Calculus III Summary

This page contains a list of tables that summarize the relationships between various ideas in Calculus III. This is information is by no means complete. (I cannot emphasize this enough.) Examples of what’s not in the summary:

{$ \def\aa{\mathbf{a}} \def\bb{\mathbf{b}} \def\hb{\mathbf{\hat{b}}} \def\cc{\mathbf{c}} \def\uu{\mathbf{u}} \def\vv{\mathbf{v}} \def\ww{\mathbf{w}} \def\nn{\mathbf{n}} \def\pp{\mathbf{p}} \def\qq{\mathbf{q}} \def\rr{\mathbf{r}} \def\FF{\mathbf{F}} \def\GG{\mathbf{G}} \def\DD{\mathbf{D}} \def\SS{\mathbf{S}} \def\TT{\mathbf{T}} \def\NN{\mathbf{N}} \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\proj{\text{proj}} \def\comp{\text{comp}} \def\orth{\text{orth}} \def\vvec#1{\left<#1\right>} $}

Dot and Cross Products

Dot Cross
{$\displaystyle \Rightarrow \cos(\theta)=\frac{\aa\dot\bb}{|\aa|\;|\bb|}$}
{$\displaystyle\Rightarrow \sin(\theta)=\frac{|\aa\cross\bb|}{|\aa|\;|\bb|}$}
Meaningprojection(oriented) area of parallelogram
Application{$W\text{ork} = \FF\dot\DD$}{$\tau\text{orque}=\rr\cross\FF$}
Properties{$\aa\dot\bb = \bb\dot\aa$}{$\aa\cross\bb = -\bb\cross\aa$}
In general, {$\aa\cross(\bb\cross\cc) \ne (\aa\cross\bb)\cross\cc$}
Orientation{$\aa\dot\bb = 0 \quad \Rightarrow \quad \aa\perp\bb$}{$\aa\cross\bb = \mathbf{0} \quad \Rightarrow \quad \aa\parallel\bb$}
ObjectsPlane: {$\nn\dot(\rr-\rr_0)=\mathbf{0}$}Line: {$\vv\cross(\rr-\rr_0)=0$}
CombinationsVolume of parallelepiped: {$\aa\dot(\bb\cross\cc) = (\aa\cross\bb)\dot\cc$}
{$(\aa\dot\bb)^2 + |\aa\cross\bb|^2 = |\aa|^2|\bb|^2 \qquad$} (follows from {$\cos^2\theta + \sin^2\theta = 1$})
{$\aa\cross(\bb\cross\cc) = (\aa\dot\cc)\bb - (\aa\dot\bb)\cc$}

Compound operators and distances

arbitrary {$\bb$} unit {$\hb =\frac{\bb}{|\bb|}$} proj and orth operators
{$\comp_\bb(\aa)$} {$\displaystyle \frac{\aa\cdot\bb}{|\bb|}$} {$\aa\cdot\hb$}
{$\proj_\bb(\aa)$} {$\displaystyle \big(\comp_\bb(\aa)\big) \frac{\bb}{|\bb|} = \frac{\aa\cdot\bb}{\bb\cdot\bb}\bb$} {$(\aa\cdot\hb)\hb$}
{$\orth_\bb(\aa)$} {$\aa - \proj_\bb(\aa)$} {$\aa - \proj_\hb(\aa)$}

You only need to remember the third column! The second one follows using the transformation {$\hb=\frac{\bb}{|\bb|}$}.

Distance between a point {$\pp$} and …
a point {$\qq$} {$|\pp-\qq| = \sqrt{(\pp-\qq)\cdot(\pp-\qq)}$}
a line through {$\qq$} in direction {$\vv$} {$|\orth_\vv(\pp-\qq)|$}
a plane through {$\qq$} with normal {$\nn$} {$|\proj_\nn(\pp-\qq)| = |\comp_\nn(\pp-\qq)|$}
an implicit surface {$g(x,y,z)=k$} minimize {$|\pp-\vvec{x,y,z}|^2$} subject to constraint {$g(x,y,z)=k$}

Note that this table refers to points using their position vectors in order to use the vector operators.

