f(x) GraphLab 3D 2D/3D Graphing Studio, Step-by-Step Matrix & Symbolic Calculus Workbench العربية

Elliptic Paraboloid Convex Bowl z = ax² + by² — Elliptical Level Curves

Strictly convex quadratic surface with positive definite Hessian H = diag(2a, 2b) and nested elliptical contour isolines.

Synchronized 3D Surface Mesh + 2D Marching-Squares Contour Map (Click Contour to Probe)
3D Perspective Mesh (Drag to Orbit) z = f(x, y)
Topographic Isolines & Gradient ∇f f(x, y) = c

Symbolic Partial Derivatives & Differential Geometry Summary

z = f(x, y)0.28*(x^2 + 1.5*y^2) - 2
∂f / ∂x0.56*x
∂f / ∂y0.84*y
Computed Elevation Range Z[-2.000, 9.200]

Explore More 3D Surfaces

Quadric Surfaces
Hyperbolic Paraboloid Saddle Surface z = a(x² − y²) & Minimax Point

Doubly-ruled negative Gaussian curvature surface curving upward along x and downward along y with canonical saddle point at (0,0,0).

z = 0.35*(x^2 - y^2)
Wave Optics & radially Symmetric Fields
3D Radial Sinc Sombrero Surface z = A·sin(kr)/(kr)

Axisymmetric 2D Fourier transform of a circular aperture, displaying central Airy-like peak surrounded by concentric annular ripples.

z = 3.2*sinc(1.8*sqrt(x^2 + y^2))
Differential Geometry
Monkey Saddle Surface z = x³ − 3xy² — Degenerate Zero-Hessian Point

Real part of complex cubic Re(z³) possessing three alternating valleys and ridges where both gradient ∇f and Hessian vanish at (0,0).

z = 0.16*(x^3 - 3*x*y^2)
Non-Convex Optimization
Himmelblau’s Optimization Benchmark f(x,y) = (x²+y−11)² + (x+y²−7)²

Multi-modal optimization test landscape featuring one central local maximum and four distinct global minima equal to zero.

z = 0.025*((x^2 + y - 11)^2 + (x + y^2 - 7)^2) - 2.5
Gradient Descent Benchmarks
Rosenbrock’s Banana Valley f(x,y) = (1 − x)² + b(y − x²)²

Narrow parabolic valley where finding the valley floor is easy, but converging to the global minimum at (1, 1) challenges first-order optimizers.

z = 0.015*((1 - x)^2 + 30*(y - x^2)^2) - 1.5
Electrostatics & Vector Calculus
Gaussian Dipole Field z = A·x·e^(−α(x² + y²)) — Paired Source & Sink

Antisymmetric smooth scalar potential exhibiting paired global maximum and minimum separated across the nodal line x = 0.

z = 3.5*x*exp(-0.45*(x^2 + y^2))

Mathematical Reference & Frequently Asked Questions

How is the tangent plane equation computed for this 3D surface?

At probe point (1.5, 1), the linear tangent plane follows z = f(x₀,y₀) + ∂f/∂x·(x − x₀) + ∂f/∂y·(y − y₀).