Notes for AKT-140214/0:08:40

The set of differential $k$-forms on a manifold $M$ (example $\mathbb {R} ^{3}$) is a vector space $\Omega ^{k}(M)$ and when $k=0$ then $\Omega ^{0}(M)$ is the set of smooth functions. Thus smooth functions are 0-forms. Now $k$-forms are integrated on $k$-manifolds. For example, a 1-form $f(x,y)\mathrm {d} x+g(x,y)\mathrm {d} y$ can be integrated on a curve $\gamma$. Also differential forms can be differentiated using the operator d called the exterior operator where $d$ acts on a $k$-form to produce a $k+1$-form and that $\mathrm {d} \circ \mathrm {d} =0$.

Now

1. if $f\in \Omega ^{0}(\mathbb {R} ^{3})$, then $\mathrm {d} f=\sum _{i}^{3}{\frac {\partial {f}}{\partial {x_{i}}}}\mathrm {d} x_{i}$ is a 1-form so that $\mathrm {d} f\in \Omega ^{1}(M)$. Thus $d:\mathrm {d} \Omega ^{0}(\mathbb {R} ^{3})\rightarrow \Omega ^{1}(\mathbb {R} ^{3})$ is the gradient operator $\mathrm {grad}$.

2. If we have a 1-form $v=v_{x}\mathrm {d} x+v_{y}\mathrm {d} y+v_{z}\mathrm {d} z$, then $\mathrm {d} v=\left({\frac {\partial {v_{z}}}{\partial {y}}}-{\frac {\partial {v_{y}}}{\partial {z}}}\right)\mathrm {d} y\wedge \mathrm {d} z+\left({\frac {\partial {v_{x}}}{\partial {z}}}-{\frac {\partial {v_{z}}}{\partial {x}}}\right)\mathrm {d} x\wedge \mathrm {d} z+\left({\frac {\partial {v_{x}}}{\partial {y}}}-{\frac {\partial {v_{y}}}{\partial {x}}}\right)\mathrm {d} x\wedge \mathrm {d} y$ which is a two form. In this case we have $d:\mathrm {d} \Omega ^{1}(\mathbb {R} ^{3})\rightarrow \Omega ^{2}(\mathbb {R} ^{3})$ is the $\mathrm {curl}$ operator.

3. If we have 2-form $\omega =(\omega _{x},\omega _{y},\omega _{z})$ then again get a 3-form $\mathrm {d} \omega =\left({\frac {\partial {\omega _{x}}}{\partial {x}}}+{\frac {\partial {\omega _{y}}}{\partial {y}}}+{\frac {\partial {\omega _{z}}}{\partial {z}}}\right)\mathrm {d} x\wedge \mathrm {d} y\wedge \mathrm {d} z$. If we think of $\mathrm {d} x\wedge \mathrm {d} y\wedge \mathrm {d} z$ as a function $f$, then again we get $d:\mathrm {d} \Omega ^{2}(\mathbb {R} ^{3})\rightarrow \Omega ^{3}(\mathbb {R} ^{3})$ is the $\mathrm {div}$ operator.