Question

Determine which of the vector fields is conservative (ie gradient vector fields). If the vector field is conservative find a function ? such that ∇? = ?⃗.

?⃗(?, ?, ?) =< 2??, ?^2+2yz^3, 3y^2z^2+1>

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) =

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

Which of the following vector fields are conservative?
(i)
F(x, y) =
(7x6y6 +
3) i +
(6x7y5 +
7) j
(ii)
F(x, y) =
(6ye6x +
sin 3y) i +
(e6x +
3x cos 3y) j
(iii)
F(x, y) =
7y2e7xy i
+ (7 +
xy) e7xy j

In the following functions: a) Find the gradient of f. , b)
Evaluate
the gradient at point P. and
c) Find the rate of change of f in P, in the direction of
vector.
1- f(x. y) = 5xy^2 - 4x^3y, P( I , 2), u = ( 5/13, 12/13 )
2- f(x, y, z) = xe^2yz , P(3, 0, 2), u = (2/3, -2/3, 1/3)

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∇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"

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 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

14a. Find the gradient vector field of ?(?, ?, ?) = (? ^2)(?)(?
^(?/z))
14b An object with mass m moves with position function ?⃗(?) = ?
sin(?) ?̂+ ? cos(?) ?̂+ ? ?̂?, 0 ≤ ? ≤ ?/2. Find the work done on
the object during this time period.
Calculus 3 question. Please help.

For each of the following vector fields F , decide whether it is
conservative or not by computing the appropriate first order
partial derivatives. Type in a potential function f (that is, ∇?=?)
with ?(0,0)=0. If it is not conservative, type N. A.
?(?,?)=(6?−6?)?+(−6?+12?)? ?(?,?)= -6xsiny + y^2 B. ?(?,?)=3??+4??
?(?,?)= N C. ?(?,?)=(3sin?)?+(−12?+3?cos?)? ?(?,?)= 0 Note: Your
answers should be either expressions of x and y (e.g. "3xy + 2y"),
or the letter "N"

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