Surfaces, gradients and gradient descent

Type a formula in x and y at the top (x^2 + 3y^2, sin(x)cos(y), x^3 - 3x + y^2, exp(-(x^2+y2))). Implicit multiplication works (2xy, 3exp(x)), as do pi, e, sqrt, exp, ln, log, the trigonometric functions and * for powers. The *view box is x from, x to, y from, y to. The function and the point you pick are shared by every tab; Book examples… opens any worked example from the book.

Surface. Three linked views: the surface (drag to rotate, double-click to reset, arrow keys when focused), the contour plot and the heat map. Hover or tap any of them and the point is marked in all three. The slider sets the number of contour levels, and the readout gives the spacing: bunched lines mean a steep surface.
Partials. Two slices through the point (a, b): y = b in red and x = a in blue. Each slice is an ordinary curve, and its slope at the point is the partial derivative. The panel lists f_x, f_y, the second partials and the mixed partials taken in both orders (Clairaut’s theorem says they agree).
Gradient. The gradient field over the contours (every arrow crosses the contours at right angles) and the dial: drag the orange unit vector u and watch D_uf = ∇f·u = |∇f| cos θ move along a cosine wave. It peaks along ∇f and is zero along the contour through the point.
Tangent plane. The plane L = f(a, b) + f_xΔx + f_yΔy touching the surface, the step (Δx, Δy) and the error f − L, with a log–log plot showing the error shrinking like (step)². The total differential and the worst-case bound |f_x dx| + |f_y dy| give the error propagation for measurements.
Critical points. Every point in the view with ∇f = 0, found by Newton’s method from a grid of starts, with the Hessian, D = f_xxf_yy − f_xy², the eigenvalues and the verdict: bowl (minimum), dome (maximum), saddle, or inconclusive when D = 0.
Lagrange. Type a constraint g(x, y) = c. The tool draws the constraint curve, finds the points where ∇f = λ∇g, and draws the contour of f that just touches the curve there. Drag along the curve (or use slide along) to watch ∇f and ∇g swing into line. λ is shown beside the measured shadow price df*/dc (the tool re-solves at c ± δ), and they agree. Move the c slider to relax the constraint.
Double integral. Columns over a rectangle, with heights from the midpoint, a corner or a random point of each cell. ▶ Refine steps n from 1 to 40 per side. The table shows how the error falls for n = 5 … 500 (about 100 times per tenfold refinement with midpoints).
Descent. Pick a landscape: round or elongated bowl, the ravine from chapter 33, the Rosenbrock banana, two minima, an egg box, or your own f. Click to place the start. Gradient descent, momentum and Adam race from the same point. Set η (logarithmic slider) and β, and press ▶ Play to animate. The loss curve is on a log scale. The stability check reads the Hessian at the lowest minimum and compares η with 2/λ_max (and 2(1 + β)/λ_max for momentum). Colour basins shades every starting point by the minimum it ends in. Ten random starts counts how many reach the global minimum. Gradient flow overlays the continuous path dx/dt = −∇f, which gradient descent follows with Euler steps of size η. The 1-D switch runs x ← x − η f′(x) on a curve, with a table of steps and the bound η < 2/f″(x*).

⛶ Full screen (or F) gives the current tab’s main view (the surface or the contour map) the whole screen — the browser’s full screen where it is allowed, otherwise a full-window layer (iPhone); it is redrawn as the window changes. Escape leaves.

Limits

Derivatives are central differences (h ≈ 10⁻⁵ for gradients, 10⁻⁴ for second derivatives, scaled to the point), so expect agreement with hand calculations to about six significant figures. Critical points and Lagrange candidates are searched for inside the view only, from a grid of starting points. A very narrow feature can be missed, and D = 0 cases are reported as inconclusive rather than classified. The 3-D views clip extreme heights (poles such as 1/x) to the bulk of the values. Stability bounds are exact for quadratic bowls and hold near the minimum for other landscapes. Adam has no simple bound. Momentum is the heavy-ball form v ← βv + ∇f, x ← x − ηv used in the book, and Adam uses β₁ = β and β₂ = 0.999.

Related: Graphing calculator for one-variable functions, Vectors for the dot product behind D_uf, Matrix transformations for the eigenvalues that set the learning rate, ODE solver for gradient flow and Euler’s method, and Neural network to see gradient descent train a model.

Companion to Multivariable Calculus and the Gradient (chapter 20), the gradient-descent teaser in Applications of Derivatives (chapter 18) and Optimisation: the art of going downhill (chapter 33) of Mathematics, Reconnected.

