Partial differential: partial scalar partial vector

In summary, this is a partial differential that is used to calculate the rate of change of a scalar function in a particular direction.
  • #1
Havik
2
0
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
 
Physics news on Phys.org
  • #2
Welcome to PF!

Hi Andreas! Welcome to PF! :smile:

(have a curly d: ∂ :wink:)
Havik said:
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). :wink:

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. :smile:
 
  • #3
Havik said:
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.
 
  • #4
Hi tiny-tim and LCKurtz,

This is exactly the explanation 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 :smile:

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!
 

1. What is a partial differential?

A partial differential is a mathematical concept used to describe the relationship between multiple variables in a system. It involves taking the partial derivatives of a function with respect to each of its variables, resulting in an equation that describes how each variable changes in relation to the others.

2. What is the difference between partial scalar and partial vector?

Partial scalar refers to taking the partial derivative of a scalar function, which has only one output variable. Partial vector, on the other hand, refers to taking the partial derivative of a vector function, which has multiple output variables.

3. How is partial differential used in science?

Partial differential equations are used in many areas of science, including physics, engineering, and economics, to model and predict the behavior of complex systems. They can also be used to solve optimization problems and analyze the rate of change of a system.

4. Can you give an example of a partial differential equation?

One example of a partial differential equation is the heat equation, which describes how heat is distributed and transferred within a system. It involves taking the partial derivative of temperature with respect to time and space variables.

5. What are the limitations of using partial differential equations?

Partial differential equations can become very complex and difficult to solve, particularly when dealing with nonlinear systems. They also require initial and boundary conditions to be known, which may not always be available or accurate. Additionally, they may not accurately model all systems and can lead to inaccurate predictions if not used properly.

Similar threads

  • Differential Equations
Replies
3
Views
1K
  • Differential Geometry
Replies
2
Views
534
  • Differential Equations
Replies
1
Views
706
Replies
3
Views
2K
Replies
14
Views
1K
  • Advanced Physics Homework Help
Replies
1
Views
875
  • Special and General Relativity
2
Replies
38
Views
4K
  • Advanced Physics Homework Help
Replies
5
Views
2K
  • Differential Equations
Replies
25
Views
2K
  • Differential Equations
Replies
6
Views
2K
Back
Top