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)

Read more about this topic:  Exterior Derivative

Famous quotes containing the word examples:

    “Histories are more full of examples of the fidelity of dogs than of friends.”
    —Alexander Pope (1688–1744)

    “In the examples that I here bring in of what I have [read], heard, done or said, I have refrained from daring to alter even the smallest and most indifferent circumstances. My conscience falsifies not an iota; for my knowledge I cannot answer.”
    —Michel de Montaigne (1533–1592)

    “There are many examples of women that have excelled in learning, and even in war, but this is no reason we should bring ‘em all up to Latin and Greek or else military discipline, instead of needle-work and housewifry.”
    —Bernard Mandeville (1670–1733)