Find the work done by the force field F(x, y, z) = 2xi+ 2yj+7k on a particle that moves along the helix r(t) = 1 cos(t)i +1 sin(t)j+ 6tk, 0

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: Find the flux of the field F = 2xy i + 2yz j + 2xz k upward across the portion of the plane x + y +…

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Q: Confirm that the force field F(x, y) = 2xy³i + 3x²y²j is conservative in some open connected region…

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Q: Find the work done by the force field F(x,y) = x² i– 2xy j in moving a particle along the quarter…

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Q: Calculate the circulation of the field F = (2xy,y²) in a counterclockwise direction and the outward…

A: Consider the given field. F=2xy,y2 And, the given curves are defined as, y=x and y=x3 The green…

Q: Find the outward flux of the field F = (x + y)î + yĵ + zk across the closed surface: S: z = 16x² -…

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Q: Find the work done by the force field F(x, y, z) = xyi+yzj+xz k on a particle that moves along the…

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Q: Find the flux of the field F = xy i - z k outward (normal away from the z-axis) through the cone z =…

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Q: Compute the work done by the force field F =< x² + 2y, 4x + y° where C is the boundary of the region…

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Q: Find the work done by the force field F(x, y, z) = 4xi + 4yj + 6k on a particle that moves along the…

A: Explanation of the answer is as follows

Q: Find the work done by the force field F(x, y, z) = xyi+ yzj+xzk or a particle that moves along the…

A: Given: Fx,y,z=xyi+yzj+xzkr(t)=ti+t2j+t3k

Q: Find the work done by the force field F(x, y, z) = 3xi + 3yj + 6k on a particle that moves along the…

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Q: Evaluate line integral of vector field F(z,9) = i+ 1- x? y2 2x along the curve x = cos(t), y =…

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Q: Find the work done by the force field F(x, y, z) = 5xi + 5yj + 2k on a particle that moves along the…

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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: Find the work done by the force field F(x,y) = 4yi + 2xj in moving a particle along acirclex^2 + y^2…

A: Step2

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 work done by the following force field F(x, y) = 3(v+ 2)'0i + 30x (v + 2)°j %3D in moving…

A: The work done is a scalar quantity, which is equal to the dot product of force applied to the body…

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: Find the work done by the force field F(x, y)=(e* – y)i +(cos y + x) j to move a particle once…

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Q: Find the work done by the force field F(x,y) = x²i – xy²j around the circle (Clockwise direction)…

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Q: Find the work done by the force field F(x, y) = xy i + x^2j on a particle moving from (0, 0) to (1,…

A:

Q: Find the work done by the force field F(x, y) = xi + (y + 7)j in moving an object along an arch of…

A: Given  Fx,y=xi+(y+7)j in the moving an object along an arch of the cycloid dr

Q: Evaluate the outward flux of the vector field F = (2x? – 5e" + sin(2²), –4ry + ln(x*), 5z – 3x²y°)…

A: Given

Q: Find the work done by the force field F(x, y) = x sin y i + y j on a particle that moves along the…

A: Explanation: Given that,  Force F and a parabola from (-1,1) to (2,4)…

Q: Find the work done by force field F on an object moving along the indicated path. F(x, y, z) = xi +…

A: To solve the following problem

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

A: The answer is given by using the formula of line integral

Q: Find the work done by the force field F(x, y) = (3x² + y²) i+ (2y²)j along the curve C given by ř(t)…

A: The given vector field is  f→x,y=3x2+y2i→+2y2j→       (1)  and the curve is  r→t=ti→+e2tj→…

Q: (a) Compute the flux of the field F = r i+yj-zk across the cylindrical surface a + z = 1, 0 <y53…

A: If E be a  solid region bounded by surface of S with positive orientation then for a vector field…

Q: I Need help for this question

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Q: Compute the work done by the force field F = 6.2y" i + 6.2a" j on a particle moving along the…

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Q: Consider the force field F = (2rze"*: – 4, cos' z, r²e*** – 3y cos? z sin z - Find the work done by…

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Q: Compute the work done by the force field F = , 0 <t<1

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Q: Find the counterclockwise circulation and the outward flux of the field F = (-sin y)i + (x cos y)j…

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Q: Is the force field F (x, y) = x³i + sec(x+ xy)j conservative?

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Q: Find the work done by the force field F(x, y) = xi + 2yj on a particle moving along the path C:x =…

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

A: Given that h(x,y)=y2i+2xy-e2yj  The parametric equation of the circular arc is  r(t)=cos(t) i+sin(t)…

Q: Confirm that the force field F(c, y) = 2xy³i + 3x?y?j is conservative in some open connected region…

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Q: d the work done by the force field F in moving a particle along the curve C. F = xyi + 3j + 9xk; C:x…

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Q: (а) Given the force field F = (z³ cos x + 2xy?)i+ (2x²y – 2)j + (3z² sin x – 4)k (i) Prove that F is…

A: Given force field is: F=z3cosx+2xy2i+2x2y−2j+3z2sinx−4k (i) To prove that F is conservative. (ii) To…

Q: (b) A particle moves once around the circular path with radius of 1. Using Greens' Theorem, compute…

A: see below the answer

Q: Consider the force field F(x, y, z) = xi + yj + zk. Compute the work done in moving a particle along…

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Q: Find the work done by the force field F(x, y) = (4 e* – y³ î + (2 cos(y) + x³ )j on a particle that…

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Q: Find the work done by the force field F(x, y, z) = 4xi + 4yj + 3k on a particle that moves along the…

A: To find the work done by the below force field on a particle that moves along the helix r(t).

Q: Find the work done by the force field F(x, y, z) = (x – y², y – z², z – x2) on a particle that moves…

A: We have to find the work done by the force field F(x,y,z)=&lt;x-y2, y-z2, z-x2&gt; on a particle…

Q: Find the work done by the force field F(x, y, z) = 5xi + 5 yj + 7k on a particle that moves along…

A: Follow next step

Q: Find the work done by the force field F(x,y)=xi+(y+5)j in moving an object along an arch of the…

A: Work done can be estimated by integrating the F( r(t) ) from t = 0 to t = 2π

Arc Length

Arc length can be thought of as the distance you would travel if you walked along the path of a curve. Arc length is used in a wide range of real applications. We might be interested in knowing how far a rocket travels if it is launched along a parabolic path. Alternatively, if a curve on a map represents a road, we might want to know how far we need to drive to get to our destination. The distance between two points along a curve is known as arc length. 

Line Integral

A line integral is one of the important topics that are discussed in the calculus syllabus. When we have a function that we want to integrate, and we evaluate the function alongside a curve, we define it as a line integral. Evaluation of a function along a curve is very important in mathematics. Usually, by a line integral, we compute the area of the function along the curve. This integral is also known as curvilinear, curve, or path integral in short. If line integrals are to be calculated in the complex plane, then the term contour integral can be used as well.     

Triple Integral

Examples: