3D Mathematical Surface Plots - Visualizing Multivariable Functions #2250962 (License: Personal Use)

The left plot shows a blue 3D surface with a single global minimum at the center, resembling a quadratic bowl (e.g., z = x² + y²). The right plot displays a more complex red surface with a toroidal saddle structure-featuring both concave and convex regions-and a central hole, typical of functions like z = (x² + y² - 1)² - x². Both use wireframe grids to emphasize curvature and are plotted over standard Cartesian domains with labeled axes.

Used in academic math/physics resources, data science tutorials, or STEM education pages to illustrate multivariable calculus concepts such as critical points, Hessians, and topological features of surfaces. Matches user intent for visual learning, lecture support, or interactive modeling examples.