In General Relativity, which relies on differential geometry and tensor calculus, a quick way to do coordinate free calculus is to use differential forms. A differential k-form, that is a form of degree k, is a smooth section of the k-th exterior power of the cotangent bundle of a smooth manifold M.

As examples, a differential 0-form is a smooth function on M, where a differential 1-form is the dual to a vector field on M. If we let U be an open set on , then there exists some smooth function f on U, which we define to be the differential 0-form. Given a vector field v on , for each v, there exists a directional derivative , which is the directional derivative in the usual sense, that is, if is the jth coordinate vector then is the partial derivative of f with respect to the jth coordinate function

By their very definition, partial derivatives depend upon the choice of coordinates: Given two coordinate systems and , the transform between them is simply:

Since any vector v is a linear combination of its components, is uniquely determined by for each j and each , which are just the partial derivatives of f on U. Since the coordinates are themselves functions on U, and so define differential 1-forms . Since , the Kronecker delta function, it follows that

The meaning of this expression is given by evaluating both sides at an arbitrary point p: on the right hand side, the sum is defined “pointwise”, so that

.

Remember, since f is an arbitrary smooth function on the dual manifold, we can define, and use, it pointwise. More generally, for any smooth functions and on U, we define the differential 1-form pointwise by coordinates as for some smooth functions on U.

The second idea leading to differential forms arises from the following question: given a differential 1-form on U, when does there exist a function f on U such that ? The above expansion reduces this question to the search for a function f whose partial derivatives are equal to n given functions . For n>1, such a function does not always exist: any smooth function f satisfies

so it will be impossible to find such an f unless .

The skew-symmetry of the left hand side in i and j suggests introducing an antisymmetric product on differential 1-forms, the wedge product, so that these equations can be combined into a single condition

where .

This is an example of a differential 2-form: the exterior derivative of [/latex]\alpha= \sum_j=f_j dx^j[/latex] is given by .

Differential forms can be multiplied together using the wedge product, and for any differential k-form α, there is a differential (k+1)-form dα called the exterior derivative of α.

Thus, I hope to have convinced you that differential forms, the wedge product and the exterior derivative are independent of a choice of coordinates. Consequently they may be defined on any smooth manifold M. If this makes you uncomfortable, you can reintroduce coordinates. One way to do this is cover M with coordinate charts and define a differential k-form on M to be a a family of differential k-forms on each chart which agree on the overlaps. However, there are more intrinsic (read: modern) definitions which make the independence of coordinates manifest. See the modern idea of tensors for a good idea what coordinate free geometry can do, and the intrinsic power of dealing with objects in a coordinate free space.