- #1
Gridvvk
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Homework Statement
Find the flux of the following fields:
F_1 = xi + yj
F_2 = -yi + xj
across the following curve: The circle r(t) = (cost) i + (sint) j
t is from [0,2pi]
Homework Equations
Flux = ∫F dot n ds = ∫M dy - N dx
The Attempt at a Solution
For F_1 I got:
M = x = cos t
N = y = sin t
dy = cost t dt
dx = -sin t dt
Flux for F_1 = ∫[0,2pi] cos^2 t + sin^2 t dt = ∫[0,2pi] dt => 2pi
For F_2 I got:
M = -y = -sin t
N = x = cos t
dy = cos t dt
dx = - sin t dt
Flux for F_2 = ∫[0, 2pi] -costsint + costsint dt = ∫0 = 0
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My answers are correct (can someone verify if method was used correctly?), but the answer key used a different method:
It said "n = <cost , sin t>, and proceeded to dot that with each field.
My concern was how n was found, I thought n = T' / |T'|, where T = r' / |r'| :
so: r' = <-sint, cos t> = T (because |r'| = 1)
T' => <-cost, -sint> = n (because |T'| = 1 as well), so how did they get their n?
Also, I'm a bit unsure on how Flux = ∫M dy - N dx is derived -- in the textbook they did:
n = T x K = (dx / ds i + dy/ ds j) x k = (dy / ds i - dx/ ds j), but if someone can spell that out for me I'd appreciate it.
Thanks