Gradient

120. Gradient

The gradient $\nabla 𝑓 (𝑥, 𝑦)$ is a vector of expressions (equations)

𝑓 : ℝ^{𝑛} \to ℝ

The gradient is always a vector in the same space as the input variables

\nabla 𝑓 (𝑥) = [\begin{matrix} \frac{𝜕 𝑓}{𝜕 𝑥_{1}} \\ \frac{𝜕 𝑓}{𝜕 𝑥_{2}} \\ ⋮ \\ \frac{𝜕 𝑓}{𝜕 𝑥_{𝑛}} \end{matrix}] \in ℝ^{𝑛}

Example

\begin{matrix} 𝑓 (𝑥, 𝑦) = 𝑥^{2} + 𝑥 𝑦 + 𝑦^{2} \\ \nabla 𝑓 (𝑥, 𝑦) = [\begin{matrix} \frac{𝜕 𝑓}{𝜕 𝑥} \\ \frac{𝜕 𝑓}{𝜕 𝑦} \end{matrix}] = [\begin{matrix} 2 𝑥 + 𝑦 \\ 𝑥 + 2 𝑦 \end{matrix}] \end{matrix}

Evaluating at a point

If you plug in numbers for $𝑥$ and $𝑦$ , each component becomes a scalar:

(𝑥, 𝑦) = (1, 2) \Rightarrow \nabla 𝑓 (1, 2) = [\begin{matrix} 2 \cdot 1 + 2 \\ 1 + 2 \cdot 2 \end{matrix}] = [\begin{matrix} 4 \\ 5 \end{matrix}]