Figure 21
Elliptical quadratic minimum
The origin is the unique global minimum: f(0,0)=0. Unequal positive curvature creates an elliptical bowl.
z = 0.08*x^2+0.18*y^2
x,y ∈ [-5, 5]; dimensionless.
Minima, Saddles & Double Wells
Three checked examples for presentations and technical articles, with equations and explanations.
Figure 14
Hyperbolic paraboloid saddle
At (0,0), z=0 and the gradient vanishes. Values increase along x and decrease along y, so the origin is a saddle.
z = 0.15*(x^2-y^2)
x,y ∈ [-5, 5]; dimensionless.
Figure 24
Double-well surface
Global minima at (−2,0) and (2,0) have z=0. The stationary point at (0,0), z=1.92, is a saddle.
z = 0.12*(x^2-4)^2+0.2*y^2
x,y ∈ [-3, 3]; dimensionless.
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The sample contains three unchanged 3000×2000 sRGB JPEGs, three opaque 3000×2000 explanatory PNG cards, captions and alt text, readable equation/settings notes, and a printable HTML guide. No SumWise purchase is needed to use these files. Preview images here are 1200×800.
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Use these graphics nonexclusively inside presentations, teaching materials, articles, websites, videos and commercial projects. Reasonable cropping, resizing and annotations are permitted. Credit is optional, not mandatory.
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Zero gradient is not enough
The bowl has positive curvature in every direction at its minimum. The hyperbolic paraboloid has opposing curvature signs at its saddle. The double well has two global minima and a stationary saddle between them. The guide checks gradients, Hessians, coordinates and finite-domain limitations.
Proposed full pack — $29
One planned topic pack, not an available SKU or completed catalog. Three foundation figures exist now. A bounded nine-concept specification proposes six additional explanatory deliverables, including boundary extrema, curvature, global proofs and degenerate/tilted examples. Those six are not yet produced as pack figures. Price and demand are unvalidated; no checkout is available.
What these pictures mean
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No generative image operation was reported. These are synthetic mathematical illustrations, not measured data or certified scientific results.