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Teaching video
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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Calculate a material derivative by adding local and advective change.
- Justify the conclusion "At x=2 the material rate is 13" using the stated assumptions.
Before you start
Chain rule and scalar fields.
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 one-dimensional field is T(x,t)=x²+t and fluid velocity is u=3. Find DT/Dt at x=2.
Why this math matters
Calculate a material derivative by adding local and advective change. 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 velocity and field are differentiable.
- Units are normalized consistently for the example.
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.
Separate change at a point from change along a moving particle
PausedQuestion: Start with the question. Paused.
Question
Start with the question
A one-dimensional field is T(x,t)=x²+t and fluid velocity is u=3. Find DT/Dt at x=2.
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
∂ₜT=1 and ∂ₓT=2x
These derivatives describe local time change and spatial variation separately.
Work through the mathematics
DT/Dt=∂ₜT+u∂ₓT=1+6x
A moving particle samples different positions as well as a changing time.
Check the conclusion
At x=2 the material rate is 13
A fixed sensor reports local rate one, while the moving particle also crosses a spatial gradient.
The result
At x=2 the material rate is 13
A fixed sensor reports local rate one, while the moving particle also crosses a spatial gradient.
Common mistakes to catch
- A steady spatial field can still change along moving trajectories.
- The material derivative is not merely the time partial derivative.
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
If u=0, what rate does the particle observe?
Show a hint
The advective contribution vanishes.
Reveal answer and explanation
One
It remains at a fixed position.
Practice 2
At x=−1 with u=3, what is the material rate?
Show a hint
Substitute into 1+6x.
Reveal answer and explanation
−5
Motion toward smaller field values outweighs the positive local time change.
Take the idea with you
Explain why a drifting thermometer and a stationary thermometer can report different rates.
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: Connect a velocity gradient with Newtonian shear stress
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