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I understand that the gradient is the direction fastest rate of change and why this is true, but just because its direction is orthogonal to the surface, doesn't mean its magnitude is that of the differential tangent plane.

I understand the differential plane being derived from a cross-product of tangent lines $(dx,\,0,\, \frac{\partial{z}}{\partial{x}}dx)$ and $(0,\,dy,\,\frac{\partial{z}} {\partial{y}}dy)$

but it never made sense why taking the magnitude of the gradient would give the same result. I guess this question could also be answered as, why is the gradient of the level surface the same cross product of the partial-derivatives?

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