Linking to an example

Share link copies a URL with your settings. You can also write one by hand: /tools/surfaces/?f=x^2+3y2&box=-4,4,-3,3#surface. The part after # picks the tab: surface, partials, gradient, tangent, critical, lagrange, integral or descent. In expressions a literal ` means plus (as typed) and spaces are ignored. A link the tool builds encodes ` as %2B, which also works. Numbers may be written as 1/3 or 2pi. When a link has example=, the other parameters override the example’s settings.

Parameter Meaning Example

Parameter	Meaning	Example
`example`	one of the book presets below (also sets the tab if the link has no `#`)	`example=ch20-partials`
`f`	f(x, y)	`f=x^3+2xy2-y`
`box`	view: x from, x to, y from, y to	`box=-0.5,2.5,0,3`
`lv`	number of contour levels (3–40)	`lv=8`
`a`, `b`	the point (a, b) for partials, gradient, tangent plane and Lagrange	`a=1&b=2`
`u`	direction of the unit vector u, degrees from the x-axis	`u=53.13`
`dx`, `dy`	tangent-plane step Δx, Δy	`dx=0.1&dy=-0.1`
`g`, `c`	Lagrange constraint g(x, y) = c	`g=x+2y&c=8`
`rect`	double-integral rectangle: x from, x to, y from, y to	`rect=0,1,0,2`
`n`	columns per side (the view draws at most 40)	`n=5`
`rule`	column height from `mid`, `ll` (lower-left), `ur` (upper-right) or `rand`	`rule=mid`
`land`	descent landscape: `bowl`, `elongated`, `ravine`, `rosenbrock`, `twomin`, `eggbox`, `custom` (uses `f` and `box`)	`land=ravine`
`sx`, `sy`	descent start	`sx=10&sy=1`
`eta`, `beta`	learning rate η and momentum β	`eta=0.07&beta=0.7`
`m`	methods to race, comma-separated: `gd`, `mom`, `adam`	`m=gd,mom`
`steps`	number of steps	`steps=40`
`basins`, `flow`	colour basins, draw gradient flow (`1`/`0`)	`basins=1`
`dim`	`1` for the one-variable mode of the Descent tab	`dim=1`
`f1`, `s1`	1-D mode: f(x) and the start x₀	`f1=(x-3)^2+1&s1=0`

example

one of the book presets below (also sets the tab if the link has no #)

example=ch20-partials

f

f(x, y)

f=x^3+2xy2-y

box

view: x from, x to, y from, y to

box=-0.5,2.5,0,3

lv

number of contour levels (3–40)

lv=8

a, b

the point (a, b) for partials, gradient, tangent plane and Lagrange

a=1&b=2

u

direction of the unit vector u, degrees from the x-axis

u=53.13

dx, dy

tangent-plane step Δx, Δy

dx=0.1&dy=-0.1

g, c

Lagrange constraint g(x, y) = c

g=x+2y&c=8

rect

double-integral rectangle: x from, x to, y from, y to

rect=0,1,0,2

n

columns per side (the view draws at most 40)

n=5

rule

column height from mid, ll (lower-left), ur (upper-right) or rand

rule=mid

land

descent landscape: bowl, elongated, ravine, rosenbrock, twomin, eggbox, custom (uses f and box)

land=ravine

sx, sy

descent start

sx=10&sy=1

eta, beta

learning rate η and momentum β

eta=0.07&beta=0.7

m

methods to race, comma-separated: gd, mom, adam

m=gd,mom

steps

number of steps

steps=40

basins, flow

colour basins, draw gradient flow (1/0)

basins=1

dim

1 for the one-variable mode of the Descent tab

dim=1

f1, s1

1-D mode: f(x) and the start x₀

f1=(x-3)^2+1&s1=0

Book presets (?example=…):

Name Shows

Name	Shows
`ch20-surface`	Explore a surface: two hills in three linked views (the chapter 20 lab)
`ch20-bowl-levels`	Level curves of the bowl x² + 3y² are ellipses
`ch20-partials`	Slices of x³ + 2xy² − y at (1, 2): f_x = 11, f_y = 7, f_xy = f_yx = 8
`ch20-gradient-field`	The gradient field on the two hills, perpendicular to the contours
`ch20-follow-arrow`	Follow the arrow: at (1, 2) towards (3, 4), D_uf = 12.2; steepest rate 13.04 at 32.5° (lab)
`ch20-tangent-plane`	Tangent plane at (1, 2): L(1.1, 1.9) = 7.4 against f = 7.373
`ch20-error-propagation`	Area xy of a 20 × 10 cm rectangle, each side ±0.1 cm: worst case dA = 3
`ch20-critical`	x³ − 3x + y²: a minimum at (1, 0) (D = 12) and a saddle at (−1, 0) (D = −12)
`ch20-saddle`	x² − 4xy + y²: a single saddle, D = −12
`ch20-flat-test`	x⁴ − y⁴: D = 0, the test is inconclusive
`ch20-gd-bowl`	Gradient descent on x² + 3y² from (3, 1) with η = 0.1: 12 → 6.24 → 3.76 → 2.37
`ch20-learning-rate`	Tune the learning rate: η = 0.34 just above 2/λ_max = 1/3 diverges (lab)
`ch20-rosenbrock`	The Rosenbrock banana valley
`ch20-basins`	Two minima, each start coloured by the minimum it reaches (lab)
`ch20-lagrange-fence`	Fence against a wall: max xy with x + 2y = 8 at (4, 2), λ = 2
`ch20-lagrange-budget`	Coffee and cake: max xy with 2x + 4y = 40 at (10, 5), λ = 2.5
`ch20-lagrange-circle`	x + y on the circle x² + y² = 2: λ = ±½
`ch20-double-integral`	∬ (x² + y) dA over [0, 1] × [0, 2] = 8/3 with 5 × 5 midpoint columns (2.66)
`ch18-gd-1d`	One-variable descent on (x − 3)² + 1 from 0 with η = 0.2: 1.2, 1.92, 2.352 …
`ch18-gd-diverge`	The same with η = 1.1: overshoots and diverges (needs η < 1)
`ch33-optimiser-race`	Gradient descent, momentum and Adam in the ravine ½(x² + 25y²) from (10, 1) (lab)
`ch33-momentum`	Forty steps of plain descent against momentum (β = 0.7, η = 0.07)
`ch33-egg-box`	Egg box: basins and ten random starts (lab)
`ch33-drink-can`	The 330 ml can: minimum area at r ≈ 3.74 cm, h ≈ 7.49 cm; λ = 2/r ≈ 0.534 cm² per ml
`ch33-gradient-flow`	Gradient descent as Euler’s method on gradient flow (η = 0.3 zig-zags off the flow)

ch20-surface

Explore a surface: two hills in three linked views (the chapter 20 lab)

ch20-bowl-levels

Level curves of the bowl x² + 3y² are ellipses

ch20-partials

Slices of x³ + 2xy² − y at (1, 2): f_x = 11, f_y = 7, f_xy = f_yx = 8

ch20-gradient-field

The gradient field on the two hills, perpendicular to the contours

ch20-follow-arrow

Follow the arrow: at (1, 2) towards (3, 4), D_uf = 12.2; steepest rate 13.04 at 32.5° (lab)

ch20-tangent-plane

Tangent plane at (1, 2): L(1.1, 1.9) = 7.4 against f = 7.373

ch20-error-propagation

Area xy of a 20 × 10 cm rectangle, each side ±0.1 cm: worst case dA = 3

ch20-critical

x³ − 3x + y²: a minimum at (1, 0) (D = 12) and a saddle at (−1, 0) (D = −12)

ch20-saddle

x² − 4xy + y²: a single saddle, D = −12

ch20-flat-test

x⁴ − y⁴: D = 0, the test is inconclusive

ch20-gd-bowl

Gradient descent on x² + 3y² from (3, 1) with η = 0.1: 12 → 6.24 → 3.76 → 2.37

ch20-learning-rate

Tune the learning rate: η = 0.34 just above 2/λ_max = 1/3 diverges (lab)

ch20-rosenbrock

The Rosenbrock banana valley

ch20-basins

Two minima, each start coloured by the minimum it reaches (lab)

ch20-lagrange-fence

Fence against a wall: max xy with x + 2y = 8 at (4, 2), λ = 2

ch20-lagrange-budget

Coffee and cake: max xy with 2x + 4y = 40 at (10, 5), λ = 2.5

ch20-lagrange-circle

x + y on the circle x² + y² = 2: λ = ±½

ch20-double-integral

∬ (x² + y) dA over [0, 1] × [0, 2] = 8/3 with 5 × 5 midpoint columns (2.66)

ch18-gd-1d

One-variable descent on (x − 3)² + 1 from 0 with η = 0.2: 1.2, 1.92, 2.352 …

ch18-gd-diverge

The same with η = 1.1: overshoots and diverges (needs η < 1)

ch33-optimiser-race

Gradient descent, momentum and Adam in the ravine ½(x² + 25y²) from (10, 1) (lab)

ch33-momentum

Forty steps of plain descent against momentum (β = 0.7, η = 0.07)

ch33-egg-box

Egg box: basins and ten random starts (lab)

ch33-drink-can

The 330 ml can: minimum area at r ≈ 3.74 cm, h ≈ 7.49 cm; λ = 2/r ≈ 0.534 cm² per ml

ch33-gradient-flow

Gradient descent as Euler’s method on gradient flow (η = 0.3 zig-zags off the flow)