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4
votes
Why is the gradient normal?
Imagine two surfaces: $S_1 : f(x,y,z) = c_1$ and $S_2 : f(x,y,z) = c_2$ where $c_1 < c_2 $. Now let's see what happens in a region $ x \in (a,a+\Delta x),y \in (b,b+\Delta y) $ in which $S_1$ is appro …