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johann1301
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Whats the difference between tensors and vectors?
A tensor is a multilinear map between Cartesian products of vector spaces and real numbers. In more colloquial terms, a tensor assigns a real number to a list of vectors, where each vector's map is linear. An example is a tensor whose input is a list of two Euclidean vectors, and whose output is the signed area of the unique parallelogram spanned by those vectors. Another example with two vector inputs and one real output, linear in both arguments, is the dot product.johann1301 said:Whats the difference between tensors and vectors?
A tensor is a mathematical object that describes the relationship between different coordinate systems. It is a generalization of a vector and can have multiple components in each dimension.
A vector is a mathematical object that has both magnitude and direction. It is often represented as an arrow in a specific coordinate system and can be used to represent physical quantities such as velocity and force.
Tensors are a generalization of vectors, meaning that all vectors can be considered tensors, but not all tensors are vectors. Vectors are a special case of tensors with only one component in each dimension.
The main difference between tensors and vectors is their number of components. Vectors have only one component in each dimension, while tensors can have multiple components in each dimension. Additionally, tensors have higher order and can describe more complex relationships between coordinate systems.
Tensors and vectors are used in science to describe and analyze physical phenomena and relationships between different quantities. They are commonly used in fields such as physics, engineering, and mathematics to model and solve problems in various areas of study, such as mechanics, electromagnetics, and fluid dynamics.