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

In summary, the conversation discusses the book "Vector Calculus, Linear Algebra and Differential Forms" by John H Hubbard and Barbara Burke Hubbard, specifically focusing on Chapter 6 and the notes following Figure 6.1.6. The reader needs clarification on the meaning of the terms "vol_2" preceding certain points, and another reader explains that it represents the area of the projection of a parallelogram onto the (x,y)-plane in three-dimensional space. The conversation ends with gratitude for the explanation.
  • #1
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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
 

Attachments

  • H&H - Notes on Figure 6.1.1 .png
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  • H&H - 1 - Geometric Meaning of k-forms ... PART 1 ... .png
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  • H&H - 2. - Geometric Meaning of k-forms ... PART 2 ... .png
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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 $P_3$
 
  • #3
steenis said:
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
 

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

1. What is a k-form in vector calculus?

A k-form is a mathematical object used to represent quantities that are dependent on both position and direction in space. It is a generalization of the concept of a vector, which is a quantity that only depends on position.

2. How is a k-form represented geometrically?

A k-form is represented geometrically as a collection of arrows or vectors that are attached to each point in space. These vectors represent the magnitude and direction of the k-form at that particular point.

3. What is the significance of k in k-forms?

The value of k in k-forms represents the number of variables or dimensions in the space the form is defined in. For example, a 2-form would have two variables (x and y) and would be defined in a two-dimensional space.

4. How are k-forms related to differential forms?

K-forms are a specific type of differential form, which are mathematical objects used to represent quantities that are dependent on both position and direction in space. K-forms are a type of differential form that have a specific number of variables or dimensions.

5. What is the role of geometric interpretation in understanding k-forms?

Geometric interpretation is crucial in understanding k-forms because it allows us to visualize and understand the behavior of these mathematical objects in a more intuitive way. By representing k-forms geometrically, we can better understand their properties and relationships with other mathematical concepts.

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