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# Multivariable Calculus watch

1. https://www0.maths.ox.ac.uk/system/f.../37/sheet8.pdf

Question 2 here. Basically I have no idea how to start the first part, any hints?

For the second, I feel like stokes theorem is useful but I'm not sure how to apply it in this sense. Ive calculated curl(F) as -A(div(R)) - (R dot del)A but am not sure if this is useful or even correct

TeeEm
2. (Original post by Gome44)
https://www0.maths.ox.ac.uk/system/f.../37/sheet8.pdf

Question 2 here. Basically I have no idea how to start the first part, any hints?

For the second, I feel like stokes theorem is useful but I'm not sure how to apply it in this sense. Ive calculated curl(F) as -A(div(R)) - (R dot del)A but am not sure if this is useful or even correct

TeeEm
This is some result from electromagnetism, which I do not remember at present.
will look at it soon
(just came back from my daily walk ...)
3. Perhaps writing |r-R|^2 = (r-R).(r-R) could help? Note also that it's a closed curve as that may be significant when evaluating the integrals
4. (Original post by A Slice of Pi)
Perhaps writing |r-R|^2 = (r-R).(r-R) could help?
Yeah i tried that, don't know how to precede after expanding it though
5. (Original post by Gome44)
Yeah i tried that, don't know how to precede after expanding it though
I tried for 1 1/2 hour and got nowhere ...
(looked at at least 20 textbooks for a similar question even left as an exercise and nothing)
I permit TSR to laugh at me as I have been beaten (really bad) by this question ...
I am giving up just for today ...
6. (Original post by Gome44)
https://www0.maths.ox.ac.uk/system/f.../37/sheet8.pdf

Question 2 here. Basically I have no idea how to start the first part, any hints?
If you ever get an answer to this, I'd be interested in seeing it.

Given that the closed curve in question is arbitrary, it seems to me that an integral of the form given can end up as an arbitrary vector - you are calculating the sum of the infinitesimal vectors around the curve weighted by the square of their distances from , so those parts of the curve where that quantity is large contribute more to the integral - so their direction will dominate in the final vector.
7. Where are the rest of the "big guns" needed to take down this problem?

Or you could try stackexchange maybe.
8. (Original post by atsruser)
If you ever get an answer to this, I'd be interested in seeing it.

Given that the closed curve in question is arbitrary, it seems to me that an integral of the form given can end up as an arbitrary vector - you are calculating the sum of the infinitesimal vectors around the curve weighted by the square of their distances from , so those parts of the curve where that quantity is large contribute more to the integral - so their direction will dominate in the final vector.
So there's 2 ways, an easy application of stokes theorem and a more impressive way that my tutor showed me (although I would never have thought of it).

1) Apply stoke's theorem F= cf where c is a constant vector and f is scalar field to obtain: ∬ grad(f) x dS = - ∫ f dr, then its fairly simple if you let r = (x,y,z) and R = (R1,R2,R3) and just expand

2) Write |r-R|^2 as a dot product, expand it out. From linear algebra F(R) is of the form x^T M x + Nv + B (quadratic form +linear form + constant).

Showing M = 0 is fairly simple, then you show N is skew symmetric by dotting with the basis vectors.
9. (Original post by Gome44)
So there's 2 ways, an easy application of stokes theorem and a more impressive way that my tutor showed me (although I would never have thought of it).

1) Apply stoke's theorem F= cf where c is a constant vector and f is scalar field to obtain: ∬ grad(f) x dS = - ∫ f dr, then its fairly simple if you let r = (x,y,z) and R = (R1,R2,R3) and just expand

2) Write |r-R|^2 as a dot product, expand it out. From linear algebra F(R) is of the form x^T M x + Nv + B (quadratic form +linear form + constant).

Showing M = 0 is fairly simple, then you show N is skew symmetric by dotting with the basis vectors.
I looked at that approach (using that result) but I discarded it thinking that if this result was to be used you would have been asked to prove it first ...

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