- #1

kostoglotov

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

Find the work done by the gravitational field

[tex]\vec{F}(\vec{x}) = -\frac{mMG}{|\vec{x}|^3}\vec{x}[/tex]

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.

[tex]f(x,y,z) = \frac{mMG}{\sqrt{x^2+y^2+z^2}}[/tex]

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

[tex]\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}[/tex]

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

[tex]|\vec{x}|^2 = x^2 + y^2 + z^2[/tex]

[tex]\sqrt{x^2+y^2+z^2}[/tex]

so shouldn't it be

[tex]f(x,y,z) = \frac{mMG}{|\vec{x}|^2} = \frac{mMG}{x^2+y^2+z^2}[/tex]

how would one end up with

[tex]f(x,y,z) = \frac{mMG}{\sqrt{x^2+y^2+z^2}}[/tex]

??

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

1. Homework Statement1. Homework Statement

Find the work done by the gravitational field

[tex]\vec{F}(\vec{x}) = -\frac{mMG}{|\vec{x}|^3}\vec{x}[/tex]

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.

[tex]f(x,y,z) = \frac{mMG}{\sqrt{x^2+y^2+z^2}}[/tex]

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

[tex]\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}[/tex]

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

[tex]|\vec{x}|^2 = x^2 + y^2 + z^2[/tex]

__not__

[tex]\sqrt{x^2+y^2+z^2}[/tex]

so shouldn't it be

[tex]f(x,y,z) = \frac{mMG}{|\vec{x}|^2} = \frac{mMG}{x^2+y^2+z^2}[/tex]

how would one end up with

[tex]f(x,y,z) = \frac{mMG}{\sqrt{x^2+y^2+z^2}}[/tex]

??

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

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