# What is the effect of axis rotation on functions defined on $\mathbb{R}^{2}$

I haven't studied multivariable calculus yet but I have a question that bothers me. Let $F$ be a function $\mathbb{R}^2 \to \mathbb{R}$. Imagine that we rotate the co-ordinate axis by an angle $\theta$. I think the shape of the function should change. How should this function change if we make a rotation of the co-ordinate axis by some angle?

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Do you mean to rotate in the plane of inputs $R^2$? If so this will only rotate the entire graph of $F$. If you want to rotate the entire $R^3$ in which the graph lies, the rotation may not even give the graph of a function. –  coffeemath Nov 4 '12 at 23:04
Yes, I mean to rotate the domain . How can we express the new function in terms of the old function and the angle $\theta$? –  Nabil Nov 5 '12 at 8:45

Let $F : \mathbb{R}^2 \to \mathbb{R}$ be the given function, and let $G_{\theta} : \mathbb{R}^2\to\mathbb{R}^2$ be rotation by $\theta$, then you want to consider the function $F\circ G_{\theta}$. Note that $G_{\theta}$ is a linear transformation and its standard matrix is a rotation matrix so we obtain
$$G_{\theta}\left(\left[\begin{array}\ x\\ y\end{array}\right]\right) = \left[\begin{array}\ \cos\theta & -\sin\theta\\ \sin\theta & \cos\theta\end{array}\right]\left[\begin{array}\ x\\ y\end{array}\right] = \left[\begin{array}\ x\cos\theta - y\sin\theta\\ x\sin\theta + y\cos\theta\end{array}\right].$$
$$(F\circ G_{\theta})(x, y) = F(G_{\theta}(x, y)) = F(x\cos\theta - y\sin\theta, x\sin\theta + y\cos\theta).$$
Note that the graph of $F\circ G_{\theta}$ is just the graph of $F$ rotated around the $z$-axis by $\theta$.