Insanity 4 years ago 1) Find the flux of the vector field F(x,y,z) = (e^-y) i - (y) j + (x sinz) k across σ wit outward orientation where σ is the portion of the elliptic cylinder r(u,v) = (2cos v) i + (sin v) j + (u) k with 0 ≤ u ≤ 5, 0 ≤ v ≤ 2pi. 2) Find the work done by the force field F(x, y, z) = (x + y) i + (xy) j - (z^2) k on a particle that moves along line segments from (0,0,0) to (1,3,1) to (2,-1,4).

1. Mathematics_pro

sorry wat 2 high 4 me

2. lalaly

$\int\limits{\int\limits{s F dS}} = ∫∫∫v ∇·F dV$ $∇·F = x \cos(z) - 1$ $= ∫∫∫v x \cos(z) - 1 dx dy dz$ $v: x² + 4y² ≤ 4 ; 0 ≤ z ≤ 5$ let u = x , v = 2y , z = z ∂(u,v)/∂(x,y) = 2 = 1/2 ∫∫∫v u cos(z) - 1 du dv dz v: u² + v² ≤ 4 ; 0 ≤ z ≤ 5 let u = r cos(theta) ; v = r sin(theta) z = z $\frac{∂(u,v)}{∂(r,θ)} = r$ $= 1/2 ∫∫∫ (r \cos(θ) \cos(z) - 1) r dr dθ dz$ ${(r,θ,z) | 0 ≤ r ≤ 2 ; 0 ≤ θ ≤ 2π ; 0 ≤ z ≤ 5}$ $= 1/3 ∫∫ (4 \cos(θ) \cos(z) - 3) dθ dz$ ${(θ,z) | 0 ≤ θ ≤ 2π ; 0 ≤ z ≤ 5}$ $= -2π ∫ dz$(z) | 0 ≤ z ≤ 5 $= -10π$

3. Mathematics_pro

oh 2 smaalllllllll

4. Insanity

you know the second one

5. lalaly

is the first one right? or u dont know tha answer?

6. Insanity

im just checking what i got wrong on my hw and yeah the answer is right

7. lalaly

oh ok im thinkin

8. Insanity

k thanks lana :)

9. lalaly

(0,0,0) -->> (1,3,1) x = t y = 3t z = t dx =dt........ dy = 3 dt...... dz = dt work done is integration of F.dr int{ (x + y)dx + (xy)dy - (z^2) dz

10. lalaly

now substitute x=t y=3t and z=t and the dx dy and dz values in terms of dt

11. lalaly

then integrate