Tensors for Beginners 0: Tensor Definition

eigenchris
9 Dec 201709:56
EducationalLearning
32 Likes 10 Comments

TLDRThis video offers an introduction to tensors, presenting multiple definitions to help viewers grasp the concept. Starting with the 'array definition,' it explains tensors as multi-dimensional arrays of numbers, including examples like scalars, vectors, and matrices. However, it clarifies that tensors are more than just arrays, possessing geometrical meaning and being 'invariant' under coordinate changes. The 'coordinate definition' describes tensors as objects with components that transform predictably. Lastly, the 'abstract definition' views tensors as combinations of vectors and covectors through the tensor product. The video also hints at tensors' role in calculus, particularly with partial derivatives and gradients.

Takeaways
  • 📚 Tensors are introduced as multi-dimensional arrays of numbers, akin to grids, with the simplest form being a scalar.
  • 🔢 Scalars, or rank zero tensors, are single numbers, which can conceptually be represented within square brackets as arrays.
  • 📈 Vectors, also known as rank one tensors, extend in one dimension and are lists of numbers, forming a one-dimensional array.
  • 📊 Matrices are rank two tensors, characterized as two-dimensional grids of numbers, extending in both vertical and horizontal directions.
  • 🧩 The concept of rank in tensors indicates the number of dimensions, with tensors capable of having ranks three and higher, forming complex multi-dimensional arrays.
  • ❌ The initial 'array definition' of tensors is identified as incorrect for serious mathematical and geometrical understanding.
  • 🌐 Tensors are geometrically meaningful objects that remain invariant under changes in coordinate systems, unlike their components.
  • 📏 The 'coordinate definition' of a tensor emphasizes that while tensors themselves do not change with coordinate systems, their components do in a predictable manner.
  • 🔄 The transformation of components between different coordinate systems is facilitated by 'forward' and 'backward' transformations, to be discussed in subsequent videos.
  • 🔗 The 'abstract definition' of a tensor is presented as a collection of vectors and covectors combined through the tensor product, a concept requiring further explanation.
  • 📚 For those with a calculus background, tensors appear in calculus as partial derivatives and gradients, transforming with the Jacobian matrix, especially relevant in general relativity.
  • 🚀 The video promises to delve into real mathematics in the next installment, starting with forward and backward transformations to fully understand tensors.
Q & A

  • What is the basic definition of a tensor given in the video?

    -The basic definition of a tensor given in the video is a multi-dimensional array of numbers, which can also be thought of as a grid of numbers.

  • What is a scalar in the context of tensors?

    -A scalar, also known as a rank zero tensor, is the simplest example of a tensor, which is essentially just a single number.

  • What is the difference between a vector and a rank 1 tensor?

    -In the context of the video, a vector and a rank 1 tensor are the same; it is a one-dimensional list of numbers extending in one dimension.

  • What is the term used for a two-dimensional grid of numbers in the context of tensors?

    -A two-dimensional grid of numbers is referred to as a matrix or a rank two tensor.

  • Why is the initial 'array definition' of a tensor considered incorrect for a serious mathematical understanding?

    -The initial 'array definition' is considered incorrect for serious mathematical understanding because tensors are not just multi-dimensional arrays of numbers; they have a real geometrical meaning that is missed in this definition.

  • What does it mean for an object to be 'invariant under a change of coordinates'?

    -An object is 'invariant under a change of coordinates' if its fundamental properties do not change regardless of the coordinate system used to describe it, such as the length or orientation of an object.

  • What is the special predictable way in which tensor components change called?

    -The special predictable way in which tensor components change under a change of coordinates is referred to as 'forward' and 'backward' transformations.

  • What is the 'coordinate definition of a tensor'?

    -The 'coordinate definition of a tensor' is an object that remains invariant under a change of coordinates, but its components change in a special predictable way that can be determined using forward and backward transformations.

  • What is the final definition of a tensor given in the video?

    -The final definition of a tensor given in the video is a collection of vectors and covectors combined together using the tensor product.

  • What is a covector and why is it significant in the final definition of a tensor?

    -A covector is a one-dimensional linear functional that maps vectors to scalars. It is significant in the final definition of a tensor because tensors are described as combinations of vectors and covectors through the tensor product.

  • How do tensors relate to calculus, as briefly mentioned in the video?

    -Tensors relate to calculus as they appear as partial derivatives and gradients that transform with the Jacobian matrix, which is important for understanding curved geometry in general relativity.

Outlines
00:00
📚 Introduction to Tensors

The video script introduces the concept of tensors as part of a series. It acknowledges the difficulty in defining tensors and proposes to offer multiple definitions to provide a comprehensive understanding. The script begins by defining tensors as multi-dimensional arrays or grids of numbers, using the terms 'scalar', 'vector', 'matrix', and higher 'rank' tensors to illustrate increasing complexity. However, it clarifies that this definition, while useful, is not entirely accurate as it overlooks the geometrical significance of tensors. The true essence of tensors is hinted at, suggesting they have more to do with geometrical invariance than mere numerical arrays.

