Consider F and C below.
F(x, y, z) = yz i + xz j + (xy + 14z) k
C is the line segment from (3, 0, −1) to (6, 4, 2)
(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.
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?
Consider F and C below.
F(x, y, z) =
2xz + y2
i + 2xy j +
x2 + 6z2
k
C: x = t2, y = t +
2, z = 3t − 1, 0 ≤ t
≤ 1
(a) Find a function f such that F =
∇f.
f(x, y, z) =
x2z+xy2+2z3+c
(b) Use part (a) to evaluate
C
∇f · dr along the given curve C.
Consider F and C below.
F(x, y, z) =
2xz + y2
i + 2xy j +
x2 + 15z2
k
C: x = t2, y = t +
2, z = 4t − 1, 0 ≤ t
≤ 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.
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.
Consider the vector fields F = ze^xz cos y i − e^xz sin y j +
xe^xz cos y k G = x^2 z i + xy^2 j + yz^3 k (a) Compute div F. (b)
Use curl to determine which of F and G is conservative. (c) Find a
function f such that your answer to part (b) is equal to ∇f. (d)
Find ∇^2f.
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...
The electrical voltage in a certain region of space is given by
the function V(x,y,z)=80+xz−sin(yz). If the directional derivative
of V at (1,1,π) in the direction 〈a,−1,π〉 is π, what is the value
of a?
Group of answer choices
π22
−π22
π2
−π2
None of the above.