The most common first approach to differential calculus in more than one variable starts by defining partial derivatives and directional derivatives, as we did. But instead of defining the differential, it simply collects the partial derivatives together as the components of a vector called the gradient of , and written .
We showed that these partial derivatives are the components of the differential (when it exists), and so there should be some connection between the two concepts. And indeed there is.
As a bilinear form, our inner product defines an isomorphism from the space of displacements to its dual space. This isomorphism sends the basis vector to the dual basis vector , since we can check both
That is, the linear functional is the same as the linear functional .
So under this isomorphism the differential corresponds to the vector
We can remove the function from this notation to write the operator on its own as
We also write the gradient vector at a given point as , where we have to remember to parse this as evaluating a function at the point rather than as applying the operator to the value .
Now, under our approach the differential is more fundamental and more useful than the gradient vector. However, there is some meaning to the geometric interpretation of the gradient as a displacement vector.
First of all, let’s ask that be a unit vector. Then we can calculate the directional derivative . But the linear functional given by the differential is the same as the linear functional . Thus we also find that . And we can interpret this inner product in terms of the length of and the angle between and :
since the length of is automatically .
The cosine term has a maximum value of when points in the same direction as so that . That is, the direction that gives us the largest directional derivative is the direction of . And then we can calculate the rate that the function increases in this direction as the length of the gradient .
So for most purposes we’ll stick to using the differential, but in practice it’s often useful to think of the gradient vector to get some geometric intuition about what the differential means.