Thread: directional derivative ...

1. ok..I have been learning for quite a bit but am having hard time visualizing how it works.
Can anyone explain me the derivation of the directional derivative ( the logic behind it ).

Thx,

2.

3. Originally Posted by newspaper
ok..I have been learning for quite a bit but am having hard time visualizing how it works.
Can anyone explain me the derivation of the directional derivative ( the logic behind it ).

Thx,
Let f be a function defined on n-space and let x be a point in n-space and let v be an n-vector. Now consider the function g of the real variable t defined by g(t)=f(x+tv). Then g'(0) is the directional derivative of f at x in the direction of v. You get it by looking at the function f restricted to a line through x in the direction of v. The set of points x+tv for fixed x and v as t varies is just that line.

4. Originally Posted by DrRocket
Originally Posted by newspaper
ok..I have been learning for quite a bit but am having hard time visualizing how it works.
Can anyone explain me the derivation of the directional derivative ( the logic behind it ).

Thx,
The real variable t defined by g(t)=f(x+tv). Then g'(0) is the directional derivative of f at x in the direction of v. You get it by looking at the function f restricted to a line through x in the direction of v. The set of points x+tv for fixed x and v as t varies is just that line.
Could you explain a little more about it, if you don't mind.
I appreciate your help. :wink:

5. Originally Posted by newspaper
Originally Posted by DrRocket
Originally Posted by newspaper
ok..I have been learning for quite a bit but am having hard time visualizing how it works.
Can anyone explain me the derivation of the directional derivative ( the logic behind it ).

Thx,
The real variable t defined by g(t)=f(x+tv). Then g'(0) is the directional derivative of f at x in the direction of v. You get it by looking at the function f restricted to a line through x in the direction of v. The set of points x+tv for fixed x and v as t varies is just that line.
Could you explain a little more about it, if you don't mind.
I appreciate your help. :wink:
Why don't you try to explain how you look at it. I have given you one way that I look at it. If I could see your ideas, correct or otherwise, then I could perhaps understand what is confusing you on what is basically a simple geometric idea.

6. Hi dr. rocket

The derivative of 'f' in the direction u at point p is :

This is the definition i have learnt. Its just that i am failing to visualize it. I have a very bad habit of not going forward until i can visualize it.

Thx,

7. Im not quite sure what you are asking to hear but what helped me understand the concept of directional derivatives was that they are the rate of change of the function f(x,y,z) in the direction of the unit vector u=<a,b,c>. So to find the rate of change of the function you would then need to take the partial derivatives of each variable and multiply it by its corresponding direction in the unit vector to find the rate of change in a certain direction.

Hope that helps!

8. Originally Posted by newspaper
Hi dr. rocket

The derivative of 'f' in the direction u at point p is :

This is the definition i have learnt. Its just that i am failing to visualize it. I have a very bad habit of not going forward until i can visualize it.

Thx,
Look at what you wrote and think of it geometrically. You fixed a point P and a direction u. Then you have a new function of and you took the derivative of that new function a the point That new function was precisely the function f restricted to a line passing through P in the direction u.

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