Question

Consider the vector field below: F ⃗=〈2xy+y^2,x^2+2xy〉 Let C be the circular arc of radius 1 starting at (1,0), oriented counter clock wise, and ending at another point on the circle. Determine the ending point so that the work done by F ⃗ in moving an object along C is 1/2.

Answer #1

Consider the vector field F = <2 x
y^3 , 3 x^2
y^2+sin y>. Compute
the line integral of this vector field along the quarter-circle,
center at the origin, above the x axis, going from the point (1 ,
0) to the point (0 , 1). HINT: Is there a potential?

Using Stoke's Thm, find the work done by the vector field F(x,
y, z) = 〈z, x, y〉, that moves an object along the triangle with
vertices P(1, 0, 0), Q(0, 1, 0), R(0, 0, 1), in a counterclockwise
manner, starting and ending at P.

given field F =[x+y, 2xy ] and c: x= y^2
calculate the line integral along (1,-1) to (4,2)

If C is the curve given by
?(?)=(1+1sin?)?+(1+3sin2?)?+(1+2sin3?)?, 0≤?≤?2 and F is the radial
vector field ?(?,?,?)=??+??+??
compute the work done by F on a particle moving along C.

Let F(x,y,z) = yzi + xzj + (xy+2z)k
show that vector field F is conservative by finding a function f
such that
and use that to evaluate
where C is any path from (1,0,-2) to (4,6,3)

Consider the vector force field given by F⃗ = 〈2x + y, 3y +
x〉
(a) Let C1 be the straight line segment from (2, 0) to (−2,
0).
Directly compute ∫ C1 F⃗ · d⃗r (Do not use Green’s Theorem or
the Fundamental Theorem of Line Integration)
(b) Is the vector field F⃗ conservative? If it is not
conservative, explain why. If it is conservative, find its
potential function f(x, y)
Let C2 be the arc of the half-circle...

Let C be the circle with radius 1 and with center (−2,1), and
let f(x,y) be the square of the distance from the point (x,y) to
the origin.
Evaluate the integral ∫f(x,y)ds

Consider the funtion T(x,y)=2xy-y^2(°C) which determines the
temperature for a metallic circular plate centered at the origin
and radius = 5, Explain why there's no direction in which the rate
of change of temperature in the point P(1,-1) is equals to 5°C.
consider units are cm.

1.) Let f(x,y) =x^2+y^3+sin(x^2+y^3). Determine the line
integral of f(x,y) with respect to arc length over the unit circle
centered at the origin (0, 0).
2.)
Let f ( x,y)=x^3+y+cos( x )+e^(x − y). Determine the line
integral of f(x,y) with respect to arc length over the line segment
from (-1, 0) to (1, -2)

Given the force field F(x, y) = (x − y, 4x + y^2 ), find the
work done to move along a line segment from (0, 0) to (2,0), along
a line segment from (2,0) to (0,1), and then along another line to
the point (−2, 0). Show your work.

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