I know how to do it only for x and y but not for x,y,z
#Potential Function
woven patio
granite hinge
You can check whether a function F of three variables is conservative by calculating its rotor ∇⨯F. If you forgot, here's what rotor looks like in cartesian coordinates for F = {F(x), F(y), F(z)}:
∇⨯F = {∂F(z)/∂y - ∂F(y)/∂z, ∂F(x)/∂z - ∂F(z)/∂x, ∂F(y)/∂x - ∂F(x)/∂y}
As long as the rotor of your function is a zero vector, the function is conservative. In other words, you need to have ∂F(z)/∂y = ∂F(y)/∂z, ∂F(x)/∂z = ∂F(z)/∂x, ∂F(y)/∂x = ∂F(x)/∂y, which was done here.
You can see that for a two-dimensional function that reduces to just ∂F(y)/∂x = ∂F(x)/∂y - a familiar condition, I'm sure.
Here's how you can calculate the rotor. Basically, you literally do it as a cross product of gradient and your function, which gives the vector I wrote above.
Remember: the rotor operator acts on a vector field and produces a vector field.
woven patio
Here's how you can calculate the rotor. Basically, you literally do it as a cros...
so i need to use this equation to find the potential function? for all x,y,z formulas that is 3 D
granite hinge
so i need to use this equation to find the potential function? for all x,y,z for...
Oh, no, that's just how you check whether F is a potential field.
To find the potential, you need to integrate F·dr.
granite hinge
so i need to use this equation to find the potential function? for all x,y,z for...
There are several ways to find the potential. I like the following.
We have F(x, y, z) = {u(x, y, z), v(x, y, z), w(x, y, z)}. So:
F·dr = u(x, y, z)dx + v(x, y, z)dy + w(x, y, z)dz
To integrate it, do the following:
- Integrate u(x, y, z)dx as usual.
- Remove the terms from v(x, y, z) that depend on x. After that, integrate v(y, z)dy.
- Remove the terms from w(x, y, z) that depend on x and y. After that, integrate w(z)dz.
- Add up the results from 1-3, then add a constant at the end.
That will be your potential. Example:
F = {2xyz^3 + ye^(xy), x^2 z^3 + xe^(xy), 3x^2 yz^2 + cos(z)}
You can verify that this is a potential field yourself. Let's find the potential.
First, we integrate (2xyz^3 + ye^(xy))dx, which gives x^2 yz^3 + e^(xy).
Then, we remove the terms from x^2 z^3 + xe^(xy) that contain x. Both terms do, so we are left with 0, so the integral is also 0.
After that, we remove terms from 3x^2 yz^2 + cos(z) that depend on x and y. The first term does, so we are left with cos(z), which integrates to sin(z).
Thus, adding all of that up and C, we get the potential to be f(x, y, z) = x^2 yz^3 + e^(xy) + sin(z) + C.
Try it for your example, it's even easier in your case.
woven patio
okay lemme try it