Question

(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.

Answer #1

2. Is the vector field F = < z cos(y), −xz sin(y), x
cos(y)> conservative? Why or why not? If F is conservative, then
find its potential function.

Problem 7. Consider the line integral Z C y sin x dx − cos x
dy.
a. Evaluate the line integral, assuming C is the line segment
from (0, 1) to (π, −1).
b. Show that the vector field F = <y sin x, − cos x> is
conservative, and find a potential function V (x, y).
c. Evaluate the line integral where C is any path from (π, −1)
to (0, 1).

a. Is F(x,y,z)= <(e^z)siny,(e^z)cosx,(e^x)siny> a
conservative vector field? Justify.
b. Is F incompressible? Explain. Is it irrotational?
Explain.
c. The vector field F(x,y,z)= < 6xy^2+e^z, 6yx^2
+zcos(y),sin(y)xe^z > is conservative. Find the potential
function f. That is, the function f such that ▽f=F. Use a
process.

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?

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...

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)

F(x, y, z) = sin y i + (x.cos y + cos z) j – y.sinz k
a) Determine whether or not the vector field is
conservative.
b) If it is conservative, find the function f such that F = ∇f
.

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...

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)

Identify the surface with parametrization x = 3 cos θ sin φ, y =
3 sin θ sin φ, z = cos φ where 0 ≤ θ ≤ 2π and 0 ≤ φ ≤ π. Hint: Find
an equation of the form F(x, y, z) = 0 for this surface by
eliminating θ and φ from the equations above. (b) Calculate a
parametrization for the tangent plane to the surface at (θ, φ) =
(π/3, π/4).

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