Multivariable Calculus · free preview

Differential Forms and the Wedge Product

Zero-forms, one-forms and two-forms as the objects that integrate over points, curves and surfaces, and the antisymmetric wedge product that builds higher forms.

1. What actually gets integrated

What actually gets integrated

2. The wedge product

The wedge product

3. Computing with wedges

Computing with wedges

4. Forms as the course's unifying dictionary

Forms as the course's unifying dictionary

5. Degree matching

Which object is the two-form $3x\, dx \wedge dy + y^2\, dx \wedge dz$ built to integrate over?

6. The antisymmetry rule

Simplify the wedge product $(y\, dx) \wedge (x\, dy)$ of these two one-forms.

7. Signed area reading

The two-form $dx \wedge dy$ is integrated over the unit square. What value does the traversal orientation determine?

8. Wedge of mixed forms

Give the coefficient of $dx \wedge dy$ in $(2\, dx) \wedge (3\, dy)$ , as a plain number.

9. Degree in three variables

In three variables, give the highest degree a nonzero form can have, as a plain number.

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