Defining the gradient

To properly understand the gradient one is required to know vectors and the dot (or scalar) product. When we have color gradients what we see is that one extreme has one color and the other extreme another color, in between a gradient that is the transition between one color to the other. In physics there exists many types of gradients, such as gradients of temperature or pressure. Gradients are important because there are certain phenomena that require strong gradients to happen, such as there is no wind or oceanic currents if there is no pressure gradient between two points. Conceptually we have some quantity that changes in intensity over space and with a particular direction. That's the gradient.

Before going on for a mathematical definition, let's look at a graph of level curves:

The rate of change over the [math]\displaystyle{ x }[/math] axis is less than over the [math]\displaystyle{ y }[/math] axis because the level curves are closer in the latter direction than on the former. If we walk along the same level we don't experience any changes in the value of [math]\displaystyle{ f(x,y) }[/math], which means that the rate of change is zero for directions parallel to a level curve. The highest rates of change are achieved when we move perpendicularly to the level curves, which is the shortest path between them. If the distance between two consecutive level curves is close to zero, it means that the function's slope is close to 90°. Otherwise, if the distance between them tends to infinity, then the slope is close to 0°.

Taking a second look at the directional derivative notice that we have a sum of terms where each one is a product between the coordinates of the vector given and the partial derivatives for that coordinate. That formula is a dot product. Conclusion? We have two vectors in it, one is the vector that gives the direction which we want to find the rate of change on, the other is

[math]\displaystyle{ \nabla f = \left(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y}\right) }[/math]

For [math]\displaystyle{ n }[/math] variables we have a gradient with [math]\displaystyle{ n }[/math] coordinates. The flipped Delta is the letter Nabla, read it "del [math]\displaystyle{ f }[/math]".

We can rewrite the directional derivative using the gradient as follows

[math]\displaystyle{ \frac{\partial f}{\partial \overrightarrow{v}}(a,b) = \left(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y}\right) \cdot (a,b) = \nabla f \cdot \overrightarrow{v} }[/math]

Physical interpretation of the gradient

A practical example would be atmospheric pressure and climbing a mountain. If we circle around the mountain we don't experience any differences in atmospheric pressure. If we climb the mountain though there is a difference in atmospheric pressure.