Tensors for Beginners 9: The Metric Tensor

This paragraph introduces the concept of the metric tensor, which is essential for measuring lengths and angles in space. It begins by explaining how to calculate the length of a vector in a two-dimensional space using the components of the vector and the Pythagorean theorem. The example given involves a vector 'v' with components that form a right triangle with the basis vectors, leading to the calculation of the vector's length as the square root of the sum of the squares of its components. The paragraph then discusses the limitations of using Pythagoras's theorem in non-orthonormal coordinate systems, highlighting that it only applies correctly in orthonormal systems where basis vectors are of length one and perpendicular to each other. The correct formula for vector length in any coordinate system is shown to be the dot product of the vector with itself, leading to a more general understanding that vector length is invariant under changes in coordinate systems.

This section delves deeper into the computation of vector lengths in different coordinate systems, emphasizing the role of the metric tensor. It explains how the squared length of a vector can be calculated using the dot products of basis vectors in a given coordinate system. The paragraph provides a step-by-step guide to transforming the basis vectors and calculating the necessary dot products to find the metric tensor in a new coordinate system. It also illustrates how the components of the metric tensor can be represented as matrices, which are crucial for computing vector lengths across various coordinate systems. The summary concludes with the insight that the metric tensor's components, derived from basis vector dot products, remain consistent in measuring vector lengths, regardless of the coordinate system used.

The focus of this paragraph is on how the metric tensor can be utilized to measure angles between vectors in space. It starts by setting up a scenario with an orthonormal basis and then introduces a new basis where the second basis vector is rotated to form an angle with the first. Using the definitions of sine and cosine in a right-angled triangle, the paragraph demonstrates how to compute the dot products of the new basis vectors, which are then used to determine the metric tensor for this rotated basis. The key takeaway is that the angle between any two vectors can be found by creating a new basis aligned with the vectors, normalizing their lengths, and then using the dot product to find the cosine of the angle between them. This method is shown to be entirely dependent on the metric tensor and the components of the vectors involved.

This paragraph explains how the components of the metric tensor transform when changing coordinate systems. It outlines the process of obtaining the metric tensor components in a new coordinate system by using dot products involving the new basis vectors, which can then be expressed in terms of the old basis using the forward transformation. The transformation rules for moving from old to new and vice versa are discussed, highlighting the use of forward and backward transformations. The paragraph confirms the invariance of vector length across coordinate systems by applying these transformation rules and demonstrating that the squared length of a vector remains constant. The summary reinforces the understanding that the metric tensor, as a (0,2)-tensor, follows specific transformation rules that ensure the physical properties it represents, such as lengths, remain consistent regardless of the coordinate system used.

The final paragraph provides a comprehensive summary of the key concepts covered in the video series about tensors. It revisits the definitions and properties of tensors, including contravariant and covariant transformation rules for basis covectors and vectors, linear maps, and the metric tensor. The paragraph clarifies the transformation rules that tensors follow when changing coordinates, emphasizing the predictable pattern of transformation based on the number of contravariant and covariant indices, which defines the tensor's type as an (M,N)-tensor. The summary concludes with an introduction to the tensor product, which is the foundation for understanding tensors as collections of vectors and covectors combined in a specific way. This sets the stage for a deeper exploration of the tensor product and its significance in the study of tensors.