Learn with Amar
Teaching video
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Browse grades and teaching videos01 · Read and understand
What you will learn
- Find two partial derivatives.
- Normalize a direction vector.
- Interpret a gradient and directional derivative.
Before you start
Polynomial derivatives, vectors, dot products, and the Pythagorean theorem.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
A model temperature field is T(x, y) = 20 + x² + 2y² degrees Celsius, with coordinates in metres. At (1, 2), what is the rate of change in direction (3, 4)?
Why this math matters
On a surface, the rate you experience depends on the direction you move. A partial derivative measures motion along one coordinate while the other stays fixed. The gradient collects these coordinate rates, and a dot product combines them for another direction. Normalizing that direction makes the result a rate per metre travelled.
Set up the model
A useful answer starts with clear assumptions:
- The coefficients carry the units needed for the displayed temperature formula.
- The smooth field is a local classroom model in a horizontal plane.
- The requested direction means movement toward increasing coordinates in the ratio three to four.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Which way does a temperature map increase fastest?
PausedQuestion: Start with the question. Paused.
Question
Start with the question
A model temperature field is T(x, y) = 20 + x² + 2y² degrees Celsius, with coordinates in metres. At (1, 2), what is the rate of change in direction (3, 4)?
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Measure coordinate-direction changes
Tₓ = 2x; Tᵧ = 4y; ∇T(1, 2) = (2, 8)
Hold y fixed for the x derivative and x fixed for the y derivative. At the chosen location, the y-direction increase is steeper.
Convert the direction to unit length
||(3, 4)|| = 5; u = (3/5, 4/5)
The vector describes direction, not a five-metre movement to be substituted directly. Unit length is necessary for a derivative per metre.
Combine the local changes
DᵤT = (2, 8) · (3/5, 4/5) = 38/5 = 7.6 °C/m
The dot product weights each coordinate rate by that coordinate's share of a unit movement.
The result
The local temperature increase in the specified direction is 7.6 degrees Celsius per metre.
The largest local increase is ||∇T|| = √68 ≈ 8.25 °C/m, in the gradient's own direction. For a short displacement, multiply the directional derivative by distance to approximate the change; curvature makes larger moves nonlinear.
Common mistakes to catch
- Using (3, 4) directly returns 38, which is not the rate per metre.
- A gradient is evaluated at a location; it generally changes as you move.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
At (2, −1), find the directional derivative in direction (0, −1).
Show a hint
The given direction already has unit length.
Reveal answer and explanation
4 °C/m
The gradient is (4, −4), and its dot product with (0, −1) is four.
Practice 2
Where does this temperature field attain its smallest value?
Show a hint
Both squared terms are nonnegative.
Reveal answer and explanation
At (0, 0), where T = 20°C
x² + 2y² cannot be negative and equals zero only at the origin, proving a global minimum.
Take the idea with you
Gradients also describe how elevation, cost, or model error changes when several inputs vary together.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: How do changing forces add up along a path?
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