Back to main menu Coordinate Systems and Formulae Cartesian Coordinates $$f = f(x,y,z), \vec{F} = (F_1, F_2, F_3)$$ $$\nabla \cdot \vec{F} = \nabla \cdot (F_1(x,y,z)\hat{i} + F_2(x,y,z)\hat{j} + F_3(x,y,z)\hat{k}) = (F_1)_x + (F_2)_y + (F_3)_z$$ $$(\nabla f)_x = f_x, \qquad (\nabla f)_y = f_y, \qquad (\nabla f)_z = f_z$$ $$(\nabla \times \vec{F})_x = \frac{\partial F_3}{\partial y} ... ...rac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y}$$ $$\nabla^2 f = f_{xx} + f_{yy} + f_{zz}$$ Cylindrical Coordinates $$f = f(r,\theta,z), \vec{F} = (F_r, F_\theta, F_z)$$ $$\nabla \cdot \vec{F} = \frac{1}{r}\frac{\partial}{\partial r... ...al F_\theta}{\partial \theta} + \frac{\partial F_z}{\partial z}$$ $$(\nabla f)_r = \frac{\partial f}{\partial r}, \qquad (\nabla... ...al \theta}, \qquad (\nabla f)_z = \frac{\partial f}{\partial z}$$ $$(\nabla \times \vec{F})_r = \frac{1}{r}\frac{\partial F_z}{\... ...ac{\partial F_r}{\partial z} - \frac{\partial F_z}{\partial r},$$ $$(\nabla \times \vec{F})_z = \frac{1}{r}\frac{\partial}{\part... ...x}(rF_\theta) - \frac{1}{r}\frac{\partial F_r}{\partial \theta}$$ $$\nabla^2 f = \frac{1}{r}\frac{\partial}{\partial r}\left( r ... ...ial^2 f}{\partial \theta^2} + \frac{\partial^2 f}{\partial z^2}$$ Spherical Coordinates $$f = f(\rho, \phi, \theta), \vec{F} = (F_\rho, F_\phi, F_\theta)$$ $$\nabla \cdot \vec{F} = \frac{1}{\rho^2} \frac{\partial}{\par... ...ac{1}{\rho \sin \phi} \frac{\partial F_\theta}{\partial \theta}$$ $$(\nabla f)_\rho = \frac{\partial f}{\partial \rho}, \qquad (... ...a = \frac{1}{\rho \sin \phi} \frac{\partial f}{\partial \theta}$$ $$(\nabla \times \vec{F})_\rho = \frac{1}{\rho \sin \phi} \lef... ...hi F_\theta) - \frac{\partial F_\phi}{\partial \theta} \right],$$ $$(\nabla \times \vec{F})_\phi = \frac{1}{\rho \sin \phi} \fra... ... \frac{1}{\rho} \frac{\partial}{\partial \rho} (\rho F_\theta),$$ $$(\nabla \times \vec{F})_\theta = \frac{1}{\rho} \left[ \frac... ...o} (\rho F_\phi) - \frac{\partial F_\rho}{\partial \phi}\right]$$ $$\nabla^2 f = \frac{1}{\rho^2}\frac{\partial}{\partial \rho}\... ...ac{1}{\rho^2 \sin^2\phi} \frac{\partial^2 f}{\partial \theta^2}$$ Formulae Involving The Del Operator If $\vec{A}$ and $\vec{B}$ are smooth vector fields and $\phi$ and $\psi$are smooth scalar functions of position (x,y,z) then 1. $\nabla(\phi + \psi) = \nabla \phi + \nabla \psi$ 2. $\nabla \cdot (\vec{A} + \vec{B}) = \nabla \cdot \vec{A} + \nabla \cdot \vec{B}$ 3. $\nabla \times (\vec{A} + \vec{B}) = \nabla \times \vec{A} + \nabla \times \vec{B}$ 4. $\nabla \cdot (\phi \vec{A}) = (\nabla \phi)\cdot \vec{A} + \phi(\nabla \cdot \vec{A})$ 5. $\nabla \times (\phi\vec{A}) = (\nabla \phi)\times \vec{A} + \phi(\nabla \times \vec{A})$ 6. $\nabla \cdot (\vec{A} \times \vec{B}) = \vec{B} \cdot (\nabla \times \vec{A}) - \vec{A} \cdot (\nabla \times \vec{B})$ 7. $\nabla \times (\vec{A} \times \vec{B}) = (\vec{B} \cdot \nabla)\vec{A} - \vec{B... ... \cdot \vec{A}) - (\vec{A} \cdot \nabla)\vec{B} + \vec{A}(\nabla \cdot \vec{B})$ 8. $\nabla (\vec{A} \cdot \vec{B}) = (\vec{B} \cdot \nabla)\vec{A} + (\vec{A} \cdot... ...\vec{B} \times (\nabla \times \vec{A}) + \vec{A} \times (\nabla \times \vec{B})$ 9. $\nabla \cdot \nabla \phi = \phi_{xx} + \phi_{yy} + \phi_{zz}$, in Cartesian coordinates 10. $\nabla \times (\nabla \phi) = 0$ 11. $\nabla \cdot (\nabla \times \vec{A}) = 0$ 12. $\nabla \times (\nabla \times \vec{A}) = \nabla (\nabla \cdot \vec{A}) - \nabla^2 \vec{A}$ Back to main menu