You stand on a hillside where the height is $h(x,y)$ . What does the gradient $\nabla h$ at your feet tell you?

Hints

The directional derivative is $\nabla h \cdot \hat{u}$ , maximized when $\hat{u}$ lines up with $\nabla h$ .
Water flows along $-\nabla h$ , and the summit may lie in a completely different direction from the locally steepest path.

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C. The direction of steepest ascent, and its slope

Why

$\nabla h$ points straight up the local slope and its length is that slope. It is a strictly local instrument, a compass for "uphill", not a map to the peak. Gradient-descent algorithms in machine learning walk against exactly this vector.

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You stand on a hillside where the height is h(x,y)h(x,y)h(x,y). What does the gradient ∇h\nabla h∇h at your feet tell you?

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You stand on a hillside where the height is $h(x,y)$ . What does the gradient $\nabla h$ at your feet tell you?