Question

Consider **F** and *C* below.

F(x, y, z) = yz i + xz j + (xy + 12z) k

C is the line segment from (2, 0, −3) to (4, 6, 3)

(a) Find a function *f* such that **F** =
∇*f*.

f(x, y, z) =

(b) Use part (a) to evaluate

C |

∇f · dr along the given curve *C*.

Answer #1

Consider F and C below.
F(x, y,
z) = yz i +
xz j + (xy +
18z) k
C is the line segment from (1, 0, −3) to (4,
4, 1)
(a) Find a function f such that F =
∇f.
f(x, y,
z) =
(b) Use part (a) to evaluate
C
∇f · dr
along the given curve C.

Find the work done by the force ﬁeld F(x,y,z) = yz i + xz j + xy
k acting along the curve given by r(t) = t3 i + t2 j + tk from the
point (1,1,1) to the point (8,4,2).

1. a) For the surface f(x, y, z) = xy + yz + xz = 3, find the
equation of the tangent plane at (1, 1, 1).
b) For the surface f(x, y, z) = xy + yz + xz = 3, find the
equation of the normal line to the surface at (1, 1, 1).

Let F(x, y, z) = (yz, xz, xy) and the path c(t) = (cos3 t,sin3
t, 0) for 0 ≤ t ≤ 2π. Evaluate R c F · ds. Hint: Identify f such
that ∇f = F.

find the work done in the force camp F(x,y,z)=<xz,xy,zy> in a
particle that moves along the curve <t^2,-t^3,t^4> for 0
<= t <= 1
THE
F is F(x,y,z)= <xz,yx,zy>

Evaluate using Stokes' theorem: a) ∬ ∇ × ? ∙ ???, if F = (xy,
yz, xz) in a cylinder z = 1-x two for 0≤ x ≤1, -2 ≤y ≤2

Evaluate the outward flux ∫∫S(F·n)dS of the vector
fieldF=yz(x^2+y^2)i−xz(x^2+y^2)j+z^2(x^2+y^2)k, where S is the
surface of the region bounded by the hyperboloid x^2+y^2−z^2= 1,
and the planes z=−1 and z= 2.

Evaluate the vector line integral F*dr of F(x,y) = <xy,y>
along the line segment K from the point (2,0) to the point (0,2) in
the xy-plane

Use the divergence theorem to calculate the flux of the vector
field F = (y +xz) i+ (y + yz) j - (2x + z^2) k upward through the
first octant part of the sphere x^2 + y^2 + z^2 = a^2.

Compute the line integral with respect to arc length of the
function f(x, y, z) = xy^2 along the parametrized curve that is the
line segment from (1, 1, 1) to (2, 2, 2) followed by the line
segment from (2, 2, 2) to (−3, 6, 8).

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