@@ -23,7 +23,7 @@
        "$$\\newcommand{\\lbrc}{\\left \\{}$$\n",
        "$$\\newcommand{\\rbrc}{\\right \\}}$$\n",
        "$$\\newcommand{\\W}{\\wedge}$$\n",
-       "$$\\newcommand{\\prm}[1]{{#1}}$$\n",
+       "$$\\newcommand{\\prm}[1]{{#1}^{\\prime}}$$\n",
        "$$\\newcommand{\\ddt}[1]{\\bfrac{d{#1}}{dt}}$$\n",
        "$$\\newcommand{\\R}{\\dagger}$$\n",
        "$$\\newcommand{\\deriv}[3]{\\bfrac{d^{#3}#1}{d{#2}^{#3}}}$$\n",
@@ -107,7 +107,7 @@
     {
      "data": {
       "text/latex": [
-       "$\\displaystyle \\mathbf{B}^{\\dagger}= - B^{xy}  \\boldsymbol{e}_{x}\\wedge \\boldsymbol{e}_{y} - B^{xz}  \\boldsymbol{e}_{x}\\wedge \\boldsymbol{e}_{z} - B^{yz}  \\boldsymbol{e}_{y}\\wedge \\boldsymbol{e}_{z} $"
+       "$\\displaystyle \\mathbf{B}^{\\dagger}=  \\begin{aligned}[t]  & - B^{xy}  \\boldsymbol{e}_{x}\\wedge \\boldsymbol{e}_{y} \\\\  &  - B^{xz}  \\boldsymbol{e}_{x}\\wedge \\boldsymbol{e}_{z} \\\\  &  - B^{yz}  \\boldsymbol{e}_{y}\\wedge \\boldsymbol{e}_{z}  \\end{aligned}  $"
       ],
       "text/plain": [
        "<IPython.core.display.Math object>"
@@ -131,7 +131,7 @@
     {
      "data": {
       "text/latex": [
-       "$\\displaystyle \\begin{equation*} \\mathbf{B} = B^{xy}  \\boldsymbol{e}_{x}\\wedge \\boldsymbol{e}_{y} + B^{xz}  \\boldsymbol{e}_{x}\\wedge \\boldsymbol{e}_{z} + B^{yz}  \\boldsymbol{e}_{y}\\wedge \\boldsymbol{e}_{z} \\end{equation*}$"
+       "$\\displaystyle \\begin{equation*} \\mathbf{B} =  \\begin{aligned}[t]  & B^{xy}  \\boldsymbol{e}_{x}\\wedge \\boldsymbol{e}_{y} \\\\  &  + B^{xz}  \\boldsymbol{e}_{x}\\wedge \\boldsymbol{e}_{z} \\\\  &  + B^{yz}  \\boldsymbol{e}_{y}\\wedge \\boldsymbol{e}_{z}  \\end{aligned}  \\end{equation*}$"
       ],
       "text/plain": [
        "<IPython.core.display.Math object>"
@@ -155,7 +155,7 @@
     {
      "data": {
       "text/latex": [
-       "$\\displaystyle \\begin{equation*}  \\mathbf{B}^{\\dagger}= - B^{xy}  \\boldsymbol{e}_{x}\\wedge \\boldsymbol{e}_{y} - B^{xz}  \\boldsymbol{e}_{x}\\wedge \\boldsymbol{e}_{z} - B^{yz}  \\boldsymbol{e}_{y}\\wedge \\boldsymbol{e}_{z} \\end{equation*}$"
+       "$\\displaystyle \\begin{equation*}  \\mathbf{B}^{\\dagger}=  \\begin{aligned}[t]  & - B^{xy}  \\boldsymbol{e}_{x}\\wedge \\boldsymbol{e}_{y} \\\\  &  - B^{xz}  \\boldsymbol{e}_{x}\\wedge \\boldsymbol{e}_{z} \\\\  &  - B^{yz}  \\boldsymbol{e}_{y}\\wedge \\boldsymbol{e}_{z}  \\end{aligned}  \\end{equation*}$"
       ],
       "text/plain": [
        "<IPython.core.display.Math object>"
@@ -167,7 +167,7 @@
     {
      "data": {
       "text/latex": [
