Flux — Question 3

Question 3

A skew parallelogram requires an oriented area vector

A parallelogram is spanned by $\mathbf a=(2,0,0)$ m and $\mathbf b=(1,3,0)$ m. Its positive orientation is $\mathbf a\times\mathbf b$. Uniform flow is $\mathbf v=(4,-2,5)$ m/s.

1. Calculate its area vector, geometric area and signed volume flux.

2. Reverse the ordering of the spanning vectors and compare the new signed flux with the unchanged amount of fluid crossing per second.

Question 3 - Solution

A skew parallelogram requires an oriented area vector

(a) Method and calculation

$$\mathbf A=\mathbf a\times\mathbf b=(0,0,6)\,\mathrm{m^2},\quad A=6.00\,\mathrm{m^2}.$$$$Q=\mathbf v\cdot\mathbf A=30.0\,\mathrm{m^3/s}.$$

(b) Interpretation and further calculation

$$\mathbf b\times\mathbf a=-\mathbf A\Rightarrow Q\prime=-30.0\,\mathrm{m^3/s}.$$$$|Q\prime|=|Q|=30.0\,\mathrm{m^3/s}.$$Orientation changes the bookkeeping sign, not the physical surface or flow.

Physical check

The area is $ab\sin\gamma$, not $ab$, because the spanning vectors are not perpendicular.