Question

Consider the vector field F(x,y,z)=〈−3y,−3x,−4z〉. Find a potential function f(x,y,z) for F which satisfies f(0,0,0)=0.

Answer #1

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F(x,y) =<3x^2+ 3y^2,6xy+ 2.>
Find the potential function for F that satisfies the condition
f(0,1) = 0.

Consider the following vector field.
F(x, y, z) =
6yz ln x i + (3x −
7yz) j +
xy8z3 k
(a)
Find the curl of F evaluated at the point
(5, 1, 4).
(b)
Find the divergence of F evaluated at the
point (5, 1, 4).

Suppose that the vector field F(x,y,z) has potential function
f(x,y,z) = y2+ 3xz and C is a path from (1,1,−1) to
(0,5,π). Compute ∫CF·dr.

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.

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

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...

Consider the vector field ?(?,?,?)=(3?+3?)?+(4?+3?)?+(4?+3?)?.
a) Find a function ? such that ?=∇? and ?(0,0,0)=0.
?(?,?,?)=
b) Suppose C is any curve from (0,0,0) to (1,1,1). Use part a)
to compute the line integral ∫C ?⋅??

Find a unit normal vector for the following function at the
point P(−1,3,−10): f(x,y)=ln(−x/(−3y−z))

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