Geometric Interpretation of k-Forms from H&H's Vector Calculus

[Math Amateur]

I am reading the book: "Vector Calculus, Linear Algebra and Differential Forms" (Fourth Edition) by John H Hubbard and Barbara Burke Hubbard.

I am currently focused on Chapter 6: Forms and Vector Calculus ...

I need some help in order to understand some notes by H&H following Figure 6.1.6 ... ...

Figure 6.1.6 and the notes following it read as follows:
View attachment 8628My question regarding the notes following Figure 6.1.1 is as follows:

What is the meaning/significance of the terms \(\displaystyle \text{ vol}_2\) preceding \(\displaystyle P_1, P_2\) and \(\displaystyle P_3\) ... indeed I can see no need for the terms at all ...

Can someone please clarify this issue ...

Peter=========================================================================================It may help MHB readers of the above post to have access to H&H's section on the Geometric Meaning of k-forms ... so I am providing the text of the same ... as follows:
View attachment 8629
View attachment 8630Hope that helps ...

Peter

 

[steenis]

Hi Peter

I do not think that the notation is superfluous, but a better notation would be $vol_2(P_3)$, etc.You have a parallelogram $P$ spanned by $\vec{v_1}$ and $\vec{v_2}$ in three dimensional real space.

$P_3$ is the projection of $P$ on the $(x,y)$-plane.

$vol_2()$ is the action to compute the area of a plane, that is,

$vol_2(P_3)$ is the area of $P_3$

 

[Math Amateur]

Hi Peter

I do not think that the notation is superfluous, but a better notation would be $vol_2(P_3)$, etc.You have a parallelogram $P$ spanned by $\vec{v_1}$ and $\vec{v_2}$ in three dimensional real space.

$P_3$ is the projection of $P$ on the $(x,y)$-plane.

$vol_2()$ is the action to compute the area of a plane, that is,

$vol_2(P_3)$ is the area of $P_3$

Thanks for the insight and help, Hugo ...

Appreciate your help ...

Peter