Proving gradient points in the direction of maximum increase

[hivesaeed4]

How do we prove that the gradient points in the direction of the maximum increase? Would it be enough to simply state that the gradient is just the derivates of a function w.r.t all the variables a function depends upon. Since the derivative of a term w.r.t a certain variable gives the maximum increase of that term and since in gradient not only do we account for all possible terms and all the variables upon which a term depends upon but also includes the direction, so it is logical to conclude that the gradient points in the direction of the maximum increase.

Note: I do hope that the above proof is adequate but somehow feel that the proof has to be mathematical. Could someone tell me whether or not my intuition is correct?

 

[arildno]

"Since the derivative of a term w.r.t a certain variable gives the maximum increase of that term ".
Does it?

 

[hivesaeed4]

Sorry. It gives the rate of change of a term.

Then how do I prove that the gradient points in the direction of the maximum increase?

 

[arildno]

Have you learned about directional derivatives yet?

 

[arildno]

Ok:
I just saw that you DO know about directional derivatives, on basis of another thread posted by you!

Now, you now know that the directional derivative is the dot/scalar product between the gradient vector and the direction vector.

Can you rewrite a dot/scalar product between two vectors in another way, a way involving the magintudesof the two vectors?

 

[hivesaeed4]

Yes. But that would involve the cos of the angle between the 2 vectors. But how does that prove that the gradient gives the us maximum rate of increase?

 

[Skrew]

I haven't done this but as a suggestion.

Make the directional derivative a function of theta and maximize it.

 

[arildno]

Yes. But that would involve the cos of the angle between the 2 vectors. But how does that prove that the gradient gives the us maximum rate of increase?

For what angle between the vectors does the cosine function receive its maximum value?

 

[hivesaeed4]

zero.

 

[arildno]

zero.

Indeed!

And, therefore:
What angle between the direction vector and the gradient vector maximizes the dot product between them?

 

[hivesaeed4]

zero?

 

[arildno]

Right!

And, if two vectors have zero angle between them, then they are parallell vectors, meaning that the direction of maximal increase is in the direction of the...?

 

[hivesaeed4]

In the direction of both vectors.

 

[hivesaeed4]

So is that the required proof?

 

[arildno]

In the direction of both vectors.

Yes.
In particular, in the direction of the gradient.

 

[arildno]

So is that the required proof?

What do you think is lacking/unsatisfactory?

 

[mathwonk]

the answer depends on your definition of the gradient. if you think the gradient is the vector that dots with a direction vector to give the directional derivative, then the previous explanation is all you need.

I you think the gradient is a vector whose entries are the partial derivatives, then you need to make sense out of that a bit more. i.e. you need the chain rule to get the previous statement.

 

[arildno]

the answer depends on your definition of the gradient. if you think the gradient is the vector that dots with a direction vector to give the directional derivative, then the previous explanation is all you need.

I you think the gradient is a vector whose entries are the partial derivatives, then you need to make sense out of that a bit more. i.e. you need the chain rule to get the previous statement.

That is why I asked OP what, if anything, he found lacking.

However, since he had already posted other threads in which he showed familiarity with the concept of the directional derivative, I assumed he was also familiar with how that is derived.
Therefore, I did not address that point explicitly.