imthebestatcod
08.07.2021 •
Mathematics
Integrate G(x,y,z)=yz over the surface of x+y+z=1 in the first octant.
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Ответ:
Parameterize the surface (I'll call it S) by
r(u, v) = (1 - u) (1 - v) i + u (1 - v) j + v k
with 0 ≤ u ≤ 1 and 0 ≤ v ≤ 1.
Take the normal vector to this surface to be
n = ∂r/∂u × ∂r/∂v = ((v - 1) i + (1 - v) j) × ((u - 1) i - u j + k) = (1 - v) (i + j + k)
with magnitude
||n|| = √3 (1 - v)
Then in the integral, we have
Alternatively, if you're not familiar with parameterizing surfaces, you can use the "projection" formula:
where I write to mean the projection of the surface onto the (x, y)-plane, and z = f(x, y). We would then use
x + y + z = 1 ==> z = f(x, y) = 1 - x - y
and is the triangle,
{(x, y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 - x}
Then the integral becomes
Ответ:
In one hour there are 60 minutes. You earn $2000 per minute so you would do 60 times $2000 to get $120,000