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Qmechanic ♦
Qmechanic ♦
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Are $v^ie_{i}$ and $v^iv_{i}$ (where v$v$ are the components and e$e$ the basis vectors) both tensors? Or only the second one?

I am studying the math of tensors, I have an understanding of the concepts of covariance, contravariance, dual spaces, Einstein notation and so on. I am a bit confused about the notation though. My quick questions:

  • Is $v^ie_{i}$ a tensor? ($v$ are the components of a vector and $e$ are the basis vectors)

    Is $v^ie_{i}$ a tensor? ($v$ are the components of a vector and $e$ are the basis vectors)

  • Is $v^iv_{i}$ a tensor?

    Is $v^iv_{i}$ a tensor?

  • If both the above are tensors, how do you distinguish between the two? If not, isn't this an abuse of notation?

    If both the above are tensors, how do you distinguish between the two? If not, isn't this an abuse of notation?

  • When talking about a tensor, does one usually mean $v^iv_{i}$, $v^ie_{i}$, or both? (I guess this depends on the answer to the first two points)

    When talking about a tensor, does one usually mean $v^iv_{i}$, $v^ie_{i}$, or both? (I guess this depends on the answer to the first two points)

Are $v^ie_{i}$ and $v^iv_{i}$ (where v are the components and e the basis vectors) both tensors? Or only the second one?

I am studying the math of tensors, I have an understanding of the concepts of covariance, contravariance, dual spaces, Einstein notation and so on. I am a bit confused about the notation though. My quick questions:

  • Is $v^ie_{i}$ a tensor? ($v$ are the components of a vector and $e$ are the basis vectors)
  • Is $v^iv_{i}$ a tensor?
  • If both the above are tensors, how do you distinguish between the two? If not, isn't this an abuse of notation?
  • When talking about a tensor, does one usually mean $v^iv_{i}$, $v^ie_{i}$, or both? (I guess this depends on the answer to the first two points)

Are $v^ie_{i}$ and $v^iv_{i}$ (where $v$ are the components and $e$ the basis vectors) both tensors? Or only the second one?

I am studying the math of tensors, I have an understanding of the concepts of covariance, contravariance, dual spaces, Einstein notation and so on. I am a bit confused about the notation though. My quick questions:

  • Is $v^ie_{i}$ a tensor? ($v$ are the components of a vector and $e$ are the basis vectors)

  • Is $v^iv_{i}$ a tensor?

  • If both the above are tensors, how do you distinguish between the two? If not, isn't this an abuse of notation?

  • When talking about a tensor, does one usually mean $v^iv_{i}$, $v^ie_{i}$, or both? (I guess this depends on the answer to the first two points)

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Andy4983948
Andy4983948
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Are $v^ie_{i}$ and $v^iv_{i}$ (where v are the components and e the basis vectors) both tensors? Or only the second one?

I am studying the math of tensors, I have an understanding of the concepts of covariance, contravariance, dual spaces, Einstein notation and so on. I am a bit confused about the notation though. My quick questions:

  • Is $v^ie_{i}$ a tensor? ($v$ are the components of a vector and $e$ are the basis vectors)
  • Is $v^iv_{i}$ a tensor?
  • If both the above are tensors, how do you distinguish between the two? If not, isn't this an abuse of notation?
  • When talking about a tensor, does one usually mean $v^iv_{i}$, $v^ie_{i}$, or both? (I guess this depends on the answer to the first two points)