I've just thought of something very very deep. You know how there is a connection between the linear and bilinear forms. They are related via currying. In words, when we keep one of the inputs constant then the bilinear forms becomes linear: it takes a vector and returns a vector. So I just noticed the same pattern when thinking about fields in physics. I am specifically talking about SCALAR fields, I want this to be clear. So the concept of a field is based on the notion of a test particle, such as a test charge q1 in case of the Coulomb's law (in the scalar form): F= k q1q2/r^2. Could we not think of this formula as some sort of a "bilinear form" in terms of q1 q2? If we drop q2 which is our test charge from this expression we will get the Electric field E=kq/r^2. Can you see now how F can be thought of as a linear form with a place holder for the test charge: F = k <q, .> / r^2.
The Electric field is now essentially a linear functional that takes a test charge and spits out the Coulomb's Force! Isn't this awesome? You can apply the same logic to any field, even the gravitational field that will eat your mass and will give you your weight! In this case the linear functional <GM/r^2, .> is conveniently known as g.