative force field F(x, y) = (6x²y? + cos(x), 4x y). a) Give a general…

A: For a vector field F=P,Q that is open and simply connected in a region with continuous first order…

Q: 4. Find the work done by the force field F (x,y) = (x + y)î + (xy)ĵ in moving a particle along the…

A: The force field is given by F→ = x+yi^+xyj^ The end points of the straight line along which the…

Q: a. Show that the line integral Sc (yz — 2z) dx + (xz + 3) dy + (xy – 2x) dz is independent of the…

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Q: 2. Verify that the vector field F(1, y, 2) = sec? y+ -32 2.r tan y + e csc r, 3e -32 cot x + sec z…

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Q: QUESTION 4 a) Let F(x, y, z)=x²7+(cos y sin z) + (sin y cos z)k be a vector space. Show that F is a…

A: Here given that  F→x, y, z=x2i^+cosy sinzj^+siny coszk^ be a vector space. We have to show that F→…

Q: Question 2 Consider the function: f(x,y) = 5xy - 7x² - y² + 3y, (a) Given that x = 6±0.6 and…

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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 path C is defined by the parametric equations (t) = t, y(t) = t°, 0<t<1, and a vector field is…

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Q: QUESTION 1 a) Evaluate the line integral F. dr in terms of a where F = cos yi + rj+ ye*k and C is…

A: Work done by the force is the line integral from initial point to terminal point. A force field will…

Q: Find the outward flux of the given field across the given cardioid. 5x i+ (e*+5 tan 1ly)j 1 + y2 F =…

A: The following formula from Green's theorem is used to find the outward flux, outwards flux=∮F·n…

Q: Find the work done by the vector field F = in moving a particle around the circle x2 +y2 = 4 from…

A: The given vector field is F=x-y, xy. The particle is moving around the circle x2+y2=4 from 2,0 to…

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

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Q: Show that the field F = (cos x)i + (- y sin y)j + (z3)k is conservative and Önd a potential function…

A: F = ( Cosx )i + ( -y Siny )j + ( z3 )k  F = gradient ( - Sinx - y Siny - Cosy + z4/4 ). Thus, we can…

Q: x²+ 4) Which is a correct path integral for the net flow of the vector field F[x, y] = [ 22 ) out of…

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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: (1) Let F(x, y) = (2y³/2, 3x√√y). (a) Show that F is a conservative vector field. (b) Find a…

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

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Q: 2. Given F(x, y) = (4x³y² - 2xy³)i + (2x¹y − 3x²y² + 4y³)j is a - conservative vector field. a. Find…

A: To find the potential function for the given conservative vector field :-

Q: Let F(x, y, z) = (y² + 2z, 2xy + 4, 2x), and let the curve C be described by x = for 1<t < 2. t, y =…

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Q: 6. Consider the vector field F = (-y,x). a. Check if the vector field is conservative. b. Consider…

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Q: Find the work done by the force field F = in moving particle along the curve C defined by 7(t) =,0…

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Q: Question 3 (2y +z z2 Give line integral of vector field v(x,y,z) = along curve y: ř(t) = | -t),t E…

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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: Determine if F = (12r – 2x cos y, 2y + x² sin y) is a conservative vector field. If it is, find a…

A: Given that F=(12x3-2xcosy)i+(2y+x2siny)j. We need to determine whether if F is conservative field or…

Q: 1. Let F =i+y^2j +z^4k. Find the general flow lines for this vector field. Present the flow line…

A: Consider the provided question,

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: If F = Pi+ Qj+Rk is a vector field on R and P, Q. and R have continuous second-order partial…

A: Given:F=Pi+Qj+Rk is a vector field  in ℝ3To show that:∇.curl F=0

Q: What is vector field and potential function? Sketch the gradient field of p(x, y) = x² + y² – k².

A: If z=f(x,y), then the variables x, y are called the input of the function, and z is called the…

Q: Consider the vector field F=(x^2+y^2,10xy). Compute the line integrals ∫c_1 F⋅dr and ∫c_2 F⋅dr,…

A: Given:

Q: A path C is defined by the parametric equations x(t)-t, y(t) = 2t, 0≤t≤ 1, and a vector field is…

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Q: Find the conservative vector field for the potential function h(x,y,z) = 16xy ln(x+y) by finding its…

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Q: a. Show that the line integral Sc (yz - 2z) dx + (xz+3) dy + (xy - 2x) dz is independent of the…

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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: b) Consider the vector field F(x,Y, z) = (2xy + z³, x², 3xz²). This vector is con- servative (you do…

A: We have to find the following conditions

Q: A path C is defined by the parametric equations z(t)=t, y(t) = 2t, 0<t<1, and a vector field is…

A: It is given that a path C is defined by the parametric equations x(t)=t, y(t)=2t, 0≤t≤1 and a vector…

Q: Consider the conservative vector field 4 8 F(x, y, z) = – 3y²cos (3nz),37y sin (3tz) – x2). 2xz, %3D…

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Q: Calculate the circulation of the field F around the closed curve C. Circulation means line integral…

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Q: Calculate the circulation of the field F around the closed curve C. Circulation means line integral…

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Q: Determine whether F(x, y) = e5 cos 5y i – e5" sin 5yj is a conservative vector field. If so, find a…

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

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Q: Calculate the line integral in scalar field ∫C (x2y2-√x)dy is the arc of the curve y=√x of (1,1) to…

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Q: Calculate the flux of the field F across the closed plane curve C. F=xi+yj, the curve C is the…

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Q: EXAMPLE 4 Line integral of a conservative vector field Evaluate |(2ry – z) i + (x² + 2z) j + (2y –…

A: The line integral of a conservative vector field is  C is a simple curve from A(-3,-2,-1) to…

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: Consider the one-form w = x' cos(y) dx + xe13% dy+4z² dz on R3. The associated vector field is: F(x,…

A: For the solution follow the next step.

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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