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  1. MHB Master
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    #1
    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:






    My 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:







    Hope that helps ...

    Peter

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  3. MHB Craftsman
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    #2
    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 $P_3$

  4. MHB Master
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    #3 Thread Author
    Quote Originally Posted by steenis View Post
    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

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