通量积分与定向

Two-sided surfaces and orientation

The cross product carries both jobs

Flux out of the unit sphere

Open surfaces and lids

A disk in the $xy$-plane sits in the field $\mathbf{F} = (0, 0, 3)$, flowing upward. What is the flux through the disk oriented upward?

Computing flux via $\iint_D \mathbf{F}(\mathbf{X}) \cdot (\mathbf{X}_u \times \mathbf{X}_v)\, du\, dv$, a student divides by $|\mathbf{X}_u \times \mathbf{X}_v|$ first. What happens?

Why is the flux of $\mathbf{F} = (x, y, z)$ out of the unit sphere computable without any parametrized integral?

A field has upward flux $7$ through a surface. Give the flux through the same surface with the opposite orientation, as a plain number.

For $\mathbf{F} = (0, 0, 1)$ through the graph of $g$ over a region $D$ of area $6$, oriented upward, give the flux, as a plain number.