Exterior Derivative - Examples

Examples

1

Consider σ = u dx1∧dx2 over a 1-form basis dx1,...,dxn. The exterior derivative is:

The last formula follows easily from the properties of the wedge product. Namely, dxi ∧ dxi = 0.

2

For a 1-form σ = u dx + v dy defined over R2. We have, by applying the above formula to each term (consider x1 = x and x2 = y) the following sum,

\mathrm{d} \sigma = \left( \sum_{i=1}^2 \frac{\partial u}{\partial x^i} \mathrm{d}x^i \wedge \mathrm{d}x \right) + \left( \sum_{i=1}^2 \frac{\partial v}{\partial x^i} \mathrm{d}x^i \wedge \mathrm{d}y \right)

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