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1. Differentiating a Vector Field
Hi,

Just need a bit of help with differentiating a vector field.

I need to find

I am unsure what to differentiate with respect to as i need to have a vector as the outcome. Do I differentiate w.r.t x for i, w.r.t y for j, and w.r.t to z for k?

Any help is appreciated.
2. Re: Differentiating a Vector Field
If is a vector then is a matrix, not a vector, where (for instance) .

If you were differentiating the -component w.r.t. and the -component w.r.t. and so on then what you'd have would be and not .
Last edited by nuodai; 18-04-2012 at 13:01.
3. Re: Differentiating a Vector Field
(Original post by nuodai)
If is a vector then is a matrix, not a vector, where (for instance) .

If you were differentiating the -component w.r.t. and the -component w.r.t. and so on then what you'd have would be and not .
Okay, thank you. Thats a big help because the question is to find the values a,b,c such that F is soleneidal. i.e.
4. Re: Differentiating a Vector Field
(Original post by HappyHammer15)
Okay, thank you. Thats a big help because the question is to find the values a,b,c such that F is soleneidal. i.e.
Right. Bear in mind that is the sum of the respective derivatives of the components, so it's not a vector. (I didn't make this clear in my last post.)
5. Re: Differentiating a Vector Field
(Original post by nuodai)
Right. Bear in mind that is the sum of the respective derivatives of the components, so it's not a vector. (I didn't make this clear in my last post.)
Okay thank you, however, in this example i do not get a variable c in the . Thank you for your help thus far.
6. Re: Differentiating a Vector Field
(Original post by HappyHammer15)
Okay thank you, however, in this example i do not get a variable c in the . Thank you for your help thus far.
So?
7. Re: Differentiating a Vector Field
(Original post by around)
So?
I get a=-0.5 and b = 2. Is c an arbitrary constant then?
8. Re: Differentiating a Vector Field
(Original post by nuodai)
Right. Bear in mind that is the sum of the respective derivatives of the components, so it's not a vector. (I didn't make this clear in my last post.)
To add to this: remember that the notation is suggestive and is formally like a dot product. i.e. think of and and where in your example and so on.

Then, you form the 'dot product'

where you interpret the 'product' of, say with in the obvious way i.e. you take the partial derivative of w.r.t.
Last edited by Jake22; 18-04-2012 at 15:22.
9. Re: Differentiating a Vector Field
(Original post by HappyHammer15)
I get a=-0.5 and b = 2. Is c an arbitrary constant then?
Exactly, if it doesn't contribute anything then you can take any value.

e.g. Find all pairs (a,b) such that a = 0.
10. Re: Differentiating a Vector Field
(Original post by Jake22)
To add to this: remember that the notation is suggestive and is formally like a dot product. i.e. think of and and where in your example and so on.

Then, you form the 'dot product'

where you interpret the 'product' of, say with in the obvious way i.e. you take the partial derivative of w.r.t.

If i do it in this way i get a very complicated solution that is quite difficult to reduce. Will the way in which i first did it suffice?, i.e differentiating w.r.t x, differentiating w.r.t y and so on.
11. Re: Differentiating a Vector Field
(Original post by HappyHammer15)
If i do it in this way i get a very complicated solution that is quite difficult to reduce. Will the way in which i first did it suffice?, i.e differentiating w.r.t x, differentiating w.r.t y and so on.
I never wrote that you should calculate it in any other way... I said to do exactly that i.e. differentiate the i component wrt to x, the j component wrt y etc. and then sum them up. All I wrote was just a convinient way of remembering what the operation gives you. You shouldn't get anything different at all.

The point is to think of as a vector and as a kind of dot product of vectors whose formula in components you are (or at least should be) very familiar with. The reason that it isn't a 'true' dot product is that is a vector in a different vector space to i.e. the dual space.

Note: there was a mistake in my initial post which I have now corrected - I put commas instead of + signs on the right hand side but this shouldn't have affected how you would calculate.
Last edited by Jake22; 18-04-2012 at 15:28.
12. Re: Differentiating a Vector Field
(Original post by Jake22)
I never wrote that you should calculate it in any other way... I said to do exactly that i.e. differentiate the i component wrt to x, the j component wrt y etc. and then sum them up. All I wrote was just a convinient way of remembering what the operation gives you. You shouldn't get anything different at all.

Note: there was a mistake in my initial post which I have now corrected - I put commas instead of + signs on the right hand side.
Okay, i see now. I was reading and i was trying to compute
This seems like such a silly mistake to make now. Thanks for all of your help though.