Skip to content
Menu
中文Sign in
← All questions

What do the blue and yellow arrows represent in the Lagrange multiplier animation?

The blue arrows represent the gradient of the objective function, ∇f\nabla f, pointing outward from the center of the level curves. The yellow arrows represent the gradient of the constraint function, ∇g\nabla g, which is normal to the constraint curve. The animation uses these vectors to show that at the extremum points, the two gradients are parallel.

Conditions

  • The animation visualizes the gradients of f(x,y)=x2+y2f(x,y) = x^2 + y^2 and g(x,y)=x2+y2/2−1=0g(x,y) = x^2 + y^2/2 - 1 = 0.
  • Blue arrows correspond to ∇f\nabla f.
  • Yellow arrows correspond to ∇g\nabla g.

Reasoning, step by step

  1. Observe the blue arrows radiating from the origin, representing ∇f\nabla f.
  2. Observe the yellow arrows perpendicular to the ellipse, representing ∇g\nabla g.
  3. Note that at the points of tangency between the level curves and the constraint, the blue and yellow arrows align (are parallel).

Example

The script states: 'Blue arrows represent the gradient of the objective function (∇f), pointing outward from the center. Yellow arrows represent the gradient of the constraint function (∇g).'

Common misconceptions

  • Thinking the arrows represent the function values themselves rather than the rates of change (gradients).
  • Believing the yellow arrows point along the constraint curve rather than normal to it.

Watch the explanation

0:23 – 0:27Watch this moment ↗

Connected concepts

Optimization →

Explore next

Answers are generated from source material and independently checked. Consult the original video or creator if something is unclear.