Coordinate systems

Conversion{$x = x$}
{$y = y$}
{$z = z$}
{$x = r\cos(\theta)$}
{$y = r\sin(\theta)$}
{$z = z$}
{$x = \rho\cos(\theta)\sin(\phi)$}
{$y = \rho\sin(\theta)\sin(\phi)$}
{$z = \rho\cos(\phi)$}
Ranges{$x \in (-\infty, \infty)$}
{$y \in (-\infty, \infty)$}
{$z \in (-\infty, \infty)$}
{$r \in [0, \infty)$}
{$\theta \in [0, 2\pi)$}
{$z \in (-\infty, \infty)$}
{$\rho \in [0, \infty)$}
{$\theta \in [0, 2\pi)$}
{$\phi \in [0, \pi]$}


Area{$dA = dx\;dy = r\;dr\;d\theta$}{$\int_D dA = A(D) =$} area
Volume{$dV = dx\;dy\;dz = r\;dr\;d\theta\;dz = \rho^2\sin\phi\;d\rho\;d\phi\;d\theta$}{$\int_E dV = V(E) =$} volume
Line{$d\rr = \rr'(t)dt = \TT ds$}
{$ds = |\rr'(t)|dt = \TT \cdot d\rr$}
{$\int_a^b d\rr = \rr(b)-\rr(a) =$} displacement
{$\int_a^b ds = $} arc length
Surface{$d\SS = \rr_u\times\rr_v dA = \nn dS$}
{$dS = |\rr_u\times\rr_v| dA = \nn \cdot d\SS$}
{$\int_S dS = A(S) =$} surface area

Curves and Surfaces

(2D) rotate tangent
(3D) {$\NN(t)=\frac{\TT'(t)}{|\TT'(t)|}$}
{$\int_C ds = \int_C |\rr'(t)|\;dt$}
{$\rr(u,v)$}{$\rr_u$}, {$\rr_v$}{$\displaystyle \nn = \frac{\rr_u\cross\rr_v}{|\rr_u\cross\rr_v|}$}{$\int_S dS = \iint_D |\rr_u\cross\rr_v|\;dA$}
Level curves{$f(x,y)=c$}{$\left<dx,dy\right>$} or
{$\left<1,\frac{dy}{dx}\right>$} or
(slope = {$\frac{dy}{dx}$})
{$\grad f$} 
Level surface{$f(x,y,z)=c$}If, implicitly, {$z=g(x,y)$},
then the tangents are {$\left<1,0,g_x\right>, \left<0,1,g_y\right>$},
where {$(g_x,g_y)=-\left(\frac{f_x}{f_z}, \frac{f_y}{f_z}\right)$}.
{$\grad f$} 

Generalizations of the Fundamental Theorem

To see the vector-inspired analogy, you should treat {$\grad$} as {$\left<\pdiff{}{x}, \pdiff{}{y}, \pdiff{}{z}\right>$}.

notation →
{$\omega$} is a 0-form, 1-form, 2-form, or 3-form {$d$} = exterior derivative on {$k$}-forms (manifests itself as {$\tgrad$}, {$\tcurl$}, and {$\tdiv$} on scalar-, vector-, and bivector-valued functions) {$d\;d\omega = 0$}
{$1$} scalar {$f$}
from previous row to next{$\grad$}
{$\vv\;\;1 = \vv$} vector {$\tgrad f = \grad f$} {$\curl \grad f = \mathbf{0}$}
from previous row to next{$\curl$}
{$\uu\cross\vv$} bivector (vector-valued) {$\tcurl \FF = \curl \FF$} {$\div \curl \FF = 0$}
from previous row to next{$\div$}
signed volume {$\ww\dot(\uu\cross\vv)$} trivector (scalar-valued) {$\tdiv \GG = \div \GG$}

notation →
Generalized Stokes' Theorem {$\int_{\Omega} d\omega = \oint_{\partial\Omega} \omega$}
{$\grad$} Fundamental Theorem
of Calculus ({$f$} univariate)
{$\int_{[a,b]} df = \int_a^b f'(x)dx = f(b) - f(a)$}
Fundamental Theorem
of Line Integrals ({$f$} multivariate)
{$\int_{C} (\grad f) \dot d\rr = f(\rr(b)) - f(\rr(a))$}
{$\curl$} Green's Theorem (2D) {$\iint_{D} \left(\pdiff{Q}{x} - \pdiff{P}{y}\right) dA = \int_{\partial D} P\;dx + Q\;dy$}
{$\iint_{D} (\curl \FF) \dot \mathbf{k} \;dA = \oint_{\partial D} \FF \dot d\rr = \oint_{\partial D} \FF \dot \TT ds$}
Stokes' Theorem (3D) {$\iint_{S} (\curl \FF) \dot d\SS = \oint_{\partial S} \FF \dot d\rr = \oint_{\partial S} \FF \dot \TT ds$}
{$\div$} Green's Theorem variant (2D) {$\iint_{D} \div \FF\;dA = \oint_{\partial D} \FF \dot \NN\;ds$}
Divergence Theorem (3D) {$\iiint_{E} \div \FF\;dV = \iint_{\partial E} \FF \dot d\SS = \iint_{\partial E} \FF \dot \nn dS$}

For each of {$\tgrad$}, {$\tcurl$}, and {$\tdiv$} above, the first fundamental theorem is a specialization of the second one that appears in the table cell below it.