Answer:
The first one isnt equivalent
Step-by-step explanation:
Ok imma not sure exactly how you would put this in but this is the answer:
Answer:
The point P shows the number of rides you have left on the card as well as the amount of money on the card
Step-by-step explanation:
On the x-axis, it says number of rides and since points are usually in a format of (x,y), the 7 would show what the x-axis is trying to represent. This is also the same for the y-axis.
400-126=274
400
126
-----
0-6 can't go into so make the 0 a 6
Then make 0-2 make 0 a 10 then a 9
Then make 4 into a 3
![\vec F(x,y,z)=y\,\vec\imath+x\,\vec\jmath+3\,\vec k](https://tex.z-dn.net/?f=%5Cvec%20F%28x%2Cy%2Cz%29%3Dy%5C%2C%5Cvec%5Cimath%2Bx%5C%2C%5Cvec%5Cjmath%2B3%5C%2C%5Cvec%20k)
is conservative if there is a scalar function
such that
. This would require
![\dfrac{\partial f}{\partial x}=y](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20x%7D%3Dy)
![\dfrac{\partial f}{\partial y}=x](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20y%7D%3Dx)
![\dfrac{\partial f}{\partial z}=3](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20z%7D%3D3)
(or perhaps the last partial derivative should be 4 to match up with the integral?)
From these equations we find
![f(x,y,z)=xy+g(y,z)](https://tex.z-dn.net/?f=f%28x%2Cy%2Cz%29%3Dxy%2Bg%28y%2Cz%29)
![\dfrac{\partial f}{\partial y}=x=x+\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%3Dx%3Dx%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)
![f(x,y,z)=xy+h(z)](https://tex.z-dn.net/?f=f%28x%2Cy%2Cz%29%3Dxy%2Bh%28z%29)
![\dfrac{\partial f}{\partial z}=3=\dfrac{\mathrm dh}{\mathrm dz}\implies h(z)=3z+C](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20z%7D%3D3%3D%5Cdfrac%7B%5Cmathrm%20dh%7D%7B%5Cmathrm%20dz%7D%5Cimplies%20h%28z%29%3D3z%2BC)
![f(x,y,z)=xy+3z+C](https://tex.z-dn.net/?f=f%28x%2Cy%2Cz%29%3Dxy%2B3z%2BC)
so
is indeed conservative, and the gradient theorem (a.k.a. fundamental theorem of calculus for line integrals) applies. The value of the line integral depends only the endpoints:
![\displaystyle\int_{(1,2,3)}^{(5,7,-2)}y\,\mathrm dx+x\,\mathrm dy+3\,\mathrm dz=\int_{(1,2,3)}^{(5,7,-2)}\nabla f(x,y,z)\cdot\mathrm d\vec r](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint_%7B%281%2C2%2C3%29%7D%5E%7B%285%2C7%2C-2%29%7Dy%5C%2C%5Cmathrm%20dx%2Bx%5C%2C%5Cmathrm%20dy%2B3%5C%2C%5Cmathrm%20dz%3D%5Cint_%7B%281%2C2%2C3%29%7D%5E%7B%285%2C7%2C-2%29%7D%5Cnabla%20f%28x%2Cy%2Cz%29%5Ccdot%5Cmathrm%20d%5Cvec%20r)
![=f(5,7,-2)-f(1,2,3)=\boxed{18}](https://tex.z-dn.net/?f=%3Df%285%2C7%2C-2%29-f%281%2C2%2C3%29%3D%5Cboxed%7B18%7D)