Tensors for Beginners 6: Covector Transformation Rules

eigenchris
16 Dec 201706:41
EducationalLearning
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TLDRThis video script explores the transformation rules of covectors in the context of dual basis vectors. It explains how to derive new dual basis vectors from the old ones using backward transformation and demonstrates that covector components transform in the same way as the basis vectors, which is covariantly. The script also clarifies why covector indices are written at the top, as they transform oppositely to basis vectors. The summary of transformation rules for basis vectors, vector components, basis covectors, and covector components is provided, highlighting the contravariant and covariant nature of their transformations.

Takeaways
  • πŸ“š All co-vectors can be expressed as linear combinations of the dual basis vectors.
  • πŸ“ Co-vector components are determined by counting how many co-vector lines the basis vector intersects.
  • πŸ”„ Co-vector components transform in the opposite way to vector components.
  • πŸ” The transformation of co-vectors themselves, not just their components, is discussed, focusing on the epsilons.
  • 🌐 The new basis vectors are constructed from the old basis vectors using forward transformation.
  • πŸ”„ To find the transformation coefficients for co-vectors, the script applies epsilon 1 tilde to e 1 and e 2, revealing the coefficients Q 1 1 and Q 1 2.
  • πŸ”„ The backward transformation is used to express the new dual basis vectors in terms of the old ones.
  • πŸ”’ The script demonstrates that the coefficients for the transformation of the dual basis are similar, leading to the conclusion that the backward transformation is used to transition from the old to the new dual basis.
  • πŸ”‘ The script proves that for any dimension, the backward transformation is used for the transition of the dual basis, with Q being equal to the backward transform B.
  • 🧠 The script explains the rationale behind writing co-vector indices at the top, as they transform oppositely to basis vectors.
  • πŸ”„ The transformation rules for co-vector components are shown to be the same as for basis vectors, indicating covariant transformation.
  • πŸ“‰ A sanity check is performed to illustrate the intuitive understanding of co-vector components transforming with the size of the basis vectors.
Q & A

  • What is the relationship between co-vectors and dual basis vectors?

    -Co-vectors can be written as linear combinations of the dual basis vectors. This relationship is fundamental to understanding how co-vectors transform in different basis representations.

  • How are co-vector components obtained?

    -Co-vector components are obtained by counting how many co-vector lines the basis vector pierces. This method provides a way to determine the components of a co-vector in a given basis.

  • What is the significance of the transformation of co-vector components being opposite to that of vector components?

    -The opposite transformation behavior between co-vector components and vector components is crucial for understanding the contravariance and covariance properties of these quantities in different coordinate systems.

  • How do we determine the transformation of co-vectors themselves, not just their components?

    -The transformation of co-vectors themselves is determined by expressing the new dual basis in terms of the old dual basis using coefficients that are found by applying the co-vector to the basis vectors and using linearity rules.

  • What is the role of the epsilon tilde (Ξ΅Μƒ) in the context of co-vectors and basis vectors?

    -Epsilon tilde (Ξ΅Μƒ) represents the new dual basis co-vectors. It is used to express the transformation between the old and new dual basis by applying these co-vectors to the basis vectors.

  • How are the coefficients for the transformation of the dual basis determined?

    -The coefficients for the transformation of the dual basis are determined by applying the epsilon tilde to the basis vectors and using the linearity of covectors to isolate each coefficient.

  • What is the backward transformation in the context of basis vectors and co-vectors?

    -The backward transformation is the process of expressing the old basis vectors (or dual basis co-vectors) in terms of the new basis vectors (or dual basis co-vectors). It is used to revert the forward transformation.

  • Why are the transformation rules for basis vectors and co-vector components opposite?

    -The transformation rules for basis vectors and co-vector components are opposite because co-vector components transform covariantly, while basis vectors transform contravariantly. This is due to the nature of their definitions and the mathematical properties of linear transformations.

  • What does it mean for co-vector components to transform covariantly?

    -For co-vector components to transform covariantly means that they change in the same way as the basis vectors when a change of basis occurs. This is in contrast to contravariant transformation, where the components change in the opposite way.

  • How do we understand the placement of indices in co-vectors and basis vectors?

    -The placement of indices in co-vectors and basis vectors is determined by their transformation properties. Co-vector components have indices on the bottom because they transform covariantly, similar to basis vectors, while the placement of indices in contravariant vectors is opposite.

  • What is the significance of the Kronecker Delta in the context of transformation rules?

    -The Kronecker Delta is used to verify the correctness of the transformation rules. It appears when applying co-vectors to basis vectors, ensuring that the transformation is consistent and that the indices are correctly placed.

  • What is the purpose of the sanity check in the script?

    -The sanity check serves to provide an intuitive understanding of how co-vector components transform when the size of the basis vectors changes. It helps to confirm that the mathematical transformation rules make sense in a practical context.

Outlines
00:00
πŸ“š Covector Transformation Basics

This paragraph introduces the concept of covector transformation, focusing on how to mathematically determine the transformation rules for covectors themselves, not just their components. It explains the process of transforming from an old basis to a new basis by using the dual basis vectors and the epsilons. The explanation includes the application of linearity rules and the backward transformation to express the new dual basis vectors in terms of the old ones. The paragraph also touches on the idea that the coefficients for the transformation are similar, suggesting a connection between the transformation of basis vectors and covectors.

05:07
πŸ” Covector Component Transformation and Intuition

The second paragraph delves into the transformation of covector components, contrasting them with the transformation of basis vectors. It provides an intuitive understanding by illustrating how an increase in the size of the basis vectors results in an increase in the covector components, reinforcing the idea that covector components transform in the same way as the basis vectors. The paragraph summarizes the transformation rules for basis vectors, vector components, basis covectors, and covector components, highlighting their contravariant and covariant nature. It concludes with a brief mention of the next topic, which is the linear map, setting the stage for the following video in the series.

Mindmap
Keywords
πŸ’‘Co-vectors
Co-vectors, also known as covectors or one-forms, are mathematical objects that act on vectors to produce scalar values. They are dual to vectors in the sense that they can be thought of as 'directional derivatives'. In the video, co-vectors are discussed in the context of their transformation rules, which are essential for understanding how they change under different basis representations.
πŸ’‘Dual Basis Vectors
The dual basis vectors are elements of the dual space that are related to the basis vectors of the original vector space. They are used to express co-vectors as linear combinations. In the script, the dual basis vectors are fundamental to understanding how co-vectors transform between different coordinate systems.
πŸ’‘Linear Combinations
A linear combination is an expression constructed from a set of vectors by multiplying each vector by a scalar coefficient and adding the results. In the context of the video, co-vectors are written as linear combinations of the dual basis vectors, which is a key step in determining their transformation properties.
πŸ’‘Co-vector Components
Co-vector components are the scalar coefficients that arise when a co-vector is expressed as a linear combination of the dual basis vectors. The script explains how these components are obtained by counting the number of lines that the basis vector pierces, and how they transform in relation to the basis vectors.
πŸ’‘Transformation Rules
Transformation rules describe how mathematical objects change when the basis of the space is altered. In the video, the focus is on the transformation rules for co-vectors and their components, which are crucial for understanding how these objects behave under changes in the coordinate system.
πŸ’‘Epsilon (Ξ΅)
In the script, 'epsilon' refers to the co-vectors themselves, denoted by Ξ΅. The transformation of these co-vectors is discussed in detail, with the script explaining how they can be expressed in terms of the new basis vectors using the backward transformation.
πŸ’‘Backward Transformation
The backward transformation is the process of expressing the old basis vectors in terms of the new basis vectors. The script uses this concept to derive the transformation rules for co-vectors, showing that the new dual basis vectors can be expressed as linear combinations of the old dual basis vectors.
πŸ’‘Kronecker Delta
The Kronecker Delta is a function that equals 1 if its two indices are the same and 0 otherwise. In the video, it is used to demonstrate the relationship between the forward and backward transformations, and to show that the transformation matrices are inverses of each other.
πŸ’‘Contravariant Transformation
A contravariant transformation is one where the components of a vector or co-vector change in the opposite way to the basis vectors. The script explains that the transformation rules for basis covectors are contravariant, meaning they transform oppositely to the basis vectors.
πŸ’‘Covariant Transformation
Covariant transformation refers to the way in which the components of a tensor, such as a co-vector, change in the same way as the basis vectors. The script concludes that co-vector components transform covariantly, which is why they are written with indices at the bottom.
πŸ’‘Linear Map
A linear map, also known as a linear transformation, is a function that maps vectors from one vector space to another while preserving the operations of vector addition and scalar multiplication. The script mentions that the next video will discuss the linear map as the third example of a tensor, indicating a continuation of the exploration of tensor properties.
Highlights

Co-vectors can be written as linear combinations of the dual basis vectors.

Co-vector components are obtained by counting the number of lines pierced by the basis vector.

Co-vector components transform oppositely to vector components.

The transformation of co-vectors involves building new dual basis vectors from the old ones.

Epsilon tilde of e1 equals Q11, indicating the transformation of the dual basis.

Alternative representation of epsilon tilde involves numerical coefficients.

Backward transformation allows writing old basis vectors in terms of new basis vectors.

Coefficients for transforming the dual basis are similar, suggesting a pattern.

Transformation from old to new dual basis uses the backward transformation.

Proof for any dimension involves dual basis definitions and transformations.

Kronecker Delta is used to establish the relationship between transformations.

Q and F are inverses, with Q being equal to the backward transformation matrix B.

Co-vector indices are written on top because they transform oppositely to basis vectors.

Co-vector components transform in the same way as basis vectors, which is covariantly.

Sanity check confirms that increasing the size of the basis increases the size of co-vector components.

Transformation rules for basis vectors, vector components, basis covectors, and co-vector components are summarized.

Co-vector components transform covariantly, written with indices on the bottom like basis vectors.

Upcoming video will discuss the linear map as the third example of a tensor.

Transcripts

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Related Tags
CovectorsLinear AlgebraTransformationBasis VectorsDual BasisCovector ComponentsMathematicsVector AnalysisKronecker DeltaContravariant