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Title: Mean-Value Property of Harmonic Functions
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Series: Partial Differential Equations
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Chapter: Laplace’s Equation
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YouTube-Title: Partial Differential Equations 4 | Mean-Value Property of Harmonic Functions
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Subtitle on GitHub: pde04_sub_eng.srt missing
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Quiz Content
Q1: Let $u: \mathbb{R}^3 \rightarrow \mathbb{R}$ harmonic with $u(0) = 0$. What is correct?
A1: $u$ vanishes everywhere.
A2: $\int_{S_1(0)} u \, d\sigma = 0$.
A3: $\int_{S_r(0)} u \, d\sigma = 1$ for every $r > 0$.
A4: $\frac{1}{4 \pi} \int_{S_r(0)} u \, d\sigma = 1$.
Q2: Let $u: \mathbb{R}^3 \rightarrow \mathbb{R}$. What does the chain rule tell us?
A1: $ \frac{d}{dr} u(x + r y)|_{r = 0} = \langle \nabla u(x), y \rangle $
A2: $ \frac{d}{dr} u(x + r y)|_{r = 0} = \langle \nabla u(x+y), y \rangle $
A3: $ \frac{d}{dr} u(x + r y)|_{r = 0} = \langle u(x), y \rangle $
A4: $ \frac{d}{dr} u(x + r y)|_{r = 0} = J_u(x) $
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Date of video: 2025-11-11
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Last update: 2025-11
