- #1
HasuChObe
- 31
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So I'm looking at Schaum's outlines for Tensors and the definition of a Contravariant vector is
<br /> \bar{T}^i=T^r\frac{\partial\bar{x}^i}{\partial x^r}<br />
Where \bar{x}^i and x^r denote components of 2 different coordinates (the superscript does not mean 'to the power of') and T^i and T^r are contravariant tensors of order 1 (aka, a vector).
Lets say you have some 2-D vector {\bf v}. It can be described as
<br /> {\bf v}=\bar{T}^1\hat{\bar{e}}_1+\bar{T}^2\hat{\bar{e}}_2=T^1\hat{e}_1+T^2 \hat{e}_2<br />
The vector {\bf v} is the same length, but the basis for each vector may be different. If the operation from (\hat{e}_1,\hat{e}_2)\rightarrow(\hat{\bar{e}}_1,\hat{\bar{e}}_2) performs elongation, then (T^1,T^2)\rightarrow(\hat{T}^1,\hat{T}^2) will shrink (and vice versa) to preserve the shape of {\bf v}. In this case, (T^1,T^2) are said to be contravariant vectors because they grow contrary to the direction that the basis grows in. However, the definition I found in Schaum's outlines seem to say the opposite.
For example, if
<br /> \bar{T}^i=2,\,T^r=1,\,\frac{\partial\bar{x}^i}{\partial x^r}=2<br />
Does that not say that going from unbarred to barred coordinates, the vector components are growing and so is the coordinate system? I must be confusing myself.
<br /> \bar{T}^i=T^r\frac{\partial\bar{x}^i}{\partial x^r}<br />
Where \bar{x}^i and x^r denote components of 2 different coordinates (the superscript does not mean 'to the power of') and T^i and T^r are contravariant tensors of order 1 (aka, a vector).
Lets say you have some 2-D vector {\bf v}. It can be described as
<br /> {\bf v}=\bar{T}^1\hat{\bar{e}}_1+\bar{T}^2\hat{\bar{e}}_2=T^1\hat{e}_1+T^2 \hat{e}_2<br />
The vector {\bf v} is the same length, but the basis for each vector may be different. If the operation from (\hat{e}_1,\hat{e}_2)\rightarrow(\hat{\bar{e}}_1,\hat{\bar{e}}_2) performs elongation, then (T^1,T^2)\rightarrow(\hat{T}^1,\hat{T}^2) will shrink (and vice versa) to preserve the shape of {\bf v}. In this case, (T^1,T^2) are said to be contravariant vectors because they grow contrary to the direction that the basis grows in. However, the definition I found in Schaum's outlines seem to say the opposite.
For example, if
<br /> \bar{T}^i=2,\,T^r=1,\,\frac{\partial\bar{x}^i}{\partial x^r}=2<br />
Does that not say that going from unbarred to barred coordinates, the vector components are growing and so is the coordinate system? I must be confusing myself.