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d1i1m1o1n [39]
3 years ago
9

Use the Divergence Theorem to compute the net outward flux of the vector field

Mathematics
1 answer:
zepelin [54]3 years ago
4 0

\vec F(x,y,z)=(x^2,-y^2,z^2) has divergence

\mathrm{div}\vec F(x,y,z)=2x-2y+2z

so that by the divergence theorem, the flux of \vec F across the boundary of D is

\displaystyle\iint_{\partial D}\vec F\cdot\mathrm d\vec S=2\iiint_D(x-y+z)\,\mathrm dV

which you can compute by subtracting [the integral of \mathrm{div}\vec F over the region bounded by 9-x-y and the coordinate axes] and [the integral of \mathrm{div}\vec F over the region bounded by 3-x-y and the coordinate axes].

=2\displaystyle\left\{\int_0^9\int_0^{9-x}\int_0^{9-x-y}-\int_0^3\int_0^{3-x}\int_0^{3-x-y}\right\}(x-y+z)\,\mathrm dz\,\mathrm dy\,\mathrm dx

=2\left(\dfrac{2187}8-\dfrac{27}8\right)=\boxed{540}

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Observation one

From the markings on the diagram <1 = 60o The left triangle is at least isosceles. Therefore equal sides produce equal angles opposite them.

Now we have accounted for 2 angles that are equal (each is 60 degrees) and add up to 120 degrees. The third angle (angle 2) is found from this equation.

<1 + 60 + <2 = 180 degrees.  All triangles have 180 degrees.

60 + 60 + <2 = 180

Observation 2

<2 = 60 degrees.

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m<2 = 180 - 120

m<2 = 60 degrees.

Observation 3

m<3 = 120

<2 and <3 are supplementary.

Any 2 angles on the same straight line are supplementary

60 + <3 = 180

<3 = 180 - 60

<3 = 120

Observation 4

m<4 = 40 degrees.

All triangles have 180 degrees. No exceptions.

m<4 + 20 +m<3 = 180

m<4 + 20 + 120 = 180

m<4 + 140 = 180          

m<4 = 180 - 140

m<4 = 40


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