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Q: d) A vector field is given by F(x,y,2) = 2xj+ (x² - y(1-y))k . 2(1-e"?) Find the work that the…

A: To solve d and e: A vector field is given by, The parameterized curve C is given by, The interval…

Q: 3. (a) Determine the value of coefficient a for which the vector field F = (ayz + 22, 2.02?, 6ryz² +…

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Q: 11. A particle moves along a curve parameterized by r(t) = 3t² i-t² j+tk from the point (0, 0, 0) to…

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Q: The force exerted by an electric charge at the origin on a charged particle at a point (x, y, z)…

A: The force exerted by an electric charge at the origin, at the point (x, y ,z) is, F=krr3 where,…

Q: funcion

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Q: 4z² + 5dz 금(122+5) xo 4a? + 5 Part 2 of 2 ewrite the integrand in terms of u so that the integral…

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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 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: Find the increase rate of the scalar V at point P from origin to the point P(2,6; 9,8 ; 9,5) where…

A: According to question given that The origin to the point P(2,6 ;9,8 ; 9,5) where V=3, 3xy+ 2, 6xyz

Q: Suppose F(x, y) = x²i + y j and C is the line segment segment from point P = (5, 5) to Q = (9,0).…

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Q: Find the flux of the vector field F = (x, z, - y) through the helicoid with parameterization r(u, v)…

A: Given :-  The flux of the vector field  F→ =  x3 , z , -y through the helicoid with 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: For n solid regions in Divergence Theorem, A. the triple integral over the whole solid region is the…

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Q: consider a particle traveling within the force field F = along the parametrized curve r(t) = from…

A: Consider the given: F=−y,x,12rt=tcost,tsint,12tFrom point 0,0,0 to 2π,0,π. The work done moving the…

Q: The magnitude of the x-component of a unit vector at the point (1, 1) that is normal to $(r) : equi-…

A: The formula to calculate the vector which is normal to the equipotential  line of the potential…

Q: Consider the vector-field (a) Show that F = (x + yeª + sin y)i + (eª + x cos y)j. is conservative.…

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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 along line segments from the origin to the points (1, 0, 0), (1, 3, 1), (0, 3, 1),…

A: Given that  A particle moves along the line segments from the origin to the points…

Q: Compute the work done performed by the force F(x, y) = (4r'y", zyt) acting on the object moving…

A: workdone   by a force F  along curve C is∫C F dr Here F=4x2y3,xy4since c is closed curve in xy…

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: F - () (3x²y²z and F = 2r³yz Q.)11.1 Be (a) Explain why vector fields are conservative. (b)…

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

A: Let's find.

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: A smooth curve is normal to a surface f(x,y,z) = c at a point of intersection if the curve's…

A: Given: A smooth curve is normal to a surface fx,y,z=c at a point of intersectionif the curve's…

Q: Compute the work done by the force along the curve (use defini- tion): 1 F = [3r – y + y] curve is…

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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: 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: 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: Consider the force field F = (3x²yz - y + z, x³z - x, x³y + x). (a) Verify that it is a conservative…

A: A vector field will be conservative if its curl is zero . If f is potential function of F then ∫abF…

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

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Q: Calculate the line integral of the vector-function F(x, y, z) = (y² +2²) i-yzj + xk along the path…

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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: 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: Find the flux integral ∫∫scurl(F).dS, where F(x,y,z) = [-y + 2cos(πy)e2x + z2, x2cos(zπ/2) -…

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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: 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: 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: 2. Let S be the portion of the paraboloid y = 9x² - z² with y ≥ 0. a) Provide a vector-valued…

A: let S be the portion of the paraboloid y=9-x2-z2 with y≥0 (a) provide a vector valued function that…

Q: Consider the force field F = (-y, r, 3rz²). Sketch each curve C and compute | F- Ť ds. Does it…

A: Consider the given  force field: F→=-y,x,3xz2

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: 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,…

Q: A solid material has thermal conductivity K in kilowatts per meter-kelvin and temperature given at…

A: A sphere of radius 1 is x2 + y2 + z2 = 1. Parametrize the sphere as follows:…

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: