Let F(x,y,z) = < z tan-1(y2),
z3 ln(x2 + 1), z >. Find the flux of F
across S, the top part of the paraboloid x2 +
y2 + z = 2 that lies above the plane z = 1 and is
oriented upward. Note that S is not a closed surface.
Let F(x, y, z) = z tan^−1(y^2)i + z^3 ln(x^2 + 7)j + zk. Find
the flux of F across S, the part of the paraboloid x2 + y2 + z = 29
that lies above the plane z = 4 and is oriented upward.
Let F= (x2 +
y + 2 + z2) i + (exp(
x2 ) + y2)
j + (3 + x) k . Let a
> 0 and let S be part of the spherical
surface x2 + y2 +
z2 = 2az + 15a2
that is above the x-y plane and the disk formed in the
x-y plane by the circular intersection between the sphere
and the plane. Find the flux of F outward across
S.
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.
Let F(x, y, z) =
(6x5 ln(5y2 + 3) +
10z4) i +
((( 10yx6)/(5y2 + 3))+
8z) j + (40xz3 +
8y − 6π sin πz) k
and let r(t) = (t3
+ 1) i + (t2 +
2) j +
t3 k , 0 ≤ t ≤
1. Evaluate
∫
C
F · dr
(please explain steps, thank you :)
The curried version of let f (x,y,z) = (x,(y,z)) is
let f (x,(y,z)) = (x,(y,z))
Just f (because f is already curried)
let f x y z = (x,(y,z))
let f x y z = x (y z)