is conservative if we can find a scalar function
such that
. This would require
![\dfrac{\partial f}{\partial x}=5y^2z^3](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20x%7D%3D5y%5E2z%5E3)
![\dfrac{\partial f}{\partial y}=10xyz^3](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20y%7D%3D10xyz%5E3)
![\dfrac{\partial f}{\partial z}=15xy^2z^2](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20z%7D%3D15xy%5E2z%5E2)
Integrate both sides of the first PDE with respect to
:
(*)
Differentiate both sides of (*) with respect to
:
![\dfrac{\partial f}{\partial y}=10xyz^3=10xyz^3+\dfrac{\partial g}{\partial y}\implies\dfrac{\partial g}{\partial y}=0\implies g(y,z)=h(z)](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20y%7D%3D10xyz%5E3%3D10xyz%5E3%2B%5Cdfrac%7B%5Cpartial%20g%7D%7B%5Cpartial%20y%7D%5Cimplies%5Cdfrac%7B%5Cpartial%20g%7D%7B%5Cpartial%20y%7D%3D0%5Cimplies%20g%28y%2Cz%29%3Dh%28z%29)
Differentiate both sides of (*) with respect to
:
![\dfrac{\partial f}{\partial z}=15xy^2z^2=15xy^2z^2+\dfrac{\mathrm dh}{\mathrm dz}\implies\dfrac{\mathrm dh}{\mathrm dz}=0\implies h(z)=C](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20z%7D%3D15xy%5E2z%5E2%3D15xy%5E2z%5E2%2B%5Cdfrac%7B%5Cmathrm%20dh%7D%7B%5Cmathrm%20dz%7D%5Cimplies%5Cdfrac%7B%5Cmathrm%20dh%7D%7B%5Cmathrm%20dz%7D%3D0%5Cimplies%20h%28z%29%3DC)
So we have
![f(x,y,z)=5xy^2z^3+C](https://tex.z-dn.net/?f=f%28x%2Cy%2Cz%29%3D5xy%5E2z%5E3%2BC)
and so
is indeed conservative.
Answer:
the answer is 30
Step-by-step explanation:
hight x base²
8 foot roll I think because 3 yards in a foot 3.42 divided by 1 is 3.42 and 3.60 divided by 8 is 0.45
Answer:
Michael Glover is a familiar name
Answer:
Step-by-step explanation:
Percentage increase = increase/original price × 100/1
Increase = final price - initial price
In this case of 2012 - 2013
Initial price = $30, 000
Final price = $31, 200
Increase = $31, 200 - $30, 000
= $1200
% increase = $1200/$30000 × 100/1
= 0.04×100
= 4%
For the year 2013-2014
Initial price = $31, 200
Final price = $32,448
Increase = $32,448-$31, 200
= $1248
% increase = $1248/$31,200 ×100/1
= 0.04×100/1
= 4%
Hence,the percentage increase in tuition year will be at the rate of 4%