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This is me doing some independent study on Tensors because I eventually hope to understand General Relativity.

My question is about the following equation which describe hoe the components of a displacement vector transform when there is a change in the coordinate system.

[itex]d{y}^1 = \frac{∂y^1}{∂x^1} dx^1 + \frac{∂y^1}{∂x^2} dx^2[/itex]

To understand with and example, I drew out a normal 2D Cartesian Coordinate system with the axes [itex]x^1[/itex] and[itex] x^2 [/itex] and drew a new coordinate system with axes [itex]y^1[/itex] and [itex]y^2[/itex]. This new coordinate system was just the x system rotated anticlockwise by 30 degrees. I drew a displacement vector which went from (1,1) to (2,2) essentially a displacement vector of [itex](dx^1,dx^2)[/itex]

Since I already know linear algebra fairly well, I used the inverse transformation of a clockwise rotation of 30 degrees

[itex]

\left( \begin{array}{ccc}

\frac{\sqrt{3}}{2} & \frac{1}{2} \\

\frac{-1}{2} & \frac{\sqrt{3}}{2} \\

\end{array} \right)

[/itex]

to find the values of [itex]y^1[/itex] and [itex]y^2[/itex] and got

[itex] dy^1 =\frac{ \sqrt{3}}{2} dx^2 + \frac{1}{2} dx^2 [/itex]

Now try as I might I cannot seem to be able to figure out how the heck [itex] \frac{∂y^1}{∂x^1} = \frac{\sqrt{3}}{2} [/itex] and [itex] \frac{∂y^1}{∂x^2} = \frac{1}{2} [/itex]

Can someone help me out?

2. Relevant equations

3. The attempt at a solution

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# Homework Help: Contravariant vectors and Transformation Equations

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