2.5 Vector Analysis

Basic Operations

$\nabla =(\frac{\partial \left(\right)}{\partial \left(x\right)}\frac{\partial \left(\right)}{\partial \left(y\right)}\frac{\partial \left(\right)}{\partial \left(z\right)})$

Coordinates

Cartesian

$𝔽=(F_x,F_y,F_z)$, $d\left(V\right)=d\left(x\right)d\left(y\right)d\left(z\right)$

$\nabla f=(f_x,f_y,f_z)$

Cylindrical

$(r,\mathrm{\theta },z)$, $x=r\cos \left(\mathrm{\theta }\right),$ $y=r\sin \left(\mathrm{\theta }\right)$

$\nabla ƒ=(f_r,\frac{1}{r},f_{\mathrm{\theta }},f_z)$ $d\left(V\right)=rd\left(r\right)$ $d\left(\mathrm{\theta }\right)d\left(z\right)$

$\nabla \cdot 𝔽=\frac{1}{r}{\partial }_r(r{F}_r)+\frac{1}{r}{\partial }_{\mathrm{\theta }}\left(F_{\theta }\right)+{\partial }_z\left(F_z\right)$

Spherical

$(\mathrm{\rho },\mathrm{\theta },\mathrm{\phi })$, $x=\mathrm{\rho }\sin \left(\mathrm{\phi }\right)\cos \left(\mathrm{\theta }\right)$

$\nabla ƒ=(f_{\mathrm{\rho }},\frac{1}{\mathrm{\rho }}{f}_{\mathrm{\theta }},\frac{1}{\mathrm{\rho }}\sin \left(\mathrm{\theta }\right)f_{\mathrm{\phi }})$

$d\left(V\right)={\mathrm{\rho }}^2\sin \left(\mathrm{\theta }\right)d\mathrm{\rho }d\mathrm{\theta }d\mathrm{\phi }$

$\nabla \cdot 𝔽=\frac{1}{{\mathrm{\rho }}^2}{\partial }_{\mathrm{\rho }}\left(({\mathrm{\rho }}^2{F}_P)\right)+\frac{1}{\mathrm{\rho }\sin \left(\mathrm{\theta }\right)}{\partial }_{\mathrm{\theta }}(\sin \left(\mathrm{\theta }\right)F_{\mathrm{\theta }})+\frac{1}{\mathrm{\rho }\sin \left(\mathrm{\theta }\right)}{\partial }_{\mathrm{\phi }}\left(F_{\Phi }\right)$

Integral Theorems

Useful Identities

$\nabla \times (\nabla ƒ)=0$

$\nabla \cdot (\nabla \times 𝔽)=0$

$\nabla \cdot (ƒ𝔽)=ƒ\nabla \cdot 𝔽+\nabla ƒ\cdot 𝔽$

$\nabla \times (ƒ𝔽)=\nabla ƒ\times 𝔽+ƒ\nabla \times 𝔽$

$\nabla \times (\nabla \times 𝔽)=\nabla (\nabla \cdot 𝔽)-\mathrm{\Delta }𝔽$

Vector Identities

$A\times (B\times C)=B(A\cdot C)-C(A\cdot B)$

$\nabla (ƒg)=ƒ\nabla g+g\nabla ƒ$

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