SumWise Visuals / Three-figure sample

Minima, Saddles & Double Wells

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Use three concrete surfaces to distinguish a minimum from a saddle. Supplied JPEG and PNG files work in ordinary presentation and publishing tools without purchasing SumWise.

What zero gradient tells us

For a differentiable function on an open region, an interior local extremum must have zero gradient. The converse is false: a stationary point can be a minimum, maximum, saddle, or a degenerate case. A positive-definite Hessian at a stationary point establishes a strict local minimum; an indefinite Hessian establishes a saddle. A semidefinite or zero Hessian alone is inconclusive.

Here the two minima claims are also global: each function is a sum of nonnegative square terms, and we can solve exactly where all terms vanish.

Figure 21

Elliptical quadratic minimum

Blue and orange elliptical quadratic bowl on black, shown in perspective with independently normalized axes.

The origin is the unique global minimum: f(0,0)=0. Unequal positive curvature creates an elliptical bowl.

Equation: 0.08*x^2+0.18*y^2. Domain: x,y ∈ [-5, 5].

Original JPEG · Explanatory PNG card · Equation/settings notes

Figure 14

Hyperbolic paraboloid saddle

Blue and orange saddle surface curving upward along one direction and downward along the other, on black.

At (0,0), z=0 and the gradient vanishes. Values increase along x and decrease along y, so the origin is a saddle.

Equation: 0.15*(x^2-y^2). Domain: x,y ∈ [-5, 5].

Original JPEG · Explanatory PNG card · Equation/settings notes

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.

Equation: 0.12*(x^2-4)^2+0.2*y^2. Domain: x,y ∈ [-3, 3].

Original JPEG · Explanatory PNG card · Equation/settings notes

Checked mathematics

A / 21: ∇f=(0.16x,0.36y), H=diag(0.16,0.36). Both eigenvalues are positive. f(1,0)=0.08 and f(0,1)=0.18. The unique global minimum is (0,0), z=0. On the displayed square the maximum is 6.5 at its four corners.

B / 14: ∇f=(0.3x,−0.3y), H=diag(0.3,−0.3). At (0,0), f=0 and ∇f=0, but the Hessian is indefinite. f(1,0)=0.15; f(0,1)=−0.15. Over the unrestricted plane the function is unbounded above and below. On the finite displayed square its minimum is −3.75 at (0,±5) and maximum 3.75 at (±5,0). These boundary extrema do not turn the origin into a minimum.

C / 24: ∇f=(0.48x(x²−4),0.4y), H=diag(1.44x²−1.92,0.4). The stationary points are (−2,0), (0,0), (2,0). At either minimum H=diag(3.84,0.4) and f=0. At the origin H=diag(−1.92,0.4), f=1.92: a saddle. On the displayed square the maximum is 4.8 at its corners.

A short worked comparison

All three functions have zero gradient at the origin. Which has a local minimum there? For a small nonzero t, A(t,0)=0.08t² and A(0,t)=0.18t² are both positive. B(t,0)=0.15t² is positive but B(0,t)=−0.15t² is negative. For C, C(t,0)−C(0,0)=−0.96t²+0.12t⁴<0 for 0<|t|<√8, while C(0,t)−C(0,0)=0.2t²>0. Answer: only A has a minimum at the origin. B and C are saddles there.

How to read and use the files

All variables and values are assumed dimensionless. Angles are in radians where applicable. Each image maps x/y independently to [−1,1] and the sampled z range to [−0.8,0.8]. These are perspective illustrations, not equal-axis metric diagrams. Do not infer slopes, angles, distances or exact critical-point locations from pixels.

The double-well mesh samples a minimum near 0.0004724, not exactly zero, because its grid does not land exactly on x=±2. The formula's exact minima remain zero. Coordinate claims refer to the function; no guessed image markers are included. Actual camera and sampled scaling are in each equation/settings note.

Three unchanged original JPEGs: 3000×2000, RGB, embedded sRGB.
Three opaque explanatory PNG cards: 3000×2000, RGB, embedded sRGB. Each source image is proportionally reduced inside the card; no upscaling or surface alteration.
Three readable equation/settings notes with captions and alt text.
This self-contained, printable HTML guide and a file manifest. No font files or software installation needed.

Python exact rational polynomial differentiation and evaluation checked the derivatives, values and Hessians; central finite differences independently checked the gradients. Fifteen preserved source-matched SumWise R3 function evaluations agree. SumWise was not asked to calculate derivatives or classify stationary points in this sample task. The global claims follow from the stated algebraic arguments.

SumWise evaluated the equations and rendered the originals using an unreleased development candidate. No generative image operation was reported. The notes document the production settings; they do not promise that the public Desktop release reproduces this export pipeline. This sample is not a complete curriculum or a scientific certification.