Tensors are mathematical objects that generalize scalars and vectors, defined not by their components alone, but by how those components transform under a change of basis, and are formally constructed from outer products of vectors in either the tangent or dual spaces, which determines their type and transformation properties.
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🎬Aminated results and Interactive web
$\gg$Mathematical Structures Underlying Physical Laws
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The core idea is that tensors are a generalization of scalars and vectors. While we often think of them in terms of their components-a set of numbers-their true definition lies in how those components transform when you change your coordinate system or basis. This transformation property is what ensures that physical laws expressed with tensors remain the same regardless of the chosen coordinate system.
Tensors of higher rank (like rank two or three) are constructed using the outer product (also known as the tensor product). Unlike the familiar cross product, which results in another vector, the outer product of two vectors, like $v \otimes w$, creates a new mathematical object that lives in a "product space." This outer product serves as a fundamental building block for all higher-rank tensors.
Tensors are categorized by their "type" based on the vector bases used in their construction. This classification is crucial for understanding how their components behave under a change of coordinates:
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The Outer Product and Tensor Transformations
Operations and Properties of Tensors
The Metric Tensor Covariant Derivatives and Tensor Densities
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A disc of mass M and radius R rotating around its symmetry axis with angular velocity
A disc of mass M and radius R rotating around its symmetry axis with angular velocity
The moment of inertia by comparing two discs rotating around different axes
The moment of inertia by comparing two discs rotating around different axes