This is pdfTeX, Version 3.141592653-2.6-1.40.22 (TeX Live 2022/dev/Debian) (preloaded format=pdflatex 2025.10.1) 19 OCT 2025 07:20 entering extended mode restricted \write18 enabled. %&-line parsing enabled. **vector-analysis.tex (./vector-analysis.tex LaTeX2e <2021-11-15> patch level 1 L3 programming layer <2022-01-21> (/usr/share/texlive/texmf-dist/tex/latex/base/book.cls Document Class: book 2021/10/04 v1.4n Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/bk12.clo File: bk12.clo 2021/10/04 v1.4n Standard LaTeX file (size option) ) \c@part=\count185 \c@chapter=\count186 \c@section=\count187 \c@subsection=\count188 \c@subsubsection=\count189 \c@paragraph=\count190 \c@subparagraph=\count191 \c@figure=\count192 \c@table=\count193 \abovecaptionskip=\skip47 \belowcaptionskip=\skