Consider the vector field F(x,y,z) =〈2xycos(z) +
1,x2cos(z) +zeyz,yeyz−x2ysin(z)〉.
Use the component test to determine if...
Consider the vector field F(x,y,z) =〈2xycos(z) +
1,x2cos(z) +zeyz,yeyz−x2ysin(z)〉.
Use the component test to determine if F is conservative. If it is,
find a potential function f.
Solutions
Expert Solution
Please thumbs up if it was helpful will be glad to know;)
Verify the Divergence Theorem for the vector field F(x, y, z) =
< y, x , z^2 > on the region E bounded by the planes y + z =
2, z = 0 and the cylinder x^2 + y^2 = 1.
By Surface Integral:
By Triple Integral:
Compute the line integral of the vector field F(x, y, z) = ⟨−y, x,
z⟩ along the curve which is given by the intersection of the
cylinder x 2 + y 2 = 4 and the plane x + y + z = 2 starting from
the point (2, 0, 0) and ending at the point (0, 2, 0) with the
counterclockwise orientation.
3. a) Consider the vector field F(x, y, z) = (2xy2 z, 2x 2 yz,
x2 y 2 ) and the curve r(t) = (sin t,sin t cost, cost) on the
interval [ π 4 , 3π 4 ]. Calculate R C F · dr using the definition
of the line integral. [5] b) Find a function f : R 3 → R so that F
= ∇f. [5] c) Verify your answer from (a) using (b) and the
Fundamental...
Let f(x,y) be a scalar function, and let F(x,y,z) be a vector
field. Only one of the following expressions is meaningful. Which
one?
a) grad f x div F
b) div(curl(grad f))
c) div(div F)
d) curl(div(grad f))
e) grad(curl F)
Compute the derivative of the given vector field F. Evaluate the line integral of
F(x,y,z) = (y+z+yz , x+z+xz , x+y+xy )over the path C consisting of line segments joining (1,1,1) to (1,1,2), (1, 1, 2) to (1, 3, 2), and (1, 3, 2) to (4, 3, 2) in 3 different ways, along the given path, along the line from (1,1,1) to (4,3,2), and finally by finding an anti-derivative, f, for F.
Consider the scalar functions
f(x,y,z)g(x,y,z)=x^2+y^2+z^2,
g(x,y,z)=xy+xz+yz,
and=h(x,y,z)=√xyz
Which of the three vector fields ∇f∇f, ∇g∇g and ∇h∇h are
conservative?
The vector field given by E
(x,y,z) = (yz – 2x)
x + xz y + xy
z may represent an electrostatic field?
Why? If so, finding the potential F a from which E may be
obtained.
Use the extended divergence theorem to compute the total flux of
the vector field
F(x, y, z) = −3x2 + 3xz − y, 2y3 − 6y, 9x2 + 4z2 − 3x outward
from the region F that lies inside the sphere x2 + y2 + z2 = 25 and
outside the solid cylinder x2 + y2 = 4 with top at z = 1 and bottom
at z = −1.