Questão 780303

\(\Delta u\ =\) \(\frac{∂^2u}{∂x^2}+\frac{∂^2u}{∂y^2}+\frac{∂^2u}{∂z^2}\) __e __ \(_{\ \ V}^{\int_{ }^{ }\int_{ }^{ }\int_{ }^{ }}\) \(\Delta u\ dx\ dy\ dz\) = \(_{\ o'}^{\int_{ }^{ }\int_{ }^{ }}\) \(\frac{∂u}{∂n}\ d\) o' \(_{\ \ V}^{\int_{ }^{ }\int_{ }^{ }\int_{ }^{ }}\) \(\Delta u\ dx\ dy\ dz\) = \(_{\ o'}^{\int_{ }^{ }\int_{ }^{ }}\) \(\frac{∂u}{∂n}\ d\) o'

\(\Delta u\) = \(\frac{∂^2u}{∂x^2}+\frac{∂^2u}{∂y^2}+\frac{∂^2u}{∂z^2}\) _e_ \(_{\ \ V}^{\int_{ }^{ }\int_{ }^{ }\int_{ }^{ }}\) \(\left(v\Delta u-u\Delta v\right)\ dx\ dy\ dz\) = \(_{\ o'}^{\int_{ }^{ }\int_{ }^{ }}\) \(\left(v\frac{∂u}{∂n}-u\frac{∂v}{∂n}\right)\ d\) o' \(_{\ \ V}^{\int_{ }^{ }\int_{ }^{ }\int_{ }^{ }}\) \(\left(v\Delta u-u\Delta v\right)\ dx\ dy\ dz\) = \(_{\ o'}^{\int_{ }^{ }\int_{ }^{ }}\) \(\left(v\frac{∂u}{∂n}-u\frac{∂v}{∂n}\right)\ d\) o'

\(\frac{∂^2u}{∂x^2}+\frac{∂^2u}{∂y^2}+\frac{∂^2u}{∂z^2}\) = 0__e __ \(_{\ \ V}^{\int_{ }^{ }\int_{ }^{ }\int_{ }^{ }}\) \(\Delta u\ dx\ dy\ dz\) = \(_{\ o'}^{\int_{ }^{ }\int_{ }^{ }}\) \(\frac{∂u}{∂n}d\) o' \(_{\ \ V}^{\int_{ }^{ }\int_{ }^{ }\int_{ }^{ }}\) \(\Delta u\ dx\ dy\ dz\) = \(_{\ o'}^{\int_{ }^{ }\int_{ }^{ }}\) \(\frac{∂u}{∂n}d\) o'

\(\frac{∂^2u}{∂x^2}+\frac{∂^2u}{∂y^2}+\frac{∂^2u}{∂z^2}\) =0__e__ \(_{\ \ V}^{\int_{ }^{ }\int_{ }^{ }\int_{ }^{ }}\) \(\left(v\Delta u-u\Delta v\right)\ dx\ dy\ dz\) = \(\left(v\ \frac{∂u}{∂n}\ -\ u\ \ \frac{∂v}{∂n}\ \right)\) \(_{\ o'}^{\int_{ }^{ }\int_{ }^{ }}\) \(\left(v\ \frac{∂u}{∂n}\ -\ u\ \ \frac{∂v}{∂n}\ \right)\) do'. \(_{\ \ V}^{\int_{ }^{ }\int_{ }^{ }\int_{ }^{ }}\) \(\left(v\Delta u-u\Delta v\right)\ dx\ dy\ dz\) = \(\left(v\ \frac{∂u}{∂n}\ -\ u\ \ \frac{∂v}{∂n}\ \right)\) \(_{\ o'}^{\int_{ }^{ }\int_{ }^{ }}\) \(\left(v\ \frac{∂u}{∂n}\ -\ u\ \ \frac{∂v}{∂n}\ \right)\) do'.

\(\Delta u=\) \(\frac{∂^2u}{∂x^2}+\frac{∂^2u}{∂y^2}+\frac{∂^2u}{∂z^2}\) __e__ \(_{\ D}^{\int_{ }^{ }\int_{ }^{ }}\) \(\left(\frac{∂Y}{∂x}+\frac{∂X}{∂y}\right)\ dx\ dy\) = \(_{\ L}^{\ \int_{ }^{ }}\) X\ dx\ +\ Y\ dy