A child living in the ry-plane climbs a ladder from (0, 0) to (1, 1), slides down to (2,0) along the parabolic path y = (x – 2)?, and then walks back to (0, 0) along the r-axis. Find the work done by the vector field F(x, y) = (2x – y, y') in moving the child along her path. Select one: 4 а. 21 b. 15

Q: How do you graph the vector field F = ⟨ƒ(x, y), g(x, y)⟩?

A: Given: vector field

Q: Find work done by the force field f = (XY, 2, 42) when a Particle moves along the straight line from…

A: The force field is given as,find work done by the force field F when a particle moves along the…

Q: Q4: Find the curl of the vector field F(x, y,2) = xJyi +j+ zk at the point(2,4).

A:

Q: Assuming that vector fields B (r") and A (r") in the zero infinity, check the correctness of the…

A:

Q: x-y, z + y + 3,z²) and the net is decribed by the equation y = 1- x² - z on

A:

Q: Q4: Find the curl of the vector field F(x, y.2) = x*/yi +j+ zk at the point(2,4).

A:

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

A: We have to find the rate of heat flow out of a sphere of radius .

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

A: Given that v=<x-y, z+y+9, z2> and the given curve is y=1-x2-z2,y≥0 To find: The value of ∬S…

Q: Find r'(t), r(t), and r'(to) for the given value of to- r(t) = (e², e²t), to = 0 r'(t) = r(to) =…

A: rt=et,e2t,t0=0

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

A:

Q: Find the curl of the vector fleld F(z, y) = 9ye" i+2ejat the point (-2, 0). Enter an exact answer.…

A: F(x, y)=9y2 ex3 i +2x2 ey jcurl F→=∇×F→curl F→=ijk∂∂x∂∂y∂∂z9y2 ex32x2 ey0         =∂∂x(2x2…

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: 9. Sketch the vector field F(x, y) : :(yi — х). Vx? + y?

A: Consider the given filed, Fx,y=1x2+y2yi−xj Further simplify. Fx,y=yx2+y2i-xx2+y2i=〈yx2+y2,-xx2+y2〉

Q: Find r'(t), r(t), and r'(to) for the given value of to. r(t) = (et, e24), to = 0 r'(t) = r(t) =…

A:

Q: A Particle moves 2 X half way around the circle y² = 4 oriented in the counter-clockwise direction…

A: Given force field: and the path of the particle moving half way around the circle  in the…

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

A:

Q: Sketch the vector field F(x, y) = (−x — y, x - y) at the points (x, y) where x, y = {-1,0, 1}.

A: given vector field F→x, y=-x-y, x-y we have to sketch for x, y∈-1, 0, 1

Q: Find the work done by the force field F(x,y) = (-y,x) in moving an object along the path y = V4 – x2…

A: Given force field Fx,y=−y,x and object moving along the path y=4−x2  from 2,0 to -2,0. We have to…

Q: Let C₁ be the line segment from the point (-4, 8) to the point (2, -4), C₂ be the arc on the…

A:

Q: Show that the vector field F = (yz² ye, rz² - e, 2xyz) is conservative, and find a potential…

A:

Q: The gradient of f(x, y) = x sin (y²) is the vector field F (x, y) = at (2,√π/6) and ending at…

A: Here we are given that the gradient of f(x,y)=xsin(y2) is the vector field F(x,y)=sin(y2)2xycos(y2)

Q: Find the work done by the vector field F = xi + 2xyj + 3xyzk on a particle that moves along

A: Introduction: The size of the region enclosed by the graph of the supplied function between two…

Q: Match the vector field with its graph. F(x, y) = yi -5+ -5+

A:

Q: Find a formula for the vector field V(x, y) = (vectors are shown as line segments, with a dot at the…

A: The vector field is a cluster of arrows, each with a specific magnitude and direction, that are…

Q: 2. Letr =√x² + y² + z². (a) For n ≥ 1 and r> 0, express V(r-n) in terms of r and the radial vector…

A: Given  solution(a): Radial vector field  solution(b):

Q: Given the vector fieldF(z,y)=(y³+422-2)i+(3zy2+2y+2)j. Evaluate the line integral SF-dr from (1,0)…

A:

Q: For the vector field F = yr³i - y²j, the work on the curve y = √x from (1, 1) to (0,0) is A. B. C.…

A:

Q: ind a unit vector in the direction in which fincreases most rapidly at P, and find the rate of…

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Q: The work done by the vector field F(x, y) = (-y², ²) along the arc of the parabola y = 2x² - 4x + 2…

A: According to question,Given Vector field,Fx, = -y2,x2Equation of pparabola,y = 2x2-4x+2

Q: Using arrows, clearly sketch the vector field u(r) = (y², x + y, 0) in the (x, y)-plane.

A: Given the vector field function u(r)=(y2,x+y,0). We need to plot the vector field in the…

Q: Let a vector field F(x,y)=sinx 1 + xy j be given. What is the integral representing the work done by…

A: Acoording to expert guidelines, I am allowed to answer first question only. Kindly repost the rest…

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: