Finding 1-forms on Plane Vectors using dx1 and x2

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SUMMARY

The discussion focuses on finding 1-forms on plane vectors using the given vectors v1 and v2 expressed in terms of x1 and x2. The 1-forms provided are w1 = dx1 and w2 = x2*dx1. The user attempts to express dx1 in terms of the basis vectors dv1 and dv2 but encounters difficulties due to the lack of information regarding x2. The key takeaway is the need to understand the relationship between the 1-forms and the basis vectors in a two-dimensional plane.

PREREQUISITES
  • Understanding of vector spaces and linear combinations
  • Familiarity with differential forms and their notation
  • Knowledge of basis vectors in a two-dimensional plane
  • Basic concepts of calculus, particularly derivatives
NEXT STEPS
  • Study the properties of differential forms in linear algebra
  • Learn about the relationship between 1-forms and vector fields
  • Explore the concept of basis transformations in vector spaces
  • Investigate the application of 1-forms in physics and engineering contexts
USEFUL FOR

Students studying advanced calculus, linear algebra, or differential geometry, particularly those working with vector spaces and differential forms.

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Homework Statement



Given three vectors on a plane (x1,x2) with coefficients a and b:

v1 = a1*x1 + b1*x2
v2 = a2*x1 + b2*x2
v3 = a3*x1 + b2*x2 (presumably not used in this part)

Find the 1-forms (below) on the vectors.

Homework Equations



The given 1-forms are:
w1 = dx1
w2 = x2*dx1

The Attempt at a Solution



I can write w1=dx1 as,

dx1 = (dx1/dv1)*dv1 + (dx1/dv2)*dv2

This would seem to state dx1 in terms of a basis dv1, dv2. However, I still don't know x2. This is a problem from our last homework assignment that I didn't know how to do. I'm only posting it in this section because I'm sure we'll have more. Any help is appreciated.
 
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Ok, so I'm pretty sure I'm not supposed to put dx in terms of dv1 since I have three vectors and it's only a 2D plane. So what does it mean to calculate the 1-form "on" a vector? Each vector obviously has a component in x1 and x2.
 

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