What is Gradient ? - 3

What is Gradient ? - Continuation -2
 

The recent wiki (29-Sept-20202) on Gradient is much deeper and more general (broader meanings from more different aspects)  but still does not explain "why the steepest (fastest, quickest)?".

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Cartesian coordinates

In the three-dimensional Cartesian coordinate system with a Euclidean metric, the gradient, if it exists, is given by:

${\displaystyle \nabla f={\frac {\partial f}{\partial x}}\mathbf {i} +{\frac {\partial f}{\partial y}}\mathbf {j} +{\frac {\partial f}{\partial z}}\mathbf {k} ,}$

where i, j, k are the standard unit vectors in the directions of the x, y and z coordinates, respectively.

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The explanation on the above has become just a part of <much deeper and more abstract > explanation.

For instance,

"
The gradient is dual to the derivative : the value of the gradient at a point is a tangent vector – a vector at each point; while the value of the derivative at a point is a cotangent vector – a linear function on vectors.^[c] They are related in that the dot product of the gradient of f at a point p with another tangent vector v equals the directional derivative of f at p of the function along v; that is, .

The gradient vector can be interpreted as the "direction and rate of fastest increase". If the gradient of a function is non-zero at a point p, the direction of the gradient is the direction in which the function increases most quickly from p, and the magnitude of the gradient is the rate of increase in that direction.
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We may be able to explain "why the steepest (fastest, quickest)?" by some other ways with deeper meanings.

sptt