Skip to content
Menu
中文Sign in
Back to exploration
Calculus / Chinese

Understanding the gradient

Charles队长 · Bilibili · 0:31

Open original
Bilibili · Charles队长Original video
READ & KEEP

The explanation, unpacked.

Saves 0

Reviewed learning material · Video analysis · English
Read the full overview

This video visually demonstrates the geometric relationship between directional derivatives and gradients using a 3D surface animation. It shows tangent lines at various angles representing rates of change in different directions, presents the formula for calculating these values, and concludes that the maximum rate of change occurs in the direction of the gradient vector.

Use the learning inspector for key ideas and moments, or open the reading tabs for the complete notes.

Chapters

0:00Title: Directional Derivatives and Gradients0:04Animation: Tangents and Directional Rates0:20Formula & Gradient Properties

Learning script

Generated from the video's visuals and explanation; not verbatim speech.

The segment introduces the topic 'Directional Derivatives and Gradients', exploring how multivariable functions change along specific paths. The visual focus is a downward-opening paraboloid grid.

At a point near the peak, multiple red tangent lines are drawn in different horizontal directions. Text on screen explains that the slope of these tangents represents the directional derivative value for each respective direction. This illustrates that the rate of change depends entirely on the chosen path orientation.

A mathematical formula appears: ∂f/lf/l = (∂f/∂x)cosθ + (∂f/∂y)sinθ, showing how partial derivatives combine based on angle θ. A yellow arrow labeled 'grad f' emerges, indicating the gradient vector. The final text states two key properties: first, the direction yielding the maximum derivative is exactly the gradient's direction; second, the magnitude of this maximum value equals the norm of the gradient vector itself. This assumes differentiability and compares unit directions. At a zero gradient all these rates are zero.

Knowledge cards

01

Concept of Directional Derivative

A directional derivative measures change along a specified unit direction, which may also be a coordinate-axis direction.

∂f∂l\frac{\partial f}{\partial l}
02

Calculation Formula

For a function of two variables, the directional derivative is computed as the dot product of the gradient and the unit direction vector, expanding into sine and cosine components. The gradient dot-product formula requires differentiability at the point; existence of the two partial derivatives alone is insufficient.

∂f∂l=∂f∂xcos⁡θ+∂f∂ysin⁡θ\frac{\partial f}{\partial l} = \frac{\partial f}{\partial x}\cos\theta + \frac{\partial f}{\partial y}\sin\theta
03

Gradient as Maximal Ascent

For a differentiable function with nonzero gradient, the gradient direction maximizes the derivative over unit directions. If the gradient is zero, every unit-direction derivative is zero and there is no unique steepest-ascent direction.

04

Magnitude Relationship

The steepest possible slope at any given point corresponds numerically to the length (norm) of the gradient vector located there.

max⁡∥u∥=1Duf=∥∇f∥\max_{\|u\|=1}D_u f=\|\nabla f\|

Explore the knowledge in this video

Open video knowledge graph →

  • Gradient Explanation
    Why this connection?

    This video visually demonstrates the geometric relationship between directional derivatives and gradients using a 3D surface animation. It shows tangent lines at various angles representing rates of change in different directions, presents the formula for calculating these values, and concludes that the maximum rate of change occurs in the direction of the gradient vector.

  • Directional derivatives Explanation
    Why this connection?

    This video visually demonstrates the geometric relationship between directional derivatives and gradients using a 3D surface animation. It shows tangent lines at various angles representing rates of change in different directions, presents the formula for calculating these values, and concludes that the maximum rate of change occurs in the direction of the gradient vector.

  • Derivatives ExplanationAt 0:21
    Why this connection?

    The reviewed calculation card explains Duf=∇f⋅uD_{\mathbf u}f=\nabla f\cdot\mathbf u for a function differentiable at the point and a unit direction u\mathbf u. Existence of partial derivatives alone is insufficient. A nonzero gradient gives the steepest-ascent direction; a zero gradient gives zero in all unit directions with no unique maximizing direction.

Connect the next step

Watch the reviewed segment · 268s
Watch the reviewed segment · 55s
Watch the reviewed segment · 0s

In another language

  • English video covering Gradient; the current video is in Chinese.

    Gradient

  • English video covering Directional derivatives; the current video is in Chinese.

    Directional derivatives
Watch the reviewed segment · 141s

  • English video covering Directional derivatives; the current video is in Chinese.

    Directional derivatives

  • English video covering Gradient; the current video is in Chinese.

    Gradient
Watch the reviewed segment · 0s

  • English video covering Gradient; the current video is in Chinese.

    Gradient
Watch the reviewed segment · 750s

A different approach

  • Whiteboard for Gradient, compared with the current visual explanation.

    Gradient

  • Whiteboard for Directional derivatives, compared with the current visual explanation.

    Directional derivatives
Watch the reviewed segment · 141s

  • Whiteboard for Directional derivatives, compared with the current visual explanation.

    Directional derivatives

  • Application of Gradient, complementing the current explanation.

    Gradient
Watch the reviewed segment · 0s

  • Whiteboard for Gradient, compared with the current visual explanation.

    Gradient
Watch the reviewed segment · 750s