Fig. 4

Where ∇Ŝ and ∇F are similarly aligned, Ŝ and F will both increase along most paths through that point. In each panel, gray shows the part of the x-plane where \(\frac{d\hat{S}}{dF} = \frac{\nabla\hat{S} \cdot d\mathbf {x}}{\nabla F \cdot d\mathbf {x}} > 0\), black where \(\frac{d\hat{S}}{dF}<0\). From left to right: ∇Ŝ and ∇F positively aligned; ∇Ŝ and ∇F orthogonal; ∇Ŝ and ∇F negatively aligned