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Partial differential: partial scalar partial vector

by Havik
Tags: differential, partial, scalar, vector
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Havik
#1
Sep30-09, 01:39 AM
P: 2
Hi,

I have a problem to find the meaning of a special partial differential: partial scalar partial vector.

i.e. dF/dn where F is a scalar and n is a i.e. normal vector. This is a partial diff.

n could be a vector consisting of partial differentials, (dT/dx,dT/dy)

I have looked in literature but found nothing.

Can someone help me?

Thank you very much
/Andreas
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tiny-tim
#2
Sep30-09, 08:31 AM
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Hi Andreas! Welcome to PF!

(have a curly d: ∂ )
Quote Quote by Havik View Post
I have a problem to find the meaning of a special partial differential: partial scalar partial vector.

i.e. dF/dn where F is a scalar and n is a i.e. normal vector. This is a partial diff.

n could be a vector consisting of partial differentials, (dT/dx,dT/dy)
No such thing you can't have d(scalar)/d(vector) or ∂(scalar)/∂(vector).

But (for example, when calculating flux) you can have ∂F/∂n or d(F.n^)/dn, where n^ is the unit vector in the normal direction, and n is the distance in that direction.
LCKurtz
#3
Sep30-09, 01:14 PM
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Quote Quote by Havik View Post
Hi,

I have a problem to find the meaning of a special partial differential: partial scalar partial vector.

i.e. dF/dn where F is a scalar and n is a i.e. normal vector. This is a partial diff.

n could be a vector consisting of partial differentials, (dT/dx,dT/dy)

I have looked in literature but found nothing.

Can someone help me?

Thank you very much
/Andreas
Perhaps you are thinking about directional derivatives. If F(x,y,z) is a scalar function (perhaps the temperature at (x,y,z)), and V is a vector, then the rate of change of F in the direction of V is:

[tex]\frac {\partial F}{\partial \hat v} = D_{\hat v}(F) = \nabla F \cdot \hat V[/tex]

where [tex] \hat V[/tex] is a unit vector in the direction of V.

Havik
#4
Oct1-09, 02:43 AM
P: 2
Partial differential: partial scalar partial vector

Hi tiny-tim and LCKurtz,

This is exactly the explaination I am looking for! I had a hard time to understand the meaning of such partial derivative. And it with [tex]\partial[/tex] it should be

It is the rate of F in the direction of some unit vector n that is normal to an arbitrary surface. I have no problem to find the growth rate of F in x and y but when it came to a other direction depending on other things, it became a problem. But now I understand how to do it!

I actually did not think of the thing that n must be a unit vector. I will make it a unit vector!

Thank you very much for your help on this problem, I have been struggling to find the answer for a long time!


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