Hilbert Book Model Project/Slide S3

hbmp	Hilbert Book Model Project	S3
	Clarifying symmetry related charges requires the application of the Template:Font The theorem builds on top of the terms of the first order partial differential equation
	Only terms that contain the spatial nabla $\overrightarrow{\mathrm{\nabla }}$ will be used
	$\phi ={\phi }_r+\overrightarrow{\phi }=\mathrm{\nabla }\psi =({\mathrm{\nabla }}_r+\overrightarrow{\mathrm{\nabla }})({\psi }_r+\overrightarrow{\psi })={\mathrm{\nabla }}_r{\psi }_r-⟨\overrightarrow{\mathrm{\nabla }},\overrightarrow{\psi }⟩+{\mathrm{\nabla }}_r\overrightarrow{\psi }+\overrightarrow{\mathrm{\nabla }}{\psi }_r\pm \overrightarrow{\mathrm{\nabla }}\times \overrightarrow{\psi }$
	Next unit vector $\overrightarrow{n}$ replaces $\overrightarrow{\mathrm{\nabla }}$
	$\overrightarrow{\mathrm{\nabla }}\psi =\overrightarrow{\mathrm{\nabla }}({\psi }_r+\overrightarrow{\psi })=-⟨\overrightarrow{\mathrm{\nabla }},\overrightarrow{\psi }⟩+\overrightarrow{\mathrm{\nabla }}{\psi }_r\pm \overrightarrow{\mathrm{\nabla }}\times \overrightarrow{\psi }⟺\overrightarrow{n}\psi =\overrightarrow{n}({\psi }_r+\overrightarrow{\psi })=-⟨\overrightarrow{n},\overrightarrow{\psi }⟩+\overrightarrow{n}{\psi }_r\pm \overrightarrow{n}\times \overrightarrow{\psi }$
	Finally we define the corresponding Template:Font
	$\iiint \overrightarrow{\mathrm{\nabla }}\psi dV=\text{∯}\overrightarrow{n}\psi dS$ $\iiint ⟨\overrightarrow{\mathrm{\nabla }},\overrightarrow{\psi }⟩dV=\text{∯}⟨\overrightarrow{n},\overrightarrow{\psi }⟩dS$ $\iiint \overrightarrow{\mathrm{\nabla }}\times \overrightarrow{\psi }dV=\text{∯}\overrightarrow{n}\times \overrightarrow{\psi }dS$ $\iiint \overrightarrow{\mathrm{\nabla }}{\psi }_{r}dV=\text{∯}\overrightarrow{n}{\psi }_{r}dS$
Prev		Next