Cummings Bates
04/25/2024 · Senior High School
prove Green's theorem \( \int_{T}\left(\varphi \nabla^{2} \phi-\phi \nabla^{2} \varphi\right) d T=\int_{\partial T}(\varphi \nabla \phi-\phi \nabla \varphi) \cdot d \partial \) \( A=\varphi \nabla \phi \) and \( B=\Phi \nabla \varphi \)
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Green's theorem states that the integral of the divergence of a vector field over a region is equal to the integral of the vector field over the boundary of that region. The proof involves applying the divergence theorem to a vector field defined as the difference of two gradient products.
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