Learn with Amar
Teaching video
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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Differentiate a composite scalar field through all moving coordinates.
- Justify the conclusion "F(r(t))=t⁴, whose derivative is 4t³" using the stated assumptions.
Before you start
Partial derivatives and parametric curves.
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
For F(x,y)=x²y and r(t)=(t,t²), compute dF(r(t))/dt.
Why this math matters
Differentiate a composite scalar field through all moving coordinates. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- The field and trajectory are differentiable.
- The parameter t is real.
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.
Track a field along a moving trajectory
PausedQuestion: Start with the question. Paused.
Question
Start with the question
For F(x,y)=x²y and r(t)=(t,t²), compute dF(r(t))/dt.
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.
Build the model
∇F=(2xy,x²), r′=(1,2t)
The gradient measures coordinate sensitivities and the velocity supplies coordinate rates.
Work through the mathematics
∇F(r(t))·r′(t)=(2t³,t²)·(1,2t)=4t³
Both coordinate contributions belong in the chain rule.
Check the conclusion
F(r(t))=t⁴, whose derivative is 4t³
Direct substitution independently verifies the multivariable calculation.
The result
F(r(t))=t⁴, whose derivative is 4t³
Direct substitution independently verifies the multivariable calculation.
Common mistakes to catch
- A partial derivative alone omits changes through other coordinates.
- Evaluate the gradient along the trajectory before taking the dot product.
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
Find the rate at t=2.
Show a hint
Evaluate 4t³.
Reveal answer and explanation
32
The composite grows at rate 4·8.
Practice 2
If F also depends explicitly on time, what term is added?
Show a hint
Keep position fixed while differentiating time.
Reveal answer and explanation
∂F/∂t
Total change includes local time change plus motion through the spatial gradient.
Take the idea with you
Distinguish a moving sensor's temperature change from the change measured at one fixed location.
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: Find the direction of fastest local increase
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