Line Integral Example - mistake or am I missing something?

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1. Jun 2, 2015

kostoglotov

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

1. The problem statement, all variables and given/known data

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.

Last edited: Jun 2, 2015
2. Jun 2, 2015

DEvens

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

3. Jun 2, 2015

kostoglotov

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 :)

Last edited: Jun 2, 2015
4. Jun 2, 2015