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I have been learning about tensors recently but I am still confused about one thing. I understand that tensors are just objects whose elements transform in a fixed way so that the object is coordinate independent.

The thing that I don't understand is the covariant derivative. According to Wikipedia, the covariant derivative is the component of the derivative along the space.(The component perpendicular to the space in an $n+1$ Euclidean space is neglected).

But what do we mean by that?

Do tensors have directions for it to have components perpendicular to the space? I have heard people tell me that tensors point in multiple directions. I understand the representation of vectors as $n×1$ matrices.

But I know how a $n×n$ matrix can be assigned directions. But if scalars don't have a direction and is still considered a tensor, then what do they mean by a tensor pointing in multiple directions?

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  • $\begingroup$ Can you clarify this question a bit? Are you asking about tensors or covariant derivatives? Tensors don't "point in multiple directions," so that's a bit of a non-starter. That's also not the definition of the covariant derivative, so if you reference a particular source then perhaps we can clarify the issue. $\endgroup$ – J. Murray Jul 23 '17 at 7:14
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I have been learning about tensors recently but I am still confused about one thing. I understand that tensors are just objects whose elements transform in a fixed way so that the object is coordinate independent.

Agreed, the physical entity represented by the tensor has a certain relationship to the other objects surrounding it, and all the tensor does is maintain that relationship for all observers. I do appreciate you know this, it's just that I think you are making life difficult for yourself in your next paragraph.

The thing that I don't understand is the covariant derivative. According to Wikipedia, the covariant derivative is the component of the derivative along the space.(The component perpendicular to the space in an n+1n+1 Euclidean space is neglected).But what do we mean by that?

In my opinion, you are overthinking it here, based on a Wikipedia post that covers a very important point about the different aspects of how you produce a new tensor (which has exactly the same straightforward job as the original, undifferentiated tensor, in the sense that everybody has to agree on the results obtained post differentation). But you know how to derive the covariant derivative tensor, (at least I assume you do) and why it has the form it does, compared to ordinary derivatives.

Do tensors have directions for it to have components perpendicular to the space? I have heard people tell me that tensors point in multiple directions.

I think you were on track until you tried to understand a Wikipedia page that is very dense, and tries to cover the subject from the physical and mathematical point of view. I would recommend, as you are heading off track at this point, (for your level, and mine, so no offence) that you read another description of covariant derivatives, such as Physicspages: covariant derivatives, rather than Wikipedia.

You had the right idea in how to treat a tensor, at the start of your post then, because ordinary derivatives can be used to indicate direction, this has led you to get bogged down in stuff that is not important, as long as you remember that all tensors do the same job, and what's "inside" them, in the matrix representation, does not really matter as much as you currently think it does.

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