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lions [1.4K]
3 years ago
10

PLEASE HELP!!! Name the postulate or theorem that you can use to prove.....

Mathematics
1 answer:
bearhunter [10]3 years ago
5 0

Answer:

HL theorem

Step-by-step explanation:

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Evaluate the surface integral S F · dS for the given vector field F and the oriented surface S. In other words, find the flux of
tresset_1 [31]

Because I've gone ahead with trying to parameterize S directly and learned the hard way that the resulting integral is large and annoying to work with, I'll propose a less direct approach.

Rather than compute the surface integral over S straight away, let's close off the hemisphere with the disk D of radius 9 centered at the origin and coincident with the plane y=0. Then by the divergence theorem, since the region S\cup D is closed, we have

\displaystyle\iint_{S\cup D}\vec F\cdot\mathrm d\vec S=\iiint_R(\nabla\cdot\vec F)\,\mathrm dV

where R is the interior of S\cup D. \vec F has divergence

\nabla\cdot\vec F(x,y,z)=\dfrac{\partial(xz)}{\partial x}+\dfrac{\partial(x)}{\partial y}+\dfrac{\partial(y)}{\partial z}=z

so the flux over the closed region is

\displaystyle\iiint_Rz\,\mathrm dV=\int_0^\pi\int_0^\pi\int_0^9\rho^3\cos\varphi\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi=0

The total flux over the closed surface is equal to the flux over its component surfaces, so we have

\displaystyle\iint_{S\cup D}\vec F\cdot\mathrm d\vec S=\iint_S\vec F\cdot\mathrm d\vec S+\iint_D\vec F\cdot\mathrm d\vec S=0

\implies\boxed{\displaystyle\iint_S\vec F\cdot\mathrm d\vec S=-\iint_D\vec F\cdot\mathrm d\vec S}

Parameterize D by

\vec s(u,v)=u\cos v\,\vec\imath+u\sin v\,\vec k

with 0\le u\le9 and 0\le v\le2\pi. Take the normal vector to D to be

\vec s_u\times\vec s_v=-u\,\vec\jmath

Then the flux of \vec F across S is

\displaystyle\iint_D\vec F\cdot\mathrm d\vec S=\int_0^{2\pi}\int_0^9\vec F(x(u,v),y(u,v),z(u,v))\cdot(\vec s_u\times\vec s_v)\,\mathrm du\,\mathrm dv

=\displaystyle\int_0^{2\pi}\int_0^9(u^2\cos v\sin v\,\vec\imath+u\cos v\,\vec\jmath)\cdot(-u\,\vec\jmath)\,\mathrm du\,\mathrm dv

=\displaystyle-\int_0^{2\pi}\int_0^9u^2\cos v\,\mathrm du\,\mathrm dv=0

\implies\displaystyle\iint_S\vec F\cdot\mathrm d\vec S=\boxed{0}

8 0
3 years ago
To prove that ΔAED ˜ ΔACB by SAS, Jose shows that StartFraction A E Over A C EndFraction = StartFraction A D Over A B EndFractio
Ira Lisetskai [31]

Answer:

A on Edg.

Step-by-step explanation:

Took the Test

5 0
3 years ago
Read 2 more answers
describe the transformations that produce the graph of g(x)=1/2(x-4)^3+5 from the graph of the parent function f(x)=x^3. Give th
SSSSS [86.1K]

We have been given a parent function f(x)=x^{3} and we need to transform this function into g(x)=\frac{1}{2}(x-4)^{3}+5.

We will be required to use three transformations to obtain the required function from f(x)=x^{3}.

First transformation would be to shift the graph to the right by 4 units. Upon using this transformation, the function will change to g(x)=(x-4)^{3}.

Second transformation would be to compress the graph vertically by half. Upon using the second transformation, the new function becomes g(x)=\frac{1}{2}(x-4)^{3}.

Third transformation would be to shift the graph upwards by 5 units. Upon using this last transformation, we get the new function as g(x)=\frac{1}{2}(x-4)^{3}+5.

6 0
3 years ago
Please help I’m really stuck
motikmotik

Answer:

A is the correct answer

have a great day

7 0
3 years ago
Joy is reading a 352 page novel for her summer reading project on Monday she reads 3/8 of the novel on Tuesday she reads 28 page
MAXImum [283]

Answer:

104 Pages

Step-by-step explanation:

on Monday she reads 3/8 of the novel which means

\frac{3}{8} x 352 = 132 pages

on Tuesday she reads 28 pages, doesn't require any calculations.

on Wednesday she reads 1/4 of the novel,

\frac{1}{4} x 352 = 88

Just add all of that,

132+28+88=248 Pages

Subtract the novel pages by the read pages value

352-248=104pages.

8 0
3 years ago
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