Question

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 · d~r C is the line segment from (1, 0) to (3, −2).

b) F~ 2(x, y) = h2xy, x2 + 2yi Z C F~ 2 · d~r C is the parabola ~r(t) = ht 2 + 2, −3ti, 0 ≤ t ≤ 1.

Answer #1

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1. a) Let F(x,y) = hcosy,−xsiny + 2yi. Show that F is
conservative, and find a function
φ such that ∇φ(x,y) = F(x,y).
b) Use the result from part a) to find
R
C F · Tds, where C is given by r(t) = ht,πti,0 ≤
t ≤ 1.

(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.

Sketch the central field F = (x /(x2 +
y2)1/2)i + (y /(x2 +
y2)1/2) j and the curve C consisting of the
parabola y = 2 − x2 from (−1, 1) to (1, 1) to determine
whether you expect the work done by F on a particle moving along C
to be positive, null, or negative. Then compute the line integral
corresponding to the work.

1.) Let f ( x , y , z ) = x ^3 + y + z + sin ( x + z ) + e^( x
− y). Determine the line integral of f ( x , y , z ) with respect
to arc length over the line segment from (1, 0, 1) to (2, -1,
0)
2.) Letf ( x , y , z ) = x ^3 * y ^2 + y ^3 * z^...

Evaluate the line integral ∫CF⋅dr, where F(x,y,z)=5xi+yj−2zk and
C is given by the vector function r(t)=〈sint,cost,t〉, 0≤t≤3π/2.

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...

Compute the line integral of f(x, y, z) = x 2 + y 2 −
cos(z) over the following paths:
(a) the line segment from (0, 0, 0) to (3, 4, 5)
(b) the helical path → r (t) = cos(t) i + sin(t) j + t k from
(1, 0, 0) to (1, 0, 2π)

(1 point)
Evaluate the line integral ∫CF⋅dr∫CF⋅dr, where F(x,y,z)=3xi+4yj-zk
and C is given by the vector function r(t)=〈sint,cost,t〉,
0≤t≤3π/2.

Verify the Divergence Theorem for the vector field F(x, y, z) =
< y, x , z^2 > on the region E bounded by the planes y + z =
2, z = 0 and the cylinder x^2 + y^2 = 1.
By Surface Integral:
By Triple Integral:

Suppose that the vector field F(x,y,z) has potential function
f(x,y,z) = y2+ 3xz and C is a path from (1,1,−1) to
(0,5,π). Compute ∫CF·dr.

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