Question

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).

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.

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 by the force field
F(x,y,z)=2xi+2yj+7kF(x,y,z)=2xi+2yj+7k
on a particle that moves along the helix
r(t)=3cos(t)i+3sin(t)j+4tk,0≤t≤2π

Find the work done by the force field F(x,y,z)=6xi+6yj+7k on a
particle that moves along the helix r(t)=5cos(t)i+5sin(t)j+7tk, 0 ≤
t≤ 2π

Let F~ (x, y, z) = x cos(x 2 + y 2 − z 2 )~i + y cos(x 2 + y 2 −
z 2 )~j − z cos(x 2 + y 2 − z 2 ) ~k be the force acting on a
particle at location (x, y, z). Under this force field, the
particle is moved from the point P = (1, 1, 1) to Q = (0, 0, √ π).
What is the work done by...

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.

(1 point) If C is the curve given by
r(t)=(1+5sint)i+(1+2sin2t)j+(1+3sin3t)k, 0≤t≤π2 and F is the radial
vector field F(x,y,z)=xi+yj+zk, compute the work done by F on a
particle moving along C.

Compute the work done by the force F= <sin(x+y), xy,
(x^2)z> in moving an object along the trajectory 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, 5) when force is
measured in Newtons and distance in meters.

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.

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