CAL-14

Gradient magnitude

|∇f| = √(fx² + fy² + fz²). Steepest-ascent rate.

Reading speed

MultivariableGradient

Governing equation

|\nabla f|=\sqrt{f_x^2+f_y^2+f_z^2}

$f_x \in \left[-20, 20\right]$

where

f_x: ∂f/∂x (—)
f_y: ∂f/∂y (—)
f_z: ∂f/∂z (—)
|\nabla f|: Gradient magnitude (—)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

|\nabla f|=\sqrt{f_x^2+f_y^2+f_z^2}

$f_x \in \left[-20, 20\right]$

$f_x = -20$ $|\nabla f| = 20.4$

Along the curve

②

Domain of the variable — blue

f_x_{\min} = -20,\quad f_x_{\max} = 20

$f_x \in \left[-20, 20\right]$

$f_x = -20$ $f_x = -20$

Along the curve

③

Response curve — green

|\nabla f|=\sqrt{f_x^2+f_y^2+f_z^2}

$f_x \in \left[-20, 20\right]$

$f_x = -20$ $|\nabla f| = 20.4$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}|\nabla f|}{\mathrm{d}f_x}

$f_x \in \left[-20, 20\right]$

$f_x = -19.75$ $|\nabla f|^{\prime} = -0.98$

Along the curve

Lecture brief

Historical brief

Newton and Leibniz (1670s), Taylor, the fundamental theorem and the trapezoid rule are how change became a number. The lab differentiates, integrates and linearises in one variable. This sheet (CAL-14 — Gradient magnitude) is the form associated with Gradient. Working symbols: $f_x$ , $f_y$ , $f_z$ $\rightarrow$ $|\nabla f|$ . df = ∇f · dr. The directional derivative in unit u is ∇f · u, max when u ∥ ∇f.

Purpose

Purpose: compute

|\nabla f|

from

f_x

f_y

f_z

in Calculus via

|\nabla f|=\sqrt{f_x^2+f_y^2+f_z^2}

|∇f| = √(fx² + fy² + fz²). Steepest-ascent rate. Use it when a real calculus question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

f_x = 3.000\,\mathrm{—}

f_y = 4.000\,\mathrm{—}

f_z = 0.000\,\mathrm{—}

, the governing relation

|\nabla f|=\sqrt{f_x^2+f_y^2+f_z^2}

yields

|\nabla f| = 5.0000\,\mathrm{—}

. Three partials at a point in R³ (set fz = 0 for planar). Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

∂f/∂x f_x3 —∂f/∂y f_y4 —∂f/∂z f_z0 —

Outputs

Gradient magnitude |\nabla f|5.0000 —

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Narration of this film

Three partials at a point in R³ (set fz = 0 for planar).

df = ∇f · dr. The directional derivative in unit u is ∇f · u, max when u ∥ ∇f.

Reading speed

Watch on YouTube