1-Forms .... Interpretation by Bachman ....

• MHB
• Math Amateur
In summary, Bachman discusses the geometric interpretation of the dot product and how it can be used to evaluate 1-forms. He also provides a helpful reminder about the scalar projection of vectors.
Math Amateur
Gold Member
MHB
I am reading David Bachman's book: "A Geometric Approach to Differential Forms" (Second Edition) ...

I need some help with some remarks Bachman makes near the start of his section on 1-forms ...

The relevant section reads as follows:View attachment 8604In the above text from Bachman we read the following:

" ... ... Recall the geometric interpretation of the dot product: You project $$\displaystyle \langle -1, 2 \rangle$$ onto $$\displaystyle \langle 2, 3 \rangle$$ and then multiply by $$\displaystyle \mid \langle 2, 3 \rangle \mid = \sqrt{13}$$. ... ... "My question is as follows:

Is there any reason Bachman has chosen the projection of $$\displaystyle \langle -1, 2 \rangle$$ onto $$\displaystyle \langle 2, 3 \rangle$$ ... ... ?

Could he just have easily chosen the projection of $$\displaystyle \langle 2, 3 \rangle$$ onto $$\displaystyle \langle -1, 2 \rangle$$ ... ... ?Can someone please clarify ...

Peter==================================================================================

It may help MHB readers of the above post to have access to Bachman's Section 3.1 ... ... so I am providing the same .. ... as follows:View attachment 8605
View attachment 8606Hope that helps ...

Peter

Attachments

• Bachman - 1 - Section 3.1 - PART 1 ... .png
26.2 KB · Views: 82
• Bachman - 1 - Section 3.0 - PART 1 ... .png
20.6 KB · Views: 80
• Bachman - 2 - Section 3.0 - PART 2 ... .png
75.5 KB · Views: 87
Peter said:
My question is as follows:

Is there any reason Bachman has chosen the projection of $$\displaystyle \langle -1, 2 \rangle$$ onto $$\displaystyle \langle 2, 3 \rangle$$ ... ... ?

Could he just have easily chosen the projection of $$\displaystyle \langle 2, 3 \rangle$$ onto $$\displaystyle \langle -1, 2 \rangle$$ ... ... ?
In this place, yes. The scalar projection $\text{proj}_vu$ of vector $u$ on vector $v$ is $\dfrac{u\cdot v}{|v|}$, from where $u\cdot v=(\text{proj}_vu)\cdot|v|$. The equality $u\cdot v=(\text{proj}_uv)\cdot|u|$ is equally valid. But the conclusion "Evaluating 1-form on a vector is the same as projecting [that vector] onto some line..." assumes that we project the vector we feed to the form, i.e., $\langle dx,dy\rangle$.

Evgeny.Makarov said:
In this place, yes. The scalar projection $\text{proj}_vu$ of vector $u$ on vector $v$ is $\dfrac{u\cdot v}{|v|}$, from where $u\cdot v=(\text{proj}_vu)\cdot|v|$. The equality $u\cdot v=(\text{proj}_uv)\cdot|u|$ is equally valid. But the conclusion "Evaluating 1-form on a vector is the same as projecting [that vector] onto some line..." assumes that we project the vector we feed to the form, i.e., $\langle dx,dy\rangle$.
Thanks Evgeny

Peter

1. What are 1-forms and how are they used in mathematics?

1-forms are mathematical objects used to describe quantities that have both magnitude and direction. They are typically used in vector calculus and differential geometry, as well as in physics and engineering. They serve as a way to represent and manipulate the concept of "directional derivatives" in multi-dimensional spaces.

2. What is the interpretation of 1-forms in the context of Bachman notation?

In Bachman notation, 1-forms can be interpreted as objects that map tangent vectors to real numbers. In other words, they take in a direction and output a "rate of change" or "slope" in that particular direction. This interpretation is useful in understanding the behavior of functions in multi-dimensional spaces.

3. How are 1-forms related to differential forms?

1-forms are a special case of differential forms, specifically 1-forms are differential 1-forms. Differential forms are generalizations of the concept of 1-forms, allowing for more complex mappings between tangent vectors and real numbers. They are widely used in various branches of mathematics and physics.

4. Can you give an example of a 1-form in real-world applications?

One example of a 1-form in real-world applications is the gradient of a scalar field, which represents the direction and magnitude of the steepest ascent at any point in the field. This is commonly used in physics and engineering to analyze the behavior of scalar fields, such as temperature or pressure distributions.

5. How can 1-forms be used to solve problems in vector calculus?

1-forms can be used in vector calculus to solve problems involving surfaces and curves in multi-dimensional spaces. They can be integrated along curves to calculate the work done by a force or to find the change in a physical quantity. They can also be used to find the flux of a vector field across a surface or to calculate line integrals and surface integrals.

• Topology and Analysis
Replies
1
Views
1K
• Topology and Analysis
Replies
2
Views
1K
• Topology and Analysis
Replies
2
Views
1K
• Topology and Analysis
Replies
4
Views
1K
• Topology and Analysis
Replies
2
Views
1K
• Topology and Analysis
Replies
2
Views
1K
• Differential Geometry
Replies
16
Views
2K
• Topology and Analysis
Replies
5
Views
2K
• Quantum Interpretations and Foundations
Replies
47
Views
2K
• Topology and Analysis
Replies
5
Views
2K