alpha is the so called rapidity (e.g. see http://en.wikipedia.org/wiki/Lorentz_transformation" )jbowers9 said:I'm currently reading DF Lawden's book Introduction to Tensor Calculus, Relativity and Cosmology. He coverss the idea of translation of inertial frames in the beginning and by using Minkowski's device for the time coordinate, x4 = ict, shows that inertial frames moving at velocity u = -ictan(alpha) in a Lorentz transformation are offset by -ictan(alpha). Doesn't this imply that there are singularities at certain points, ie tan90? thank you.
Tensors are mathematical objects that describe how quantities such as scalars, vectors, and matrices change in different coordinate systems. They are important in science because they provide a powerful tool for representing and manipulating physical quantities, making it easier to solve complex problems in fields such as physics, engineering, and computer science.
Yes, DF Lawden's book is an excellent introduction to tensors for beginners. It covers the basics of tensor algebra and calculus in a clear and easy-to-understand manner, making it a great resource for students and scientists who are new to this topic.
Yes, there are many real-world applications of tensors. They are used in fields such as mechanics, electromagnetism, fluid dynamics, and general relativity to model and analyze physical phenomena. They are also widely used in machine learning and data analysis to process and manipulate large datasets.
Yes, tensors can be visualized in three-dimensional space. For example, a second-order tensor (matrix) can be represented as a set of three orthogonal axes, with the length and direction of each axis representing the magnitude and direction of the tensor's components. Higher-order tensors can also be visualized in a similar manner, although it becomes more complex as the number of dimensions increases.
Tensors and matrices are both mathematical objects that involve multiple dimensions. However, tensors are more general than matrices, as they can have any number of dimensions and can represent a wider range of physical quantities. Matrices, on the other hand, are limited to two dimensions and can only represent linear transformations. In other words, tensors can be thought of as a generalization of matrices to higher dimensions.