Graduate · Critical norm scaling · 449 of 650
Critical norm scaling · Final spatial factor λ=4; Lebesgue exponent p=4.5
Evaluate at the current scale λ=2.8. Find the norm-scaling exponent, the Lᵖ norm ratio, and the spatial volume ratio. Givens: Final spatial factor λ=4; Lebesgue exponent p=4.5.
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01 · Make a prediction
Your practice question
Let u be a nonzero smooth compactly supported field on R³, with uλ(x)=λu(λx). Use Lᵖ exponent p=4.5 and an animation final scale of 4. Evaluate at the current scale λ=2.8. Find the norm-scaling exponent, the Lᵖ norm ratio, and the spatial volume ratio.
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A starting point
A change of variables gives λ⁻³ for volume. Taking the pth root after including amplitude λᵖ yields λ^(1−3/p).
Work through the reasoning
Step 1
Identify the model and target
Let u be a nonzero smooth compactly supported field on R³, with uλ(x)=λu(λx). Use Lᵖ exponent p=4.5 and an animation final scale of 4. Evaluate at the current scale λ=2.8. The governing relation is uλ(x)=λu(λx); ||uλ||ₚ/||u||ₚ=λ^(1−3/p). A change of variables gives λ⁻³ for volume. Taking the pth root after including amplitude λᵖ yields λ^(1−3/p).
Step 2
Substitute and calculate
The exponent is 1−3/4.5=0.333333. At λ=2.8, the norm ratio is (2.8)^(0.333333)=1.40946; the volume ratio is (2.8)^(−3)=0.045554.
Step 3
Check the mathematical meaning
The exponent is 1−3/4.5=0.333333. The norm ratio 1.40946 has the expected positive exponent; the associated volume factor is λ⁻³=0.045554. Current scale λ: 2.8; Norm scaling exponent: 0.333; Lᵖ norm ratio: 1.409; Volume ratio: 0.046. Decimal values are rounded, so use unrounded intermediate values.
The answer
The exponent is 1−3/4.5=0.333333. At λ=2.8, the norm ratio is (2.8)^(0.333333)=1.40946; the volume ratio is (2.8)^(−3)=0.045554. The exponent is 1−3/4.5=0.333333. The norm ratio 1.40946 has the expected positive exponent; the associated volume factor is λ⁻³=0.045554. Animation check: Current scale λ: 2.8; Norm scaling exponent: 0.333; Lᵖ norm ratio: 1.409; Volume ratio: 0.046. Decimal displays are rounded; retain the original parameters and exact π until the final step.
Compare the method as well as the result. A different valid method may reach the same answer. Keep exact values until the last step when the question asks for rounding.
02 · Explore the model
See the mathematical relationship move.
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Watch the relationship
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The mathematical idea
Changing amplitude and spatial concentration together affects different norms differently. In three dimensions, the change of variables contributes λ⁻³ to volume, while the amplitude contributes λᵖ to the pth-power integral. Their balance makes p=3 invariant under this particular spatial scaling. Let u be a nonzero smooth compactly supported field on R³, with uλ(x)=λu(λx). Use Lᵖ exponent p=4.5 and an animation final scale of 4. Evaluate at the current scale λ=2.8. The requested state occurs at 60% playback; use the exact target stated in the question for your calculation.
uλ(x)=λu(λx); ||uλ||ₚ/||u||ₚ=λ^(1−3/p)
03 · Reflect and transfer
Explain what changes and why.
Why is p=3 critical for this spatial scaling, and why is a scaling identity not a theorem proving fluid regularity?
This is a distinct guided scenario using a reusable mathematical model. Similar-looking diagrams can represent different given values and conclusions; they are not different mathematical theories.