-       "$\\displaystyle \\begin{equation*}  \\mathbf{\\nabla B} = \\left ( - \\partial_{y} B^{xy}  - \\partial_{z} B^{xz} \\right ) \\boldsymbol{e}_{x} + \\left ( \\partial_{x} B^{xy}  - \\partial_{z} B^{yz} \\right ) \\boldsymbol{e}_{y} + \\left ( \\partial_{x} B^{xz}  + \\partial_{y} B^{yz} \\right ) \\boldsymbol{e}_{z} + \\left ( \\partial_{z} B^{xy}  - \\partial_{y} B^{xz}  + \\partial_{x} B^{yz} \\right ) \\boldsymbol{e}_{x}\\wedge \\boldsymbol{e}_{y}\\wedge \\boldsymbol{e}_{z} \\end{equation*}$"
+       "$\\displaystyle \\begin{equation*}  \\mathbf{\\nabla B} =  \\begin{aligned}[t]  & \\left ( - \\partial_{y} B^{xy}  - \\partial_{z} B^{xz} \\right ) \\boldsymbol{e}_{x} \\\\  &  + \\left ( \\partial_{x} B^{xy}  - \\partial_{z} B^{yz} \\right ) \\boldsymbol{e}_{y} \\\\  &  + \\left ( \\partial_{x} B^{xz}  + \\partial_{y} B^{yz} \\right ) \\boldsymbol{e}_{z} \\\\  &  + \\left ( \\partial_{z} B^{xy}  - \\partial_{y} B^{xz}  + \\partial_{x} B^{yz} \\right ) \\boldsymbol{e}_{x}\\wedge \\boldsymbol{e}_{y}\\wedge \\boldsymbol{e}_{z}  \\end{aligned}  \\end{equation*}$"
       ],
       "text/plain": [
        "<IPython.core.display.Math object>"
@@ -193,7 +193,7 @@
     {
      "data": {
       "text/latex": [
-       "$\\displaystyle  \\frac{\\partial^{3}}{\\partial x^{2}\\partial y} \\bs{B}= \\frac{\\partial^{3}}{\\partial^{2} x\\partial y}  B^{xy} {\\left (x,y,z \\right )} \\boldsymbol{e}_{x}\\wedge \\boldsymbol{e}_{y} + \\frac{\\partial^{3}}{\\partial^{2} x\\partial y}  B^{xz} {\\left (x,y,z \\right )} \\boldsymbol{e}_{x}\\wedge \\boldsymbol{e}_{z} + \\frac{\\partial^{3}}{\\partial^{2} x\\partial y}  B^{yz} {\\left (x,y,z \\right )} \\boldsymbol{e}_{y}\\wedge \\boldsymbol{e}_{z} $"
+       "$\\displaystyle  \\frac{\\partial^{3}}{\\partial x^{2}\\partial y} \\bs{B}=  \\begin{aligned}[t]  & \\frac{\\partial^{3}}{\\partial^{2} x\\partial y}  B^{xy} {\\left (x,y,z \\right )} \\boldsymbol{e}_{x}\\wedge \\boldsymbol{e}_{y} \\\\  &  + \\frac{\\partial^{3}}{\\partial^{2} x\\partial y}  B^{xz} {\\left (x,y,z \\right )} \\boldsymbol{e}_{x}\\wedge \\boldsymbol{e}_{z} \\\\  &  + \\frac{\\partial^{3}}{\\partial^{2} x\\partial y}  B^{yz} {\\left (x,y,z \\right )} \\boldsymbol{e}_{y}\\wedge \\boldsymbol{e}_{z}  \\end{aligned}  $"
       ],
       "text/plain": [
        "<IPython.core.display.Math object>"
@@ -218,7 +218,7 @@
     {
      "data": {
       "text/latex": [
-       "$\\displaystyle  \\frac{\\partial^{3}}{\\partial x^{2}\\partial y} \\bs{B}= \\partial^{2}_{x}\\partial_{y} B^{xy}  \\boldsymbol{e}_{x}\\wedge \\boldsymbol{e}_{y} + \\partial^{2}_{x}\\partial_{y} B^{xz}  \\boldsymbol{e}_{x}\\wedge \\boldsymbol{e}_{z} + \\partial^{2}_{x}\\partial_{y} B^{yz}  \\boldsymbol{e}_{y}\\wedge \\boldsymbol{e}_{z} $"
+       "$\\displaystyle  \\frac{\\partial^{3}}{\\partial x^{2}\\partial y} \\bs{B}=  \\begin{aligned}[t]  & \\partial^{2}_{x}\\partial_{y} B^{xy}  \\boldsymbol{e}_{x}\\wedge \\boldsymbol{e}_{y} \\\\  &  + \\partial^{2}_{x}\\partial_{y} B^{xz}  \\boldsymbol{e}_{x}\\wedge \\boldsymbol{e}_{z} \\\\  &  + \\partial^{2}_{x}\\partial_{y} B^{yz}  \\boldsymbol{e}_{y}\\wedge \\boldsymbol{e}_{z}  \\end{aligned}  $"
