Vector operations

“del” (or “nabla”) operator

\[\nabla =\frac{\partial}{\partial x}\mathbf{i}+\frac{\partial}{\partial y}\mathbf{j}+\frac{\partial}{\partial z}\mathbf{k}\]

Gradient of a scalar:

\[\nabla s=\frac{\partial s}{\partial x}\mathbf{i}+\frac{\partial s}{\partial y}\mathbf{j}+\frac{\partial s}{\partial z}\mathbf{k}\]

Directional derivative:

\[\frac{\mathrm{d}s}{\mathrm{d}l}=\nabla s\cdot\mathbf{e}_l=\Vert\nabla s\Vert\cos(\nabla s,\mathbf{e}_l)\]

Divergence of a vector:

\[\nabla\cdot\mathbf{v}=\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}\]

Laplacian of a scalar s:

\[\nabla\cdot(\nabla s)=\nabla^2 s=\frac{\partial^2 s}{\partial x^2}+\frac{\partial^2 s}{\partial y^2}+\frac{\partial^2 s}{\partial z^2}\]

Laplacian of a vector:

\[\nabla^2 \mathbf{v}=(\nabla^2 u)\mathbf{i}+(\nabla^2 v)\mathbf{j}+(\nabla^2 w)\mathbf{k}\]

Curl of a vector:

\[\nabla\times\mathbf{v}=\left(\frac{\partial}{\partial x}\mathbf{i}+\frac{\partial}{\partial y}\mathbf{j}+\frac{\partial}{\partial z}\mathbf{k}\right)\times(u\mathbf{i}+v\mathbf{j}+w\mathbf{k})=\left\vert\begin{matrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ u & v & w \end{matrix}\right\vert\]

Divergence of a vector with its gradient:

\[[(\mathbf{v}\cdot\nabla)\mathbf{v}]=(u\mathbf{i}+v\mathbf{j}+w\mathbf{k})\cdot(u\mathbf{i}+v\mathbf{j}+w\mathbf{k})\]

Tensor operations

Dyadic product of a vector by itself: ->tensor

\[\begin{aligned} \{\mathbf v\mathbf v\}= & (u\mathbf{i}+v\mathbf{j}+w\mathbf{k})(u\mathbf{i}+v\mathbf{j}+w\mathbf{k}) \\ = & uu\mathbf{i}\mathbf{i}+uv\mathbf{i}\mathbf{j}+uw\mathbf{i}\mathbf{k}+ \\ & vu\mathbf{j}\mathbf{i}+vv\mathbf{j}\mathbf{j}+vw\mathbf{j}\mathbf{k}+ \\ & wu\mathbf{k}\mathbf{i}+wv\mathbf{k}\mathbf{j}+ww\mathbf{i}\mathbf{k} \\ = & \left[\begin{matrix} uu & uv & uw \\ vu & vv & vw \\ wu & wv & ww \\ \end{matrix}\right] \end{aligned}\]

Gradient of a vector: ->tensor

\[\begin{aligned} \{\nabla\mathbf v\}= & (\frac{\partial}{\partial x}\mathbf i+\frac{\partial}{\partial y}\mathbf j+\frac{\partial}{\partial z}\mathbf k)(u\mathbf{i}+v\mathbf{j}+w\mathbf{k}) \\ = & \mathbf i\mathbf i\frac{\partial}{\partial x}+\mathbf i\mathbf j\frac{\partial}{\partial y}+\mathbf i\mathbf k\frac{\partial}{\partial z}+ \\ & \mathbf j\mathbf i\frac{\partial}{\partial x}+\mathbf j\mathbf j\frac{\partial}{\partial y}+\mathbf j\mathbf k\frac{\partial}{\partial z}+ \\ & \mathbf k\mathbf i\frac{\partial}{\partial x}+\mathbf k\mathbf j\frac{\partial}{\partial y}+\mathbf k\mathbf k\frac{\partial}{\partial z}+ \\ = & \left[\begin{matrix} \frac{\partial u}{\partial x} & \frac{\partial v}{\partial x} & \frac{\partial w}{\partial x} \\ \frac{\partial u}{\partial y} & \frac{\partial v}{\partial y} & \frac{\partial w}{\partial y} \\ \frac{\partial u}{\partial x} & \frac{\partial v}{\partial y} & \frac{\partial w}{\partial z} \\ \end{matrix}\right] \end{aligned}\]

Dot product of a tensor $\mathbf\tau$ by a vector $\mathbf v$: ->vector

\[\{\mathbf\tau\cdot\mathbf v\}= \left[\begin{matrix} \tau_{xx} & \tau_{xy} & \tau_{xz} \\ \tau_{yx} & \tau_{yy} & \tau_{yz} \\ \tau_{zx} & \tau_{zy} & \tau_{zz} \\ \end{matrix}\right] \left[\begin{matrix} u \\ v \\ w \\ \end{matrix}\right] =...\]

Divergence of a tensor:

Double dot product of two tensors: