Line Integral Example - mistake or am I missing something?

[kostoglotov]

This is an example at the beginning of the section on the Fundamental Theorem for Line Integrals.

1. Homework Statement

Find the work done by the gravitational field

$$\vec{F}(\vec{x}) = -\frac{mMG}{|\vec{x}|^3}\vec{x}$$

in moving a particle from the point (3,4,12) to (2,2,0) along a piece wise smooth curve

Now, I think that there's a mistake in the solution given...but this textbook is pretty good, and plenty of times in the past I've thought it had made a mistake and really I was mistaken.

So, I understand all the concepts (edit: I obviously didn't), all good there. It's here where I'm scratching my head.

$$f(x,y,z) = \frac{mMG}{\sqrt{x^2+y^2+z^2}}$$

So, we need to get the scalar function of f, call it the potential function, we know that in a conservative vector field $\vec{F} = \nabla f$, no worries. However, shouldn't it be

$$\vec{F}(\vec{x}) = -\frac{mMG}{|\vec{x}|^3}\vec{x} = -\frac{mMG|\vec{x}|}{|\vec{x}|^3}\vec{u} = -\frac{mMG}{|\vec{x}|^2}\vec{u}$$

And so converting the vector form of F into a scalar field from which we can compute the grad vector, doesn't

$$|\vec{x}|^2 = x^2 + y^2 + z^2$$

not

$$\sqrt{x^2+y^2+z^2}$$

so shouldn't it be

$$f(x,y,z) = \frac{mMG}{|\vec{x}|^2} = \frac{mMG}{x^2+y^2+z^2}$$

how would one end up with

$$f(x,y,z) = \frac{mMG}{\sqrt{x^2+y^2+z^2}}$$

??

edit: both the 6th and 7th editions have the same example...so I'm guessing that I'm missing something.

 

[DEvens]

Can you work out the gradient of the f(x,y,z) you suggest? Does it give the right force?

 

[kostoglotov]

Can you work out the gradient of the f(x,y,z) you suggest? Does it give the right force?

No, it gives N not N m.

I'm still not sure what's going on here though. Is it the case that we're NOT trying to just find the magnitude of the gravity vectors on a given domain, but rather searching for some other related function, that will produce a grad vec that when dot producted with a path vector will result in the dimensions that we need in our answer?

I still don't know how they got to their answer.

edit: Oh wait, I think I see what's happening...f(x,y,z) is essentially an antiderivative ?? Thus, we need to be able to get to the grad vec from that f(x,y,z)...??

edit2: yep, that worked, and it made sense...thanks :)

 

[Ray Vickson]

This is an example at the beginning of the section on the Fundamental Theorem for Line Integrals.

1. Homework Statement

Find the work done by the gravitational field

$$\vec{F}(\vec{x}) = -\frac{mMG}{|\vec{x}|^3}\vec{x}$$

in moving a particle from the point (3,4,12) to (2,2,0) along a piece wise smooth curve

Now, I think that there's a mistake in the solution given...but this textbook is pretty good, and plenty of times in the past I've thought it had made a mistake and really I was mistaken.

So, I understand all the concepts (edit: I obviously didn't), all good there. It's here where I'm scratching my head.

$$f(x,y,z) = \frac{mMG}{\sqrt{x^2+y^2+z^2}}$$Standard result: if $\vec{r} = x\vec{e}_x + y\vec{e}_y + z \vec{e}_z$ with magnitude $r = \sqrt{x^2 + y^2 + z^2}$, then
$$\vec{ \nabla} \frac{1}{r} = -\frac{\vec{r}}{r^3}$$
Just look at $\partial r^{-1} / \, \partial x$ for example.