Question

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 of radius 2 from (2, 0) to (−2, 0) with y ≥ 0. Compute ∫ C2 F⃗ · d⃗r

Explain your work and how your answer relates to the answer to part (a).

Answer #1

Sketch the vector field F⃗ (x,y)=−5i and calculate the line
integral of F⃗ along the line segment from (−5,3) to (0,4).

(a) Is the vector field F = <e^(−x) cos y, e^(−x) sin y>
conservative?
(b) If so, find the associated potential function φ.
(c) Evaluate Integral C F*dr, where C is the straight line path
from (0, 0) to (2π, 2π).
(d) Write the expression for the line integral as a single
integral without using the fundamental theorem of calculus.

For each vector field F~ (x, y) = hP(x, y), Q(x, y)i, find a
function f(x, y) such that F~ (x, y) = ∇f(x, y) = h ∂f ∂x , ∂f ∂y i
by integrating P and Q with respect to the appropriate variables
and combining answers. Then use that potential function to directly
calculate the given line integral (via the Fundamental Theorem of
Line Integrals):
a) F~ 1(x, y) = hx 2 , y2 i Z C F~ 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?

. a. [2] Compute the divergence of vector field F = x 3y 2 i +
yj − 3zx2y 2k
b. [7] Use divergence theorem to compute the outward flux of the
vector field F through the surface of the solid bounded by the
surfaces z = x 2 + y 2 and z = 2y

Suppose F⃗ (x,y)=〈x^2+5y,7x−3y^2〉. Use Green's Theorem to
calculate the circulation of F⃗ around the perimeter of the
triangle C oriented counter-clockwise with vertices (10,0), (0,5),
and (−10,0).

2. Consider the line integral I C F · d r, where the vector
field F = x(cos(x 2 ) + y)i + 2y 3 (e y sin3 y + x 3/2 )j and C is
the closed curve in the first quadrant consisting of the curve y =
1 − x 3 and the coordinate axes x = 0 and y = 0, taken
anticlockwise.
(a) Use Green’s theorem to express the line integral in terms of
a double...

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.

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)

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.

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