Finding d\rr on Rectangular Paths

\(\newcommand{\vf}[1]{\mathbf{\boldsymbol{\vec{#1}}}} \renewcommand{\Hat}[1]{\mathbf{\boldsymbol{\hat{#1}}}} \let\VF=\vf \let\HAT=\Hat \newcommand{\Prime}{{}\kern0.5pt'} \newcommand{\PARTIAL}[2]{{\partial^2#1\over\partial#2^2}} \newcommand{\Partial}[2]{{\partial#1\over\partial#2}} \newcommand{\tr}{{\mathrm tr}} \newcommand{\CC}{{\mathbb C}} \newcommand{\HH}{{\mathbb H}} \newcommand{\KK}{{\mathbb K}} \newcommand{\RR}{{\mathbb R}} \newcommand{\HR}{{}^*{\mathbb R}} \renewcommand{\AA}{\vf{A}} \newcommand{\BB}{\vf{B}} \newcommand{\CCv}{\vf{C}} \newcommand{\EE}{\vf{E}} \newcommand{\FF}{\vf{F}} \newcommand{\GG}{\vf{G}} \newcommand{\HHv}{\vf{H}} \newcommand{\II}{\vf{I}} \newcommand{\JJ}{\vf{J}} \newcommand{\KKv}{\vf{Kv}} \renewcommand{\SS}{\vf{S}} \renewcommand{\aa}{\VF{a}} \newcommand{\bb}{\VF{b}} \newcommand{\ee}{\VF{e}} \newcommand{\gv}{\VF{g}} \newcommand{\iv}{\vf{imath}} \newcommand{\rr}{\VF{r}} \newcommand{\rrp}{\rr\Prime} \newcommand{\uu}{\VF{u}} \newcommand{\vv}{\VF{v}} \newcommand{\ww}{\VF{w}} \newcommand{\grad}{\vf{\nabla}} \newcommand{\zero}{\vf{0}} \newcommand{\Ihat}{\Hat I} \newcommand{\Jhat}{\Hat J} \newcommand{\nn}{\Hat n} \newcommand{\NN}{\Hat N} \newcommand{\TT}{\Hat T} \newcommand{\ihat}{\Hat\imath} \newcommand{\jhat}{\Hat\jmath} \newcommand{\khat}{\Hat k} \newcommand{\nhat}{\Hat n} \newcommand{\rhat}{\HAT r} \newcommand{\shat}{\HAT s} \newcommand{\xhat}{\Hat x} \newcommand{\yhat}{\Hat y} \newcommand{\zhat}{\Hat z} \newcommand{\that}{\Hat\theta} \newcommand{\phat}{\Hat\phi} \newcommand{\LL}{\mathcal{L}} \newcommand{\DD}[1]{D_{\textrm{$#1$}}} \newcommand{\bra}[1]{\langle#1|} \newcommand{\ket}[1]{|#1/rangle} \newcommand{\braket}[2]{\langle#1|#2\rangle} \newcommand{\LargeMath}[1]{\hbox{\large$#1$}} \newcommand{\INT}{\LargeMath{\int}} \newcommand{\OINT}{\LargeMath{\oint}} \newcommand{\LINT}{\mathop{\INT}\limits_C} \newcommand{\Int}{\int\limits} \newcommand{\dint}{\mathchoice{\int\!\!\!\int}{\int\!\!\int}{}{}} \newcommand{\tint}{\int\!\!\!\int\!\!\!\int} \newcommand{\DInt}[1]{\int\!\!\!\!\int\limits_{#1~~}} \newcommand{\TInt}[1]{\int\!\!\!\int\limits_{#1}\!\!\!\int} \newcommand{\Bint}{\TInt{B}} \newcommand{\Dint}{\DInt{D}} \newcommand{\Eint}{\TInt{E}} \newcommand{\Lint}{\int\limits_C} \newcommand{\Oint}{\oint\limits_C} \newcommand{\Rint}{\DInt{R}} \newcommand{\Sint}{\int\limits_S} \newcommand{\Item}{\smallskip\item{$\bullet$}} \newcommand{\LeftB}{\vector(-1,-2){25}} \newcommand{\RightB}{\vector(1,-2){25}} \newcommand{\DownB}{\vector(0,-1){60}} \newcommand{\DLeft}{\vector(-1,-1){60}} \newcommand{\DRight}{\vector(1,-1){60}} \newcommand{\Left}{\vector(-1,-1){50}} \newcommand{\Down}{\vector(0,-1){50}} \newcommand{\Right}{\vector(1,-1){50}} \newcommand{\ILeft}{\vector(1,1){50}} \newcommand{\IRight}{\vector(-1,1){50}} \newcommand{\Partials}[3] {\displaystyle{\partial^2#1\over\partial#2\,\partial#3}} \newcommand{\Jacobian}[4]{\frac{\partial(#1,#2)}{\partial(#3,#4)}} \newcommand{\JACOBIAN}[6]{\frac{\partial(#1,#2,#3)}{\partial(#4,#5,#6)}} \newcommand{\LLv}{\vf{L}} \newcommand{\OOb}{\boldsymbol{O}} \newcommand{\PPv}{\vf{P}_\text{cm}} \newcommand{\RRv}{\vf{R}_\text{cm}} \newcommand{\ff}{\vf{f}} \newcommand{\pp}{\vf{p}} \newcommand{\tauv}{\vf{\tau}} \newcommand{\Lap}{\nabla^2} \newcommand{\Hop}{H_\text{op}} \newcommand{\Lop}{L_\text{op}} \newcommand{\Hhat}{\hat{H}} \newcommand{\Lhat}{\hat{L}} \newcommand{\defeq}{\overset{\rm def}{=}} \newcommand{\absm}{\vert m\vert} \newcommand{\ii}{\ihat} \newcommand{\jj}{\jhat} \newcommand{\kk}{\khat} \newcommand{\dS}{dS} \newcommand{\dA}{dA} \newcommand{\dV}{d\tau} \renewcommand{\ii}{\xhat} \renewcommand{\jj}{\yhat} \renewcommand{\kk}{\zhat} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \definecolor{fillinmathshade}{gray}{0.9} \newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}} \)

Section 6.2 Finding $d\rr$ on Rectangular Paths

In the activity below, you will construct the vector differential $d\rr$ in rectangular coordinates. This vector differential is the building block used to construct multi-dimensional integrals, including flux, surface, and volume integrals, so long as they are expressed in rectangular coordinates.

Activity 6.2.1. The Vector Differential in Rectangular Coordinates.

The arbitrary infinitesimal displacement vector in Cartesian coordinates is:

\begin{equation} d\rr=dx\,\xhat + dy\,\yhat +dz\,\zhat\tag{6.2.1} \end{equation}

Given the cube shown below, find $d\rr$ on each of the three paths, leading from $a$ to $b\text{.}$ Notice that the vector differential in Equation (6.2.1) has both a magnitude and a direction.

Path 1: $d\rr=$

Path 2: $d\rr=$

Path 3: $d\rr=$

Prev Top Next

THE GEOMETRY OF STATIC FIELDS

Section 6.2 Finding \(d\rr\) on Rectangular Paths

Activity 6.2.1. The Vector Differential in Rectangular Coordinates.