       ],
       "text/plain": [
        "<IPython.core.display.Math object>"
@@ -265,7 +265,7 @@
     {
      "data": {
       "text/latex": [
-       "$\\displaystyle \\grade{B}{2} = B^{xy}  \\boldsymbol{e}_{x}\\wedge \\boldsymbol{e}_{y} + B^{xz}  \\boldsymbol{e}_{x}\\wedge \\boldsymbol{e}_{z} + B^{yz}  \\boldsymbol{e}_{y}\\wedge \\boldsymbol{e}_{z} $"
+       "$\\displaystyle \\grade{B}{2} =  \\begin{aligned}[t]  & B^{xy}  \\boldsymbol{e}_{x}\\wedge \\boldsymbol{e}_{y} \\\\  &  + B^{xz}  \\boldsymbol{e}_{x}\\wedge \\boldsymbol{e}_{z} \\\\  &  + B^{yz}  \\boldsymbol{e}_{y}\\wedge \\boldsymbol{e}_{z}  \\end{aligned}  $"
       ],
       "text/plain": [
        "<IPython.core.display.Math object>"
@@ -354,10 +354,12 @@
     {
      "data": {
       "text/latex": [
-       "\\begin{equation*} V = V^{x} \\boldsymbol{e}_{x} + V^{y} \\boldsymbol{e}_{y} + V^{z} \\boldsymbol{e}_{z} \\end{equation*}"
+       "$\\displaystyle V =  \\begin{aligned}[t]  & V^{x} \\mathbf{e}_{x} \\\\  &  + V^{y} \\mathbf{e}_{y} \\\\  &  + V^{z} \\mathbf{e}_{z}  \\end{aligned} $"
       ],
       "text/plain": [
-       "V__x*e_x + V__y*e_y + V__z*e_z"
+       " V__x*e_x\n",
+       " + V__y*e_y\n",
+       " + V__z*e_z"
       ]
      },
      "execution_count": 12,
@@ -391,7 +393,7 @@
     {
      "data": {
       "text/latex": [
-       "$\\displaystyle  \\left ( L_{xx} L_{yy} - L_{xy} L_{yx}\\right ) \\boldsymbol{e}_{x}\\wedge \\boldsymbol{e}_{y} $"
+       "$\\displaystyle  \\left ( {L^{x}}_{x} {L^{y}}_{y} - {L^{x}}_{y} {L^{y}}_{x}\\right ) \\boldsymbol{e}_{x}\\wedge \\boldsymbol{e}_{y} $"
       ],
       "text/plain": [
        "<IPython.core.display.Math object>"
@@ -414,8 +416,7 @@
     {
      "data": {
       "text/latex": [
-       "$\\displaystyle  \\left \\{ \\begin{array}{ll} L \\left ( \\boldsymbol{e}_{x}\\right ) =& 2 \\boldsymbol{e}_{y} \\\\ L \\left ( \\boldsymbol{e}_{y}\\right ) =&  \\boldsymbol{e}_{x} + \\boldsymbol{e}_{y}  \\end{array} \\right \\} \n",
-       " $"
+       "$\\displaystyle  \\left\\{ \\begin{aligned} \\boldsymbol{e}_{x} &\\mapsto 2 \\boldsymbol{e}_{y} \\\\ \\boldsymbol{e}_{y} &\\mapsto  \\begin{aligned}[t]  &  \\boldsymbol{e}_{x} \\\\  &  + \\boldsymbol{e}_{y}  \\end{aligned}  \\end{aligned} \\right\\} $"
       ],
       "text/plain": [
        "<IPython.core.display.Math object>"
@@ -512,7 +513,7 @@
  ],
  "metadata": {
   "kernelspec": {
-   "display_name": "Python 3",
+   "display_name": "Python 3 (ipykernel)",
    "language": "python",
    "name": "python3"
   },
@@ -526,7 +527,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.7.3"
+   "version": "3.11.8"
   }
  },
  "nbformat": 4,