05:05
🔍 Deep Dive into Tensor Definitions

This paragraph delves deeper into the accurate definition of tensors, emphasizing their invariance under changes of coordinates and the predictable transformation of their components. It uses the analogy of a pencil pointing towards a door to illustrate the concept of invariance, where the pencil's length and orientation remain constant regardless of the coordinate system used. The script then explains how components, measured in different coordinate systems, can vary, which is crucial for understanding tensors. It introduces the idea of 'forward' and 'backward' transformations between coordinate systems and their corresponding components. The paragraph concludes with the final, abstract definition of a tensor as a collection of vectors and covectors combined through the tensor product, which requires a foundational understanding of these concepts. Additionally, a brief mention of tensors in calculus is made, relating them to partial derivatives and gradients that transform with the Jacobian matrix, a topic that is set to be explored further in the video series.

Mindmap
Keywords
💡Tensor
A tensor is a fundamental concept in the video, defined as a multi-dimensional array of numbers with a geometrical meaning. It is not merely a collection of numbers but an object that retains its essence under different coordinate systems. The video script uses tensors to illustrate concepts like 'rank' and 'invariance', providing examples of scalars, vectors, and matrices as tensors of different ranks.
💡Scalar
A scalar is a type of tensor, specifically a rank zero tensor, which is essentially a single number. In the context of the video, scalars are the simplest form of tensors and are used to introduce the concept of tensors without the complexity of multiple dimensions.
💡Vector
A vector, or rank 1 tensor, is a one-dimensional array of numbers extending in one direction. The script uses vectors to demonstrate the progression from scalars to more complex tensors, emphasizing how vectors maintain their properties across different coordinate systems.
💡Matrix
A matrix, also known as a rank two tensor, is a two-dimensional grid of numbers. The video script uses matrices to further build on the concept of tensors, showing how they can be represented in a grid format and how they relate to the metric tensor in general relativity.
💡Rank
Rank in the context of tensors refers to the number of dimensions in the array. The video script explains that tensors can have various ranks, starting from rank zero for scalars up to higher ranks for more complex structures like three-dimensional grids or 'rank 3 tensors'.
💡Invariant
Invariance in the video script refers to the property of an object, such as a tensor, to remain unchanged under a change of coordinates. This concept is crucial for understanding the geometrical significance of tensors, as it highlights their intrinsic properties that do not depend on the coordinate system used.
💡Coordinate System
A coordinate system in the video is a framework used to describe positions and transformations in space. The script discusses how different coordinate systems can affect the components of a tensor but not the tensor itself, emphasizing the distinction between invariant tensors and their coordinate-dependent components.
💡Components
Components in the script are the individual elements of a tensor that can be measured within a specific coordinate system. The video explains how these components change predictably when the coordinate system is altered, which is a key aspect of tensor transformation.
💡Transformation
Transformation in the video refers to the process of converting from one coordinate system to another and the corresponding change in tensor components. The script introduces the concepts of 'forward' and 'backward' transformations to explain how tensor components adapt when the coordinate system is changed.
💡Tensor Product
The tensor product, mentioned in the script, is an operation that combines vectors and covectors to form tensors. Although not elaborated upon in detail in the provided transcript, it is alluded to as part of the final definition of a tensor, suggesting a more abstract and comprehensive understanding of tensors.
💡Covector
A covector, also known as a one-form, is a concept related to vectors but operates in the dual space. The video script briefly mentions covectors in the context of the tensor product, indicating that tensors can be constructed from both vectors and covectors, although it does not delve into the specifics within the provided transcript.
💡Jacobian Matrix
The Jacobian matrix is introduced in the script as a way to describe how tensors, specifically as partial derivatives and gradients, transform under changes in variables. This concept is particularly relevant for understanding tensors in the context of calculus and their application in fields like general relativity.
Highlights

Introduction to the concept of tensors, emphasizing their importance in understanding advanced mathematical concepts.

Tensors are defined as multi-dimensional arrays of numbers, with the simplest example being a scalar.

Vectors and matrices are examples of rank 1 and rank 2 tensors, respectively.

Higher rank tensors can extend to three or more dimensions, forming complex grids of numbers.

The initial 'array definition' of tensors is identified as potentially misleading without understanding their geometrical meaning.

Tensors are fundamentally objects that remain invariant under changes in coordinates.

Components of tensors change predictably when the coordinate system is altered, a key aspect of their geometrical significance.

Illustration of the concept of invariance using a pencil pointing towards a door, emphasizing the independence of physical properties from coordinate systems.

Demonstration of how the components of an object, like a pencil, vary with different coordinate systems.

Introduction of the 'coordinate definition' of a tensor, focusing on the predictability of component transformations.

The 'abstract definition' of a tensor is presented as a collection of vectors and covectors combined through the tensor product.

Clarification that understanding the abstract definition requires knowledge of vectors, covectors, and the tensor product.

Tensors in calculus are discussed, relating them to partial derivatives and gradients, and their transformation with the Jacobian matrix.

Tensors' relevance in the context of general relativity and curved geometry is briefly mentioned.

A preview of upcoming content, focusing on 'forward' and 'backward' transformations in the next video.

The video concludes with a summary of the importance of understanding tensors for delving into more complex mathematical topics.

Transcripts

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TensorsEducationGeometryMathematicsMultidimensionalArraysVectorsMatricesCalculusRelativity