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Undergraduate · Directional derivatives

Directional derivatives: A uniformly gentle field

Directional derivatives: investigate a uniformly gentle field with x² coefficient a = 0.5; y² coefficient b = 0.5.

Predict what will happen, press Play, then pause and explain what changed. Every control also works without playback.

Watch the relationship

Paused
Directional derivatives: A uniformly gentle field. Direction radians: 0. Directional derivative: 1. Maximum unit-direction rate: 1.414A unit direction samples the gradient-1.411.4103.146.28directional derivativedirection angle (radians) → · labeled axes rescale to this model
The point is fixed at (1,1); only direction varies during playback. The quadratic coefficients are positive. A zero directional derivative is a local first-order statement and does not make the function constant along that whole line.

Starts paused. Play once, pause anywhere, or use Step to inspect the mathematics. Playback stops when this panel leaves the screen.

Make it your experiment

Change one value. Notice what follows.

The controls adjust the model. Numbers below describe the current frame. Decimals are rounded.

Direction radians
0
Directional derivative
1
Maximum unit-direction rate
1.414

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Understand what you are seeing

The idea behind the motion.

The directional derivative is a dot product with a unit direction, so it depends on orientation as well as gradient length. Rotating a unit vector through all angles reveals a maximum equal to the gradient norm and a minimum equal to its negative. Perpendicular directions have zero first-order change. This investigation starts with x² coefficient a = 0.5; y² coefficient b = 0.5. Predict the result before playing, then change one parameter while keeping the other fixed to test the reason for the change.

A relationship to keep

f=ax²+by²; ∇f(1,1)=(2a,2b); Dᵤf=∇f·u

Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.

  1. STEP 1

    Set up the mathematical model

    Compute the gradient at the fixed point (1,1), keeping the evaluation point separate from the direction. The starting case is “A uniformly gentle field.”

  2. STEP 2

    Follow the changing quantity

    Rotate a unit direction around the circle and plot the directional derivative against its angle.

  3. STEP 3

    Explain and test the result

    Compare the largest rate with 2√(a²+b²). Explain why using a nonunit direction would change the rate scale.

Your turn to explain

Make a prediction. Test your reasoning.

Keep x² coefficient a = 0.5; y² coefficient b = 0.5. Pause the timeline at 40%. Given direction radians = 2.513, calculate directional derivative, maximum unit-direction rate. Show the substitution into the displayed formula.

Use the values specified in the question. Reset restores the initial values for this investigation.

Compare your explanation

The gradient at (1,1) is (1,1). Its dot product with (cos(2.513274),sin(2.513274)) is -0.221232. Cauchy–Schwarz bounds all unit-direction rates by √(1²+1²)=1.414214. Results: Directional derivative: -0.221; Maximum unit-direction rate: 1.414. Decimal values are rounded; retain the original parameters when checking.

Work through a full lesson

Connect the animation to a worked example and practice questions.

Undergraduate · Calculus

Find the direction of fastest local increase

Maximize a directional derivative under a unit-length constraint.