Gradients and Directional Derivatives — Cheat sheet

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Core rule

Duf=∇f⋅u(∥u∥=1),D_uf=\nabla f\cdot u\quad(\|u\|=1), max⁡∥u∥=1Duf=∥∇f∥.\max_{\|u\|=1}D_uf=\|\nabla f\|.

Watch for

Normalize directions. A zero gradient gives no preferred first-order direction and does not imply the function is locally constant.