Find the work done by the force field F(x, y) = (y^2)i − (x+y)j in moving a particle along the straight line path from (-2,0) to (0,2). b) Does the work done depend on the path? Justify your answer

Q: Calculate the work done by a particle moving under the influence of a force F = -(2z + y)ê + x²yŷ +…

A: The general form of a vector-valued function in three dimensions is given by F=F1x^+F2y^+F3z^…

Q: Find the work done by the force field F(x, y, =)=(: - =', y + 3:, = + x*) in moving a particle alon,…

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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…

A: Consider the provided information, The work done by force F in moving particle along the path C is…

Q: Find the flux through through the boundary of the rectangle 0 < x < 6, 0 < y < 5 for fluid flowing…

A: Given F→=x5+y  , y cos3x flux=∫cF→.n^  ds =∫∫Ddiv F dA

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: A particle moves along line segments from the origin to the points (3, 0, 0), (3, 5, 1), (0, 5, 1),…

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Q: Find the flux through through the boundary of the rectangle 0 < x < 6,0 < y < 3| for fluid flowing…

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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: A particle moves along line segments from the origin to the points (1, 0, 0), (1, 5, 1), (0, 5, 1),…

A: Obtain the equation of the plane passing through the 4 points (1,0,0), (1,5,1), (0,5,1) and the…

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: A particle moves along line segments from the origin to the points (1, 0, 0), (1, 5, 1), (0, 5, 1),…

A: First obtain the equation of the plane passing through the 4 points (1,0,0), (1,5,1), (0,5,1) and…

Q: Show that the force F = (y^²2² - 6xz² ) ₁² + ²xyz²³ j^² + (3 xy² z ²-6x²²Z) k IS Conservative, hence…

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Q: Find the work done by the vector field ⟨4x+yx,x2+6⟩ on a particle moving along the boundary of the…

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

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

A: Given the force field F by, F=xi+x3+3xy2j Work done by the force F in moving a particle along a path…

Q: A particle moves in a velocity field V(x, y) = (x2, x + y2). If it is at position (x, y) = (7, 2) at…

A:

Q: A particle moves along line segments from the origin to the points (3, 0, 0), (3, 5, 1), (0, 5, 1),…

A: Given:Fx, y, z=z2i^+2xyj^+5y2k^Particle moves from line segments from the origin to the points 3, 0,…

Q: Find the flux through through the boundary of the rectangle 0 < x < 6,0 < y < 5 for fluid flowing…

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Q: Compute the work done by the force field F(r, y) = (2, 3r) acting on an object as it moves along the…

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Q: b) Show the force field F(x, y, z) = (x²y + In(yz)) i+ 3 y cos(2:) ) j+( - y° sin( sin(2z) ) k %3D…

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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: A particle moves along line segments from the point (2,0,0) to the points (2,2,2) (0,2,2), and back…

A: Let C covers all the line segments. Consider C1 be the line joining the points (2,0,0) and (2,2,2).…

Q: Find the work done by the force field F(x, y) = (x^2-lny)i + (e^x^2+y^2)j on a particle that travels…

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Q: A particle moves along line segments from the origin to the points (3, 0, 0), (3, 3, 1), (0, 3, 1),…

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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: A particle starts at the point  (−4, 0),  moves along the x-axis to (4, 0), and then along the…

A: Consider the given force field:

Q: Why is the upward flux of a vertical vector field F = ⟨0, 0, 1⟩ across a surface equal to the area…

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Q: A particle moves along line segments from the origin to the points (2, 0, 0), (2, 4, 1), (0, 4, 1),…

A: A particle moves along line segments from the origin to the points 2, 0, 0, 2, 4, 1, 0, 4, 1 and…

Q: Find the work done in moving a particle in a force field F = 3xyi - 5xyzj + 10xk along: straight…

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Q: Find potential 9, for the conservati ve field the vector F (x, y,a)- (2xe? - y*sinx) î * sinx) î +…

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Q: A particle moves along line segments from the origin to the points (2, 0, 0), (2, 4, 1), (0, 4, 1),…

A: Given Fx,y,z=z2i+5xyj+2y2k  Let r=xi+yj+zk⇒dr=dxi+dyj+dzk Now particle moves from origin O0,0,0 to…

Q: Find the work done by the force field f(x,y,x) = on a particle that moves along the line segment…

A: Given, the force field = F(x , y, z) = &lt;x+y,xy,-z2&gt; And this force field moves along the line…

Q: A particle moves along line segments from the origin to the points (1, 0, 0), (1, 4, 1), (0, 4, 1),…

A: Here A particle moves along the line segments from the origin to the points (1,0,0),(1,4,1),(0,4,1)…

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

A:

Q: A particle moves along line segments from the origin to the points (3, 0, 0), (3, 3, 1), (0, 3, 1),…

A: From stokes theorem ∮F.dr=∯∇×FdS.  Thus, calculate the curl of F=z2i+3xyj+4y2k:…

Q: Find the flux of the vector field F = across the part of the plane | z = 2 + x + 3y above the…

A: Given: F¯=y,-z,x z=2+x+3y The formula to determine the flux is: ∫S∫F·nds=∫D∫(-agx-bgy+c)dA Here…

Q: A particle moves along line segments from the origin to the points (3, 0, 0), (3, 5, 1), (0, 5, 1),…

A:

Q: the work done by the given force F(x, y) in moving a particle along the given curve C in the…

A: Given force F(x,y)=y3i-x3j The objective is to find the work done along the curve C. The formula for…

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: A particle moves along line segments from the origin to the points (3, o, 0), (3, 5, 1), (o, 5, 1),…

A: Apply Stokes theorem

Q: Determine the total work, by Green's theorem, done by the force field F(x, y) =, on a particle…

A:

Q: Find the gradient vector field (F(r, y, z)) of f(x, y, z) = x² sin(3yz) F(z, y, 2) = (

A:

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

A:

Q: A particle moves along line segments from the origin to the points (2, 0, 0), (2, 4, 1), (0, 4, 1),…

A: Solution Given Force field F(x,y,z)=z2i+3xyj+4y2k.the origin points are(2,0,0)  (2,4,1) ,(0,4,1)Use…

Q: Compute the outward flux of the vector field F = (5xy, 5y²) for the region enclosed by the curves y…

A: Concept:   The calculus helps in understanding the changes between values that are related by a…

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

A: The solution is given below:

Q: A particle moves along line segments from the origin to the points (3, 0, 0), (3, 3, 1), (0, 3, 1),…

A: Given points, (0,0,0) (3,0,0) (3,3,1) (0,3,1) field Fx,y,z=z2i+4xyj+3y2k Now, Curl…

Q: Find the work dr done by the force field F = (y,x,z) on a particle that moves along the line segment…

A:

Q: A particle moves along line segments from the origin to the points (2, 0, 0), (2, 5, 1), (0, 5, 1),…

A: Given - A particle moves along line segments from the origin to the points (2, 0, 0), (2, 5, 1), (0,…

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: