# partial differential: partial scalar partial vector

by Havik
Tags: differential, partial, scalar, vector
 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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Hi Andreas! Welcome to PF!

(have a curly d: ∂ )
 Quote by Havik 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.
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 Quote by Havik 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:

$$\frac {\partial F}{\partial \hat v} = D_{\hat v}(F) = \nabla F \cdot \hat V$$

where $$\hat V$$ is a unit vector in the direction of V.

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## 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 $$\partial$$ 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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