PlanetPhysics/Vector Identities

It is difficult to get anywhere in physics without a firm understanding of [[../Vectors/|vectors]] and their common [[../Cod/|operations]]. Here, we will give vector identities as a reference. Basic terminology to keep straight.

{\mathbf Operation}	{\mathbf Symbol} [[../Gradient/|Gradient]] || $\mathrm{\nabla }f$
[[../LaplaceOperator/|Laplacian]]	${\mathrm{\nabla }}^2$
[[../DivergenceOfAVectorField/|divergence]]	$\mathrm{\nabla }\cdot$
[[../Curl/|curl]]	$\mathrm{\nabla }\times$

{\mathbf [[../Vector/|Vector Magnitude]]

$A=\left|𝐀\right|=\sqrt{{A_x}^2+{A_y}^2+{A_z}^2}$\\ $A=\sqrt{𝐀\cdot 𝐀}$

{\mathbf [[../DotProduct/|scalar product]] (Dot Product)}

$𝐀\cdot 𝐁=A_x{B}_x+A_y{B}_y+A_z{B}_z$ \\ $𝐀\cdot 𝐁=\left|𝐀\right|\left|𝐁\right|\cos \theta$

{\mathbf [[../VectorProduct/|vector product]] (Cross Product)}

$𝐀\times 𝐁=(A_y{B}_z-A_z{B}_y)\widehat{𝐢}+(A_z{B}_x-A_x{B}_z)\widehat{𝐣}+(A_x{B}_y-A_y{B}_x)\widehat{𝐤}$

It can be easier to remember with [[../Determinant/|determinant]] formulation

</math> \mathbf{A} \times \mathbf{B} = \left| \begin{matrix} \mathbf{\hat{i}} & \mathbf{\hat{j}} & \mathbf{\hat{k}} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{matrix}\right| = \left ( A_y B_z - A_z B_y \right ) \mathbf{\hat{i}} + \left ( A_z B_x - A_x B_z \right ) \mathbf{\hat{j}} + \left ( A_x B_y - A_y B_x \right ) \mathbf{\hat{k}}$\mathbf{[}[../BACKCAB/|VectorTripleProduct]],aka.BACCAB<math>𝐀\times (𝐁\times 𝐂)=𝐁(𝐀\cdot 𝐂)-𝐂(𝐀\cdot 𝐁)$

{\mathbf [[../Vectors/|scalar]] Triple Product}

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

{\mathbf [[../Gradient/|Gradient]]}

$\mathrm{\nabla }f=\frac{\partial f}{\partial x}\widehat{𝐢}+\frac{\partial f}{\partial y}\widehat{𝐣}+\frac{\partial f}{\partial z}\widehat{𝐤}$

{\mathbf Gradient [[../Cod/|identities]]}

$\mathrm{\nabla }(f+g)=\mathrm{\nabla }f+\mathrm{\nabla }g$ \\ $\mathrm{\nabla }\left(\alpha f\right)=\alpha \mathrm{\nabla }f$ \\ $\mathrm{\nabla }\left(fg\right)=f\mathrm{\nabla }g+g\mathrm{\nabla }f$ \\ $\mathrm{\nabla }(f/g)=\frac{(g\mathrm{\nabla }f-f\mathrm{\nabla }g)}{g^2}$ \\

{\mathbf [[../Divergence/|Divergence]]}

</math> \nabla \cdot \mathbf{A} = \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} +\frac{\partial A_z}{\partial z} $𝐃ivergenceofthecrossproduct<math>\mathrm{\nabla }\cdot (𝐀\times 𝐁)=𝐁\cdot (\mathrm{\nabla }\times 𝐀)-𝐀\cdot (\mathrm{\nabla }\times 𝐁)$

{\mathbf Divergence of the curl}

$\mathrm{\nabla }\cdot (\mathrm{\nabla }\times 𝐀)=0$

{\mathbf Laplacian Identities}

$\mathrm{\nabla }\times (\mathrm{\nabla }\times 𝐀)=\mathrm{\nabla }(\mathrm{\nabla }\cdot 𝐀)-{\mathrm{\nabla }}^2𝐀$