Question

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) = (y^{2} − 8x)i + 2xyj

Answer #1

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) =

Consider the vector field →F=〈3x+7y,7x+5y〉F→=〈3x+7y,7x+5y〉
Is this vector field Conservative? yes or no
If so:
Find a function ff so that →F=∇fF→=∇f
f(x,y) =_____ + K
Use your answer to evaluate ∫C→F⋅d→r∫CF→⋅dr→ along the curve C:
→r(t)=t2→i+t3→j, 0≤t≤3r→(t)=t2i→+t3j→, 0≤t≤3

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.

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

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.

(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

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"

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)

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...

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