Given two points A(3, 0) and B(), find the work done by the following force field in moving a particle along x² + y² = 9 from the point A to B: F = (x + y)ỉ + x²j

Q: A particle is under the influence of the force F= (-cosh (4x4) + xy)i + (e-y + x)j. The corner…

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Q: Let A be the vector potential and B the magnetic field of the infinite solenoid of radius R. Then

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Q: A net is dipped in a river. Determine the flow rate of water across the net if the velocity vector…

A: Adding the plane surface y=0 to close the surface and apply the gauss divergence theorem.

Q: If r(t) is the position vector of a particle in the plane at time t, find the indicated vector. Find…

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Q: Find the component of the force 5i+7j-k in the direction of the displacement PQ, where P(3,0,1) and…

A: The given force is

Q: Let the velocity vector be v(t) = (6 cos(3t), 8 exp(2t), 3t2). Compute the acceleration vector a(t).

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Q: Find T(t) ,N(t),B(t), and K(t) for the vector position r(t) = (e2t cos t) i + (e2t sin t)j + 2k when…

A: Given rt=e2tcosti+e2tsintj+2k

Q: Let f(x, y, z) = xy sin(xz). The rate of change of f at the point P(1, -2, π) in the direction of…

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Q: ar(t) = cos(-10t)i+sin(-10t)j-10tk, compute A. The velocity vector v(t) = B. The acceleration vector…

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Q: If r(t) is the position vector of a particle in the plane at time t, find the acceleration vector.…

A: given,  r(t)=(6cost)i+(3sint)j

Q: Let A be the vector potential and B the magnetic field of the infinite solenoid of radius R. Then…

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Q: Find the work done by the force F = 2i - 7j in moving an object from the point (2, 1) to the point…

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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: