In my homework problem I have a function $f(x, y) = z$ and a point $P = (x_{0}, y_{0})$, and am asked in which of four direction vectors the function $f$ does not change in $P$. My plan is to calculate the directional derivative $$ D_{\vec{u}} f(x_{0}, y_{0}) = f_{x} (x_{0}, y_{0}) \cdot u_{0} + f_{y} (x_{0}, y_{0}) \cdot u_{1}, $$ where $\vec{u} = (u_{0}, u_{1})$ is a direction vector, evaluate this in the four given directions and look which equals zero; then pick the related direction vector as the answer. Is this correct?
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