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Omni-MATH / Let h(x,y) be a real-valued function that is twice continuously differentiable throughout R², and define ρ(x,y)=yh_(x)−xh_(y). Prove or disprove: For any positive constants d and r…

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problem

Let h(x,y)h(x,y) be a real-valued function that is twice continuously differentiable throughout R2\mathbb{R}^2, and define ρ(x,y)=yhx−xhy.\rho(x,y) = yh_x - xh_y. Prove or disprove: For any positive constants dd and rr with d>rd>r, there is a circle S\mathcal{S} of radius rr whose center is a distance dd away from the origin such that the integral of ρ\rho over the interior of S\mathcal{S} is zero.
Plain-text mathematical notation (without MathML)
Let h(x,y) be a real-valued function that is twice continuously differentiable throughout R², and define
ρ(x,y)=yh_(x)−xh_(y).
Prove or disprove: For any positive constants d and r with d>r, there is a circle S of radius r whose center is a distance d away from the origin such that the integral of ρ over the interior of S is zero.
Original LaTeX notation
Let $h(x,y)$ be a real-valued function that is twice continuously differentiable throughout $\mathbb{R}^2$, and define
\[\rho(x,y) = yh_x - xh_y.\]
Prove or disprove: For any positive constants $d$ and $r$ with $d>r$, there is a circle $\mathcal{S}$ of radius $r$ whose center is a distance $d$ away from the origin such that the integral of $\rho$ over the interior of $\mathcal{S}$ is zero.

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Source and history

task_id
c633eb05-9036-5fd4-a69e-f6df45dd18dc
task_key
test--c633eb05-9036-5fd4-a69e-f6df45dd18dc
task_revision_id
1
config / split
/ test

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