Differential Geometry

This is that phenomenally fascinating subject that answers many of the questions you might have had lingering from Multivariable Calculus and many more you never thought to ask.

{$ \def\pp{\mathbf{p}} \def\uu{\mathbf{u}} \def\vv{\mathbf{v}} \def\FF{\mathbf{F}} \def\GG{\mathbf{G}} \def\ww{\mathbf{w}} \def\RR{\mathbb{R}} \def\st{\; \mid \;} \def\cross{\times} \def\dot{\cdot} \def\grad{\nabla} \def\curl{\grad\cross} \def\div{\grad\dot} \def\tgrad{\text{grad}\;} \def\tcurl{\text{curl}\;} \def\tdiv{\text{div}\;} \def\pdiff#1#2{\frac{\partial #1}{\partial #2}} \def\vvec#1{\left<#1\right>} \newcommand{\la}[1]{\xleftarrow{\ \ #1\ \ }} \newcommand{\ra}[1]{\xrightarrow{\ \ #1\ \ }} \newcommand{\las}[1]{\xleftarrow{\ \ \smash{#1}\ \ }} \newcommand{\ras}[1]{\xrightarrow{\ \ \smash{#1}\ \ }} \newcommand{\da}[1]{\bigg\downarrow\rlap{\scriptstyle#1}} \newcommand{\uda}[1]{\bigg\updownarrow\rlap{\scriptstyle#1}} $}

Basics

Curves

Type Description
curve (trajectory) Differentiable function {$\alpha : I \rightarrow \RR^3$}, {$I\subset \RR$} is open
Curve (set of points; implicit) {${\pp \st f(\pp)=0}$} in {$\RR^2$}

From Calculus: * {$\beta(s) = \alpha(h(s)) \quad\Rightarrow\quad \beta'(s) = \alpha'(h(s)) h'(s)$} * {$\alpha'(t)[f] \quad=\quad df(\alpha'(t)) \quad=\quad \grad f(\alpha(t)) \cdot \alpha'(t) \quad=\quad \frac{d}{dt}\left(f(\alpha(t))\right)$}

Differential 1-form

The total differential {$df = f_x dx + f_y dy + f_z dz$} from multivariable calculus is actually a 1-form: a linear function that maps tangent vectors to real numbers so that

If a differential geometry vector is thought of as a linear algebra column vector, a 1-form is like a row vector at each point. This table summarizes the relationship in {$\RR^3$} (it generalizes easily to {$\RR^n$}).

(Tangent) Vector Field1-form
Def{$V: \RR^3 \rightarrow T_\pp \RR^3$}
(assigns {$\vv_\pp$} to each {$\pp\in\RR^3$})
{$\phi:T_\pp \RR^3 \rightarrow \RR$}
(linear at each {$\pp\in\RR^3$})
Basis{$\{U_1(\pp), U_2(\pp), U_3(\pp)\}$}{$\{dx, dy, dz\}$} or {$\{dx_1, dx_2, dx_3\}$}
Basis interaction
{$dx_i(U_j) = \left\{\begin{matrix}1 \text{ if } i=j \\0 \text{ if } i \ne j\end{matrix}\right. \qquad \Rightarrow \qquad \left[\begin{smallmatrix}dx_1\\dx_2\\dx_3\end{smallmatrix}\right] \left[\begin{smallmatrix}U_1 & U_2 & U_3\end{smallmatrix}\right] = I$}
Like a dot product
{$\underbrace{(3dx+4dy+dz)}_{\left[\begin{smallmatrix}3&4&1\end{smallmatrix}\right] \left[\begin{smallmatrix}dx_1\\dx_2\\dx_3\end{smallmatrix}\right]}\underbrace{(U_1+3U_2+U_3)}_{\left[\begin{smallmatrix}U_1 & U_2 & U_3\end{smallmatrix}\right]\left[\begin{smallmatrix}1\\3\\1\end{smallmatrix}\right]} = \left[\begin{smallmatrix}3&4&1\end{smallmatrix}\right] \left[\begin{smallmatrix}1\\3\\1\end{smallmatrix}\right] = 16$}
Projecting to basis{$\vv = \sum_i dx_i(\vv) U_i$}{$\phi = \sum_i \phi(U_i) dx_i$}
Chain rule{$\frac{d}{dt}(f(\alpha(t))) = \alpha'(t)[f] = df(\alpha'(t))$}{$d(h(f)) = h'(f) df$}

Further insights into curves, 1-forms, and how to integrate 1-forms along a curve.

Differential {$k$}-forms

It is easier to understand {$k$}-forms in terms of {$k$}-vectors. In {$\RR^3$}, we have 0-forms, 1-forms, 2-forms, and 3-forms. These can be identified with scalars, vectors, oriented areas, and signed volumes, respectively. The specific case of the 1-form and its interpretation as a row vector in matrix multiplication is shown in the previous section. The wedge product ({$\wedge$}) combines lower forms to give higher ones, but it is orientation-sensitive like the cross-product. It is an alternating multilinear operator:

{$k$} Calculus 3 {$k$}-vectors (Clifford algebra) {$k$}-form {$k$}-form basis
0 1 (scalar) 1 (0-vector) {$f$} {${1}$}
1 {$\vv$} (length vector) {$\vv$} (vector) {$\phi$} {${dx, dy, dz}$}
2 {$\vv\cross\ww$}
(area vector)
{$\vv\wedge \ww$}
(bivector, oriented area)
{$\phi\wedge\psi$} {${dy dz, dz dx, dx dy}$}
3 {$\vv\cross\ww\cdot\uu$}
(volume scalar)
{$\vv\wedge \ww\wedge \uu$}
(trivector, signed volume)
{$\phi\wedge\psi\wedge\theta$} {${dx dy dz}$}

As one can imagine, based on the analogy with Calculus 3 objects above, the wedge ({$\wedge$}) operator computes a determinant ({$k$}-dimensional volume). Thus:

just like

Exterior Derivative

The exterior derivative operator {$d$} is a linear differential operator that takes a {$k$}-form to a {$(k+1)$}-form and is defined via:

Product rule: {$d(\phi\wedge\psi) = d\phi\wedge\psi + (-1)^k \phi\wedge d\psi$} (where {$\phi$} is a {$k$}-form)

Properties in {$\RR^3$} (prove these!)
{$k$}-formexterior derivativeCalculus 3 analog
{$f$}{$df = f_x dx + f_y dy + f_z dz$}{$\grad f = \vvec{f_x, f_y, f_z}$}
{$\phi = f dx + g dy + h dz$}{$d\phi = \left\|\begin{matrix}dydz&dzdx&dxdy\\\pdiff{}{x}&\pdiff{}{y}&\pdiff{}{z}\\f&g&h\end{matrix}\right\|$}{$\curl\vvec{f,g,h}=\left\|\begin{matrix}U_1&U_2&U_3\\\pdiff{}{x}&\pdiff{}{y}&\pdiff{}{z}\\f&g&h\end{matrix}\right\|$}
{$\phi = f dydz + g dzdx + h dxdy$}{$d\phi = (f_x+g_y+h_z)dz dy dz$}{$\div\vvec{f,g,h} = f_x+g_y+h_z$}
{$k \ge 3$}{$d\phi=0$}


RecallAny dimension
{$\curl\grad f=\mathbf{0}$}{$\div\curl\FF = 0$}{$dd\phi = 0$}
Conservative vector field {$\FF$}: satisfies {$\FF = \grad f$} for some {$f$}Curl-exact vector field {$\GG$}: satisfies {$\GG = \curl \FF$} for some {$\FF$}Exact form {$\phi$}: satisfies {$\phi = d\psi$} for some {$\psi$}
Curl-free: satisfies {$\curl \FF = \mathbf{0}$}Diverence-free: satisfies {$\div \GG = 0$}Closed form {$\phi$}: satisfies {$d\phi = 0$}
Conservative ⇒ curl-freeCurl-exact ⇒ divergence-freeExact ⇒ closed
In a simply-connected domain, the reverse (⇐) is also true.