Archive for February 1st, 2010

February 1st, 2010

Differentiantion of 1-forms

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 \mathbb{R}^n, then there exists some smooth function f on U, which we define to be the differential 0-form. Given a vector field v on \mathbb{R}^n, for each v, there exists a directional derivative \partial_v f, which is the directional derivative in the usual sense, that is, if v=e_j is the jth coordinate vector then \partial_v f 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 x^n and y^n, the transform between them is simply:

\frac{\partial f}{\partial x^j} = \sum_{i=1}^n\frac{\partial y^i}{\partial x^j}\frac{\partial f}{\partial y^i}

Since any vector v is a linear combination \sum v^j e_j of its components, df is uniquely determined by d_f p(e^j) for each j and each p\in U, which are just the partial derivatives of f on U. Since the coordinates x^n are themselves functions on U, and so define differential 1-forms dx^n. Since \frac{\partial x^i}{\partial x^j} = \delta^{i}_{j}, the Kronecker delta function, it follows that

df = \sum_{i=1}^n \frac{\partial f}{\partial x^i} dx^i.

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

d f_p = \sum_{i=1}^n \frac{\partial f}{\partial x^i}(p) (dx^i)_p.

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 g_i and h_i on U, we define the differential 1-form \alpha = \sum_1 g_i dh^i pointwise by coordinates as \alpha = \sum_{i=1}^n f_i d x^i for some smooth functions f_i on U.

The second idea leading to differential forms arises from the following question: given a differential 1-form \alpha on U, when does there exist a function f on U such that \alpha = df? The above expansion reduces this question to the search for a function f whose partial derivatives \frac{\partial f}{\partial x^i} are equal to n given functions f_i. For n>1, such a function does not always exist: any smooth function f satisfies \frac{\partial^2 f}{\partial x^i \partial x^j} = \frac{\partial^2 f}{\partial x^j \partial x^i}

so it will be impossible to find such an f unless \frac{\partial f_j}{\partial x^i} - \frac{\partial f_i}{\partial x^j}=0 \forall i,j.

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 \sum_{i,j=1}^n \frac{\partial f_j}{\partial x^i} dx^i \wedge dx^j = 0

where dx^i \wedge dx^j = -dx^j \wedge dx^i.

This is an example of a differential 2-form: the exterior derivative d\alpha of [/latex]\alpha= \sum_j=f_j dx^j[/latex] is given by d\alpha = \sum_{j=1}^n d f_j \wedge dx^j = \sum_{i,j=1}^n \frac{\partial f_j}{\partial x^i} dx^i \wedge dx^j.

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.

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