An application of vector fields is when the vectors represent force. Let F(x,y,z) be a force field. We say that the work done by the force field F(x,y,z) over the curve C is given by the equation Work = ∫CF(r)⋅dr∫CF(r)⋅dr∫CF(r)⋅dr∫CF(r)⋅dr Where r(t) is some parameterization of C. Let F(x,y,z) be given by the function F(x,y,z)=(x−y^2)i+(y−z^2)j +(z−x^2)k. Suppose that a particle moves through the force field F(x,y,z) along the line segment running from the point (1, 1, 1) to the point (-1, 2, -1). Using the parameterization r(t)=(1−t)(1,1,1)+t(−1,2,−1), at what time t will the force field F(x,y,z) have done 2.5 units of work on the particle? Round your answer to two decimal places.
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:
An application of vector fields is when the vectors represent force. Let F(x,y,z) be a force field. We say that the work done by the force field F(x,y,z) over the curve C is given by the equation
Work = ∫CF(r)⋅dr∫CF(r)⋅dr∫CF(r)⋅dr∫CF(r)⋅dr
Where r(t) is some parameterization of C. Let F(x,y,z) be given by the function F(x,y,z)=(x−y^2)i+(y−z^2)j +(z−x^2)k. Suppose that a particle moves through the force field F(x,y,z) along the line segment running from the point (1, 1, 1) to the point (-1, 2, -1). Using the parameterization r(t)=(1−t)(1,1,1)+t(−1,2,−1), at what time t will the force field F(x,y,z) have done 2.5 units of work on the particle? Round your answer to two decimal places.
(Hint: you will want to use the FindRoot command in Mathematica in order to solve this problem)
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