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Let $T(\theta): \Bbb R^3 \to \Bbb R^3$ be the rotation on the angle $\theta$ around $z$-axis. Write it in terms of coordinates $x, y, z$ and compute its Jacobian matrix.

I know the rotation matrix in terms of $\theta$, but how can I write it in terms of $x, y$ and $z$, is that $\cos \theta = x/y$?

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    The wording is confusing, but I believe they just want you to write out the matrix for $T(\theta)$:$$\begin{bmatrix}t_{11}&t_{12}&t_{13}\\t_{21}&t_{22}&t_{23}\\t_{31}&t_{32}&t_{33}\end{bmatrix}$$ where the $t_{ij}$ are expressions in terms of $\theta$, not $x, y, z$. In any case, $\cos \theta \ne x/y$. Perhaps you are confusing the cosine ($\cos$) with cotangent ($\cot$)? – 2017-02-08
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    no, I know the matrix in terms of theta, but they asked for coordinates in terms of x, y, z... – 2017-02-08
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    In terms of the coordinates of what? Coordinates measure something, and the only measurement defined in the problem (as you have given it here) is $\theta$. Either you've left something out of the problem statement, or "expressing it in terms of coordinates" as you are interpreting it makes no sense. That is why I assumed that it was a poorly worded request for the rotation matrix for $\theta$. If that isn't the case, is there more to the problem you didn't give? – 2017-02-08
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    I am also confused... but that's all the question say – 2017-02-08
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    I still suspect my interpretation is correct, and that the point of this problem is to compute the Jacobian of the rotation matrix. But if not, you need to ask your teacher to explain the question. – 2017-02-08

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