Question

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 .

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).

Consider the vector field.
F(x, y, z) =
7ex sin(y), 7ey sin(z), 8ez sin(x)
(a) Find the curl of the vector field.
curl F =
(b) Find the divergence of the vector field.
div F =

Consider the vector field.
F(x, y,
z) =
6ex
sin(y),
7ey
sin(z),
5ez
sin(x)
(a) Find the curl of the vector field.
curl F =
(b) Find the divergence of the vector field.

Consider the vector field. F(x, y, z) = 9ex sin(y), 9ey sin(z),
2ez sin(x) (a) Find the curl of the vector field. curl F = (b) Find
the divergence of the vector field.

4.Given F(x,y,z)=(cos(y))i+(sin(y))j+k, find divF and curlF at
P0(π/4,π,0) divF(P0)=? curlF(P0)= ?

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.

(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.

Let F~ (x, y, z) = x cos(x 2 + y 2 − z 2 )~i + y cos(x 2 + y 2 −
z 2 )~j − z cos(x 2 + y 2 − z 2 ) ~k be the force acting on a
particle at location (x, y, z). Under this force field, the
particle is moved from the point P = (1, 1, 1) to Q = (0, 0, √ π).
What is the work done by...

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)

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