In one dimension, the slope of a function, f(x), is described by a single number, df/dx.
In higher dimensions, the slope depends on the direction. For example, if f=x+2y, moving one unit x-ward increases f by 1 so the slope in the x direction is 1, but moving one unit y-ward increases f by 2 so the slope in the y direction is 2.
It turns out that we can describe the slope in n dimensions with just n numbers, the partial derivatives of f.
To calculate them, we differentiate with respect to one coordinate, while holding all the others constant. They are written using a ∂ rather than d. E.g.
Notice this is almost the same as the definition of the ordinary derivative.
If we move a small distance in each direction, we can combine three equations like 1 to get
The change in f after a small displacement is the dot product of the displacement and a special vector
This vector is called the gradient of f. It points up the direction of steepest slope. We will be using this vector quite frequently.