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}

where

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

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 —

Reading speed

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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