- #1

etotheipi

2019 Award

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## Main Question or Discussion Point

We have a basis {##\mathbf{e}_1##, ##\mathbf{e}_2##, ##\dots##} and the corresponding dual basis {##\mathbf{e}^1##, ##\mathbf{e}^2##, ##\dots##}. I learned that a vector ##\vec{V}## can be expressed in either basis, and the components in each basis are called the contravariant and covariant

But I wondered why I often see the terms "covariant vector" and "contravariant vector"? I thought any vector ##\vec{V}## has covariant and contravariant components. As an example, Wikipedia says that velocity is a contravariant vector, but can't velocity have covariant components?

So I hoped someone could explain the distinction here; sorry if this is confused! Thanks!

*components*respectively. So if $$\vec{V} = v^i \mathbf{e}_i = v_j \mathbf{e}^j$$ then the contravariant components are ##(v^1, v^2, \dots)## and the covariant components are ##(v_1, v_2, \dots)##, and you have to transform these differently if you want to express ##\vec{V}## in terms of another basis.But I wondered why I often see the terms "covariant vector" and "contravariant vector"? I thought any vector ##\vec{V}## has covariant and contravariant components. As an example, Wikipedia says that velocity is a contravariant vector, but can't velocity have covariant components?

So I hoped someone could explain the distinction here; sorry if this is confused! Thanks!

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