Question

Determine whether or not the vector field is conservative. If it is conservative, find a function f such that F = ∇f. (If the vector field is not conservative, enter DNE.)

F(x, y, z) = 8xyi + (4x2 + 10yz)j + 5y2k

Find: f(x, y, z) =

Answer #1

Determine whether or not F is a conservative
vector field. If it is, find a function f such that
F = ∇f. (If the vector field is not
conservative, enter DNE.)
F(x, y) = (y2 − 8x)i + 2xyj

Determine whether or not F is a conservative
vector field. If it is, find a function f such that
F = ∇f. (If the vector field is not
conservative, enter DNE.)
F(x, y) = (y2 − 8x)i + 2xyj

1. (a) Determine whether or not F is a conservative vector
field. If it is, find the potential function for F.
(b) Evaluate R C1 F · dr and R C2 F · dr where C1 is the
straight line path from (0, −1) to (3, 0), while C2 is the union of
two straight line paths: first piece from (0, −1) to (0, 0) and
then second piece from (0, 0) to (3, 0). (When applicable, use the
Fundamental...

For each of the following vector fields F , decide whether it is
conservative or not by computing curl F . Type in a potential
function f (that is, ∇f=F). If it is not conservative, type N. A.
F(x,y)=(−4x+3y)i+(3x+16y)j f(x,y)= B. F(x,y)=−2yi−1xj f(x,y)= C.
F(x,y,z)=−2xi−1yj+k f(x,y,z)= D. F(x,y)=(−2siny)i+(6y−2xcosy)j
f(x,y)= E. F(x,y,z)=−2x2i+3y2j+8z2k

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.

(1 point) For each of the following vector fields F , decide
whether it is conservative or not by computing curl F . Type in a
potential function f (that is, ∇f=F). If it is not conservative,
type N. A. F(x,y)=(10x+7y)i+(7x+10y)j f(x,y)= 10 B. F(x,y)=5yi+6xj
f(x,y)= N C. F(x,y,z)=5xi+6yj+k f(x,y,z)= D.
F(x,y)=(5siny)i+(14y+5xcosy)j f(x,y)= E.
F(x,y,z)=5x2i+7y2j+5z2k

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.

For the following vector fields F , decide
whether it is conservative or not by computing curl
F . Type in a potential function f (that is,
∇f=F∇f=F). If it is not conservative, type N.
F(x,y,z)=−2x2i+3y2j+8z2k

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 of the
following vector fields F , decide whether it is
conservative or not by computing curl F . Type in
a potential function f (that is, ∇f=F∇f=F
). If it is not conservative, type N.
A.
F(x,y)=(−10x+3y)i+(3x+10y)jF(x,y)=(−10x+3y)i+(3x+10y)j
f(x,y)=f(x,y)=
B.
F(x,y)=−5yi−4xjF(x,y)=−5yi−4xj
f(x,y)=f(x,y)=
C.
F(x,y,z)=−5xi−4yj+kF(x,y,z)=−5xi−4yj+k
f(x,y,z)=f(x,y,z)=
D.
F(x,y)=(−5siny)i+(6y−5xcosy)jF(x,y)=(−5siny)i+(6y−5xcosy)j
f(x,y)=f(x,y)=
E.
F(x,y,z)=−5x2i+3y2j+5z2kF(x,y,z)=−5x2i+3y2j+5z2k
f(x,y,z)=f(x,y,z)=
Note: Your answers should be either expressions of x, y and z
(e.g. "3xy + 2yz"), or the letter "N"

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