Learn with Amar
Teaching video
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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Use an absolute determinant as an area scaling factor.
- Justify the conclusion "Image area=|det J|·1=5" using the stated assumptions.
Before you start
Linear transformations and double integrals.
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
Map (u,v) to (x,y)=(2u+v,u+3v). What area comes from the unit square?
Why this math matters
Use an absolute determinant as an area scaling factor. 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 square is [0,1]².
- The map is used once and is one-to-one here.
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.
Measure how a coordinate map changes area
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Map (u,v) to (x,y)=(2u+v,u+3v). What area comes from the unit square?
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
J=[[2,1],[1,3]]
Columns show the images of unit coordinate directions.
Work through the mathematics
det J=6−1=5
The determinant measures signed area scaling.
Check the conclusion
Image area=|det J|·1=5
The invertible linear map sends the square to a parallelogram with five times the area.
The result
Image area=|det J|·1=5
The invertible linear map sends the square to a parallelogram with five times the area.
Common mistakes to catch
- Area uses the absolute determinant.
- A variable Jacobian must remain inside the integral.
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
What if the two output coordinates were swapped?
Show a hint
A row swap changes determinant sign.
Reveal answer and explanation
Area remains five
Absolute determinant removes orientation sign.
Practice 2
What would determinant zero imply?
Show a hint
The two image directions become dependent.
Reveal answer and explanation
Area collapses to zero
The map cannot give an invertible two-dimensional coordinate change.
Take the idea with you
Relate a mesh-cell transformation to the area weight used in numerical integration.
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: Replace a path integral with endpoint values
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