Parameterize the surface (call it
) that has
as its boundary by
![\vec s(u,v)=u\cos v\,\vec\imath+u\sin v\,\vec\jmath+(81-u^2)\,\vec k](https://tex.z-dn.net/?f=%5Cvec%20s%28u%2Cv%29%3Du%5Ccos%20v%5C%2C%5Cvec%5Cimath%2Bu%5Csin%20v%5C%2C%5Cvec%5Cjmath%2B%2881-u%5E2%29%5C%2C%5Cvec%20k)
with
and
.
Take the normal vector to
to be
![\vec n=\dfrac{\partial\vec s}{\partial u}\times\dfrac{\partial\vec s}{\partial v}=2u^2\cos v\,\vec\imath+2u^2\sin v\,\vec\jmath+u\,\vec k](https://tex.z-dn.net/?f=%5Cvec%20n%3D%5Cdfrac%7B%5Cpartial%5Cvec%20s%7D%7B%5Cpartial%20u%7D%5Ctimes%5Cdfrac%7B%5Cpartial%5Cvec%20s%7D%7B%5Cpartial%20v%7D%3D2u%5E2%5Ccos%20v%5C%2C%5Cvec%5Cimath%2B2u%5E2%5Csin%20v%5C%2C%5Cvec%5Cjmath%2Bu%5C%2C%5Cvec%20k)
Compute the curl of
. We have
![\nabla\times\vec F(x,y,z)=4y\,\vec\imath-4x\,\vec\jmath](https://tex.z-dn.net/?f=%5Cnabla%5Ctimes%5Cvec%20F%28x%2Cy%2Cz%29%3D4y%5C%2C%5Cvec%5Cimath-4x%5C%2C%5Cvec%5Cjmath)
![\implies\nabla\times\vec F(u,v)=4u\sin v\,\vec\imath-4u\cos v\,\vec\jmath](https://tex.z-dn.net/?f=%5Cimplies%5Cnabla%5Ctimes%5Cvec%20F%28u%2Cv%29%3D4u%5Csin%20v%5C%2C%5Cvec%5Cimath-4u%5Ccos%20v%5C%2C%5Cvec%5Cjmath)
Then by Stoke's theorem, the line integral is
![\displaystyle\int_C\vec F\cdot\mathrm d\vec r=\iint_S(\nabla\times\vec F(x,y,z))\cdot\mathrm d\vec S](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint_C%5Cvec%20F%5Ccdot%5Cmathrm%20d%5Cvec%20r%3D%5Ciint_S%28%5Cnabla%5Ctimes%5Cvec%20F%28x%2Cy%2Cz%29%29%5Ccdot%5Cmathrm%20d%5Cvec%20S)
![=\displaystyle\iint_S(\nabla\times\vec F(u,v))\cdot\vec n\,\mathrm du\,\mathrm dv](https://tex.z-dn.net/?f=%3D%5Cdisplaystyle%5Ciint_S%28%5Cnabla%5Ctimes%5Cvec%20F%28u%2Cv%29%29%5Ccdot%5Cvec%20n%5C%2C%5Cmathrm%20du%5C%2C%5Cmathrm%20dv)
![=\displaystyle\int_0^{2\pi}\int_0^90\,\mathrm du\,\mathrm dv=\boxed{0}](https://tex.z-dn.net/?f=%3D%5Cdisplaystyle%5Cint_0%5E%7B2%5Cpi%7D%5Cint_0%5E90%5C%2C%5Cmathrm%20du%5C%2C%5Cmathrm%20dv%3D%5Cboxed%7B0%7D)
We can verify this result by computing the line integral directly. Parameterize
by
![\vec r(t)=9\cos t\,\vec\imath+9\sin t\,\vec\jmath](https://tex.z-dn.net/?f=%5Cvec%20r%28t%29%3D9%5Ccos%20t%5C%2C%5Cvec%5Cimath%2B9%5Csin%20t%5C%2C%5Cvec%5Cjmath)
with
. Then
![\displaystyle\int_C\vec F(x,y,z)\cdot\mathrm d\vec r=\int_0^{2\pi}\vec F(t)\cdot\frac{\mathrm d\vec r}{\mathrm dt}\,\mathrm dt](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint_C%5Cvec%20F%28x%2Cy%2Cz%29%5Ccdot%5Cmathrm%20d%5Cvec%20r%3D%5Cint_0%5E%7B2%5Cpi%7D%5Cvec%20F%28t%29%5Ccdot%5Cfrac%7B%5Cmathrm%20d%5Cvec%20r%7D%7B%5Cmathrm%20dt%7D%5C%2C%5Cmathrm%20dt)
![=\displaystyle\int_0^{2\pi}(9\cos t\,\vec\imath+9\sin t\,\vec\jmath+162\,\vec k)\cdot(-9\sin t\,\vec\imath+9\cos t\,\vec\jmath)\,\mathrm dt](https://tex.z-dn.net/?f=%3D%5Cdisplaystyle%5Cint_0%5E%7B2%5Cpi%7D%289%5Ccos%20t%5C%2C%5Cvec%5Cimath%2B9%5Csin%20t%5C%2C%5Cvec%5Cjmath%2B162%5C%2C%5Cvec%20k%29%5Ccdot%28-9%5Csin%20t%5C%2C%5Cvec%5Cimath%2B9%5Ccos%20t%5C%2C%5Cvec%5Cjmath%29%5C%2C%5Cmathrm%20dt)
![=\displaystyle\int_0^{2\pi}0\,\mathrm dt=\boxed{0}](https://tex.z-dn.net/?f=%3D%5Cdisplaystyle%5Cint_0%5E%7B2%5Cpi%7D0%5C%2C%5Cmathrm%20dt%3D%5Cboxed%7B0%7D)
"Porportional Relationships" means that the rate in which it increases is the same.
For example: 1, 3, 5, 7, 9 is a porportional relationship, because it increases each time by (+2).
Now, look at the first one.
From y to x, y being 2, x being 10, you divided 5 from x to get 2. Next, you divided 6 from 30 to get 5. Because the number dividing is different, it is not proportional
NOT PROPORTIONAL
Look at the second one.
From y to x, y being 12, x being 10, you divided 1.2
Now, look at the next one. 30/25 = 1.2
Finally, the last one. 48/40 = 1.2
Because all of them are 1.2, the answer is proportional
PROPORTIONAL
~
Answer:
there will be £1240
Step-by-step explanation:
1000/6=60
4 years so 60x4=240
240+1000=1240
Hello.
The answer to your question is =
12, 3 x 4 and 6 x 2.
24, 3 x 8 and 6 x 4.
36, 3 x 12 and 6 x 6.
48, 3 x 16 and 6 x 8.
So it's pretty much everything except for 32.
Have a great day!
Answer:
480 Litres
Step-by-step explanation:
Percentage increase = 20% × 400
New value =
400 + Percentage increase =
400 + (20% × 400) =
400 + 20% × 400 =
(1 + 20%) × 400 =
(100% + 20%) × 400 =
120% × 400 =
120 ÷ 100 × 400 =
120 × 400 ÷ 100 =
48,000 ÷ 100 =
480