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Thread: What’s the difference between a tensor and a vector?

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    What’s the difference between a tensor and a vector?

    I realize that all vectors are tensors, but it sure seems to me that all tensors are vectors too. But perhaps I’m missing something.

    SLD

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    Learn something new every day. The dot product is a tensor, it maps product of two vectors. I always thought of it simplistically as multiple dimensions. A generalization where 2d and 3d vectors are a simplified cases.

    https://en.wikipedia.org/wiki/Tensor

    In mathematics, a tensor is an algebraic object related to a vector space and its dual space that can take several different forms, for example, a scalar, a tangent vector at a point, a cotangent vector (dual vector) at a point, or a multi-linear map between vector spaces. Euclidean vectors and scalars (which are often used in elementary physics and engineering applications where general relativity is irrelevant) are the simplest tensors.[1] While tensors are defined independent of any basis, the literature on physics often refers to them by their components in a basis related to a particular coordinate system.

    An elementary example of mapping, describable as a tensor, is the dot product, which maps two vectors to a scalar. A more complex example is the Cauchy stress tensor T, which takes a directional unit vector v as input and maps it to the stress vector T(v), which is the force (per unit area) exerted by material on the negative side of the plane orthogonal to v against the material on the positive side of the plane, thus expressing a relationship between these two vectors, shown in the figure (right). The cross product, where two vectors are mapped to a third one, is strictly speaking not a tensor, because it changes its sign under those transformations that change the orientation of the coordinate system. The totally anti-symmetric symbol ε i j k {\displaystyle \varepsilon _{ijk}} {\displaystyle \varepsilon _{ijk}} nevertheless allows a convenient handling of the cross product in equally oriented three dimensional coordinate systems.

    Definition[edit]

    Although seemingly different, the various approaches to defining tensors describe the same geometric concept using different language and at different levels of abstraction.

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    Veteran Member skepticalbip's Avatar
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    ^^^
    That seems a bit overly detailed for the question.
    A tensor with only magnitude (no direction) is a scalar.
    A tensor with magnitude and a direction is a vector.
    And then there are higher order tensors .

    But a direct answer to the question is that all vectors are tensors but not all tensors are vectors. Sorta like all oaks are trees but not all trees are oaks.

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    Mazzie Daius fromderinside's Avatar
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    Now an Oaks thread.

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    Contributor barbos's Avatar
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    I suggest to ignore what steve_bank wrote.
    Dot product is not a a tensor, it is represented by a tensor which is redundant because pretty much everything is represented by tensors in linear algebra.

    Anyway, tensor is NOT a vector. I mean not all tensors are vectors. Tensor can be viewed as direct product of n=0,1,2,3.... vectors.
    In simplest non-trivial form n is 2, so it's basically direct product of two vectors (v1)x(v2).
    Vector has one index, direct product of two vectors has 2 indices. In reality of course Tensor is a space of all direct products of all vectors and their sums. So not all (rank 2) tensors can be represented as a pair of vectors, most of them are sums of direct products like (V1)x(V2) + (V3)x(V4)+....

    Also, Tensors are introduced after co-vectors and inner product, I suspect you skipped over that to tensors.

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