What Does the Gradient Operator Really Mean?

[utkarshakash]

Homework Statement

I need some help regarding the gradient operator. I recently came across this statement while reading Griffith's Electrodynamics
"The gradient ∇T points in the direction of maximum increase of the function
T."

Wolfram Alpha also states that "The direction of ∇f is the orientation in which the directional derivative has the largest value and |∇f| is the value of that directional derivative. "

I'm finding it difficult to absorb this thing without any reasonable explanation. Can someone help?

Homework Equations

The Attempt at a Solution

 

[vanhees71]

Take a scalar field, $T(\vec{x})$ and calculate the directional derivative in direction of the unit vector $\vec{n}$ at a fixed point $\vec{x}=\vec{x}_0$. How is it related to the gradient? In which direction do have to choose $\vec{n}$ such that (modulus of) the directional derivative becomes maximal?

 

[Infinitum]

I'm unsure whether you know what a gradient is, but if you wish to get a better intuition of why it points to the direction of greatest increase of a function, I've found these articles very useful.

http://betterexplained.com/articles/vector-calculus-understanding-the-gradient/
http://betterexplained.com/articles/understanding-pythagorean-distance-and-the-gradient/

For an actual proof, of course, you can proceed as vanhees71 suggested.

 

[utkarshakash]

I'm unsure whether you know what a gradient is, but if you wish to get a better intuition of why it points to the direction of greatest increase of a function, I've found these articles very useful.

http://betterexplained.com/articles/vector-calculus-understanding-the-gradient/
http://betterexplained.com/articles/understanding-pythagorean-distance-and-the-gradient/

For an actual proof, of course, you can proceed as vanhees71 suggested.

Thank you very much! This article explains it very nicely.