What Is the Potential Function U for a Given Gradient?

[SammyS]

So $f(y,z) = y^4 z^2 + y^6 / 3 + y^2 z^4 + g(z)$?
And then do the same thing with z?

At this point, you may be able guess what g(z) is.

- and to quote fresh 42, "Don't forget the constant ..."

 

[ehild]

Homework Statement

$\nabla U = 2 r^4 \vec r$ Find U.

The method you follow will result in the potential function, but there is a simpler way when the gradient is given in terms of the position vector $\vec r$: $\nabla U = 2 ((\vec r)^2)^2 \vec r$
You did not include among the "Relevant equations" how the gradient of a function is defined.
So the total differential of a function U is $dU= \nabla (U) \cdot \vec{dr}$. ($\nabla (U)$ is a vector.)
You get the change of U from point O (the origin) to point P if you integrate both sides between O and P.
$\Delta U = U(P)-U(O)=\int_O^P{\nabla U}\cdot \vec{dr}$. In this problem $\nabla U = 2 r^4 \vec r$,
so you have the integral $\int_O^P{2 r^4 \vec r \cdot \vec {dr}}$.
You can choose the integration path arbitrary, the integral will be the same. See figure: You can reach P from O along several paths, why not along the straight line connecting them. In this case, $\vec r$ and $\vec {dr}$ have the same direction, so $\vec r \cdot \vec {dr}=r dr$, and the integral becomes $\int_0^{r(P)}{2 r^4 r dr}$.

 

[ehild]

Homework Statement

$\nabla U = 2 r^4 \vec r$ Find U.

The Attempt at a Solution

$\nabla U = 2 (x^2 + y^2 + z^2)^2 (x \hat i + y \hat j + z \hat j)$

I multiplied everything out,

$\nabla U = (2 x^4 + 4 x^2 y^2 + 4 x^2 z^2 + 4 y^2 z^2 + 2 y^4 + 2 z^4)x\hat i + (2 x^4 + 4 x^2 y^2 + 4 x^2 z^2 + 4 y^2 z^2 + 2 y^4 + 2 z^4)y\hat j + (2 x^4 + 4 x^2 y^2 + 4 x^2 z^2 + 4 y^2 z^2 + 2 y^4 + 2 z^4)z\hat k$

.

It was not necessary to expand the square.
You know that
$\frac{\partial U}{\partial x} = 2 x(x^2 + y^2 + z^2 )^2$.
Integral it with respect to x. Notice that you can do u-substitution with u= x²+y²+z². What do you get? Include the integration constant, which is a function of y and z: (f(yz).
So your integral with respect to x becomes $\frac{(x^2+y^2+z^2)^3 }{3}+f(yz)$
Take the partial derivative of the above expression with respect y. It must be equal to $2 y(x^2 + y^2 + z^2 )^2$. So what is the partial derivative of f(x,y) with respect to y?

 

[Amrator]

.

It was not necessary to expand the square.
You know that
$\frac{\partial U}{\partial x} = 2 x(x^2 + y^2 + z^2 )^2$.
Integral it with respect to x. Notice that you can do u-substitution with u= x²+y²+z². What do you get? Include the integration constant, which is a function of y and z: (f(yz).
So your integral with respect to x becomes $\frac{(x^2+y^2+z^2)^3 }{3}+f(yz)$
Take the partial derivative of the above expression with respect y. It must be equal to $2 y(x^2 + y^2 + z^2 )^2$. So what is the partial derivative of f(x,y) with respect to y?

f(x, y)? Don't you mean g(z)?

 

[ehild]

f(x, y)? Don't you mean g(z)?

A stupid error! I meant f(y,z) and then g(z).

 

[Amrator]

.

It was not necessary to expand the square.
You know that
$\frac{\partial U}{\partial x} = 2 x(x^2 + y^2 + z^2 )^2$.
Integral it with respect to x. Notice that you can do u-substitution with u= x²+y²+z². What do you get? Include the integration constant, which is a function of y and z: (f(yz).
So your integral with respect to x becomes $\frac{(x^2+y^2+z^2)^3 }{3}+f(yz)$
Take the partial derivative of the above expression with respect y. It must be equal to $2 y(x^2 + y^2 + z^2 )^2$. So what is the partial derivative of f(x,y) with respect to y?

I don't understand. How do you find f(y, z) using this method?

 

[SammyS]

I don't understand. How do you find f(y, z) using this method?

What is your result for $\displaystyle \ \frac{\partial f(y,z)}{\partial y} \$ if you use ehild's method ?

 

[Amrator]

What is your result for $\displaystyle \ \frac{\partial f(y,z)}{\partial y} \$ if you use ehild's method ?

${\partial U} / {\partial y} = 2 y (x^2 + y^2 + z^2)^2 + {\partial f(y,z)} / {\partial y}$
Setting that equal to the $\hat j$ component will give you ${\partial f(y,z)} / {\partial y} = 0$.

 

[Ray Vickson]

I don't understand. How do you find f(y, z) using this method?

You find $f(y,z)$ by first finding $\partial f(y,z) /\partial y$ and $\partial f(y,z) /\partial z$ . So, go ahead and do that: find these partial derivatives, using the method that has been explained to you many times already.

 

[Amrator]

You find $f(y,z)$ by first finding $\partial f(y,z) /\partial y$ and $\partial f(y,z) /\partial z$ . So, go ahead and do that: find these partial derivatives, using the method that has been explained to you many times already.

$\partial f(y,z) /\partial y = 0$
$\partial f(y,z) /\partial z = 0$
Yeah, that doesn't seem right to me.

 

[Amrator]

$\nabla U = 2 x (x^2 + y^2 + z^2)^2 \hat i + 2 y (x^2 + y^2 + z^2)^2 \hat j +2 z (x^2 + y^2 + z^2)^2 \hat k$
$U = \int \partial U /\partial x dx = (x^2 + y^2 + z^2)^3 / 3 + f(y,z)$
$\partial U /\partial y = 2 y (x^2 + y^2 + z^2)^2 + \partial f(y,z) /\partial y$
$2 y (x^2 + y^2 + z^2)^2 + \partial f(y,z) /\partial y = 2 y (x^2 + y^2 + z^2)^2$
$\partial f(y,z) /\partial y = 0$

 

[Ray Vickson]

$\partial f(y,z) /\partial y = 0$
$\partial f(y,z) /\partial z = 0$
Yeah, that doesn't seem right to me.

Why not? It IS what you get!

So, assuming you believe your own work, what do those two formulas above tell you about $f(y,z)$?

 

[Amrator]

Why not? It IS what you get!

So, assuming you believe your own work, what do those two formulas above tell you about $f(y,z)$?

f(y, z) = g(z)?

 

[vela]

f(y, z) = g(z)?

Are you just guessing?

 

[Amrator]

Are you just guessing?

Well the integral of 0 is a constant of integration.

 

[fresh_42]

Well the integral of 0 is a constant of integration.

Yes. And what does it mean to be constant w.r.t. y and z?

 

[Amrator]

Yes. And what does it mean to be constant w.r.t. y and z?

Oh right, f(y,z) = x since y and z have constant slopes.

 

[fresh_42]

Oh right, f(y,z) = x since y and z have constant slopes.

But f(y,z) isn't a function of x. How can this be?

 

[Amrator]

But f(y,z) isn't a function of x. How can this be?

My mistake. f(y,z) = y + g(z).
Although, what about the z?
Yesterday, I had f(y,z) = $y^4 z^2 + y^6 / 3 + y^2 z^4 + g(z)$.

 

[fresh_42]

You already have had it. Just add up what you've said:
$f(y,z)$ does not have $x$ as variable and it's dependency on $y$ is constant, as you have said. But the partial derivative w.r.t. $z$ is also zero. And you have said that means $f$ does not change if $z$ does. So it has to be?

 

[fresh_42]

My mistake. f(y,z) = y + g(z).
Although, what about the z?
Yesterday, I had f(y,z) = $y^4 z^2 + y^6 / 3 + y^2 z^4 + g(z)$.

That's been a different $f$. In both cases $f$ is just a term within a calculation. 2 calculations, different terms within.
Imagine a polynomial, like $a_{390567} z^{28768} y^{9808} + ... + a_{57657} z^2 + a_{764} y + 5$. The numbers are just to prevent you from calculating with it! Now, if you differentiate this w.r.t. $y$ and receive $0$ and then w.r.t $z$ and receive $0$ again? How does it look like?

 

[Amrator]

That's been a different $f$. In both cases $f$ is just a term within a calculation. 2 calculations, different terms within.
Imagine a polynomial, like $a_{390567} z^{28768} y^{9808} + ... + a_{57657} z^2 + a_{764} y + 5$. The numbers are just to prevent you from calculating with it! Now, if you differentiate this w.r.t. $y$ and receive $0$ and then w.r.t $z$ and receive $0$ again? How does it look like?

Oh, then it's C.

 

[fresh_42]

ευρηκα!

 

[Amrator]

Therefore $U = (x^2 + y^2 + z^2)^3 / 3 + C = r^3 / 3 + C$

Question: How would I write "integral of f(y,z) w.r.t. y and z = C" symbolically? What would the notation look like?

 

[fresh_42]

Therefore $U = (x^2 + y^2 + z^2)^3 / 3 + C = r^3 / 3 + C$

Question: How would I write "integral of f(y,z) w.r.t. y and z = C" symbolically? What would the notation look like?

Firstly, if I remember the definition of $r$ right, then $U(x,y,z) = U(r)= \frac{1}{3}r^6 +C$.
Secondly, $f$ itself is constant, not its integral. So I would write $f(y,z) = C$. It is the direct consequence of the equations $\frac{\partial }{\partial y} f(y,z) = 0$ and $\frac{\partial }{\partial z} f(y,z) = 0$ holding both.

Thirdly, if you look up what Chestermiller said $\frac{dU}{dr} = 2 r^5$ you could have gotten the result immediately.
Nevertheless, it's been a good exercise. And to be honest, I've done the long way first as well.

 

[Amrator]

Firstly, if I remember the definition of $r$ right, then $U(x,y,z) = U(r)= \frac{1}{3}r^6 +C$.
Secondly, $f$ itself is constant, not its integral. So I would write $f(y,z) = C$. It is the direct consequence of the equations $\frac{\partial }{\partial y} f(y,z) = 0$ and $\frac{\partial }{\partial z} f(y,z) = 0$ holding both.

Thirdly, if you look up what Chestermiller said $\frac{dU}{dr} = 2 r^5$ you could have gotten the result immediately.
Nevertheless, it's been a good exercise. And to be honest, I've done the long way first as well.

Sorry, I meant to say the integral of PD of f.

By what Chestermiller said, you mean spherical coordinates? I'll go learn gradients using spherical coordinates right now then.

 

[fresh_42]

Sorry, I meant to say the integral of PD of f.

I don't know whether there is an elegant short notation. Basically it is the same as we did before:

$\frac{\partial}{\partial y}f(y,z) = 0 ⇒ f(y,z) = f(z) + C' ⇒ 0 = \frac{\partial}{\partial z}f(y,z) = \frac{\partial}{\partial z} (f(z) + C') = \frac{\partial}{\partial z} f(z) ⇒ f(y,z) = f(z) = f = constant = C$

 

[Amrator]

Alright, thanks for the help everyone.

 

[fresh_42]

By what Chestermiller said, you mean spherical coordinates? I'll go learn gradients using spherical coordinates right now then.

In this case you should definitely look up ehild's first post (#32). The one with the figure. (S)he explained it very well. Mainly how $r^4 \vec r$ becomes $r^5$.

 

[SammyS]

${\partial U} / {\partial y} = 2 y (x^2 + y^2 + z^2)^2 + {\partial f(y,z)} / {\partial y}$
Setting that equal to the $\hat j$ component will give you ${\partial f(y,z)} / {\partial y} = 0$.

Yes. That's correct !

This means ƒ(y,z) is not actually a function of y.

$\partial f(y,z) /\partial y = 0$
$\partial f(y,z) /\partial z = 0$
Yeah, that doesn't seem right to me.

Also, it appears that you found a similar result regarding ƒ(y,z) not depending on z.

What does that leave you with?

ƒ(y,z) is just an ordinary constant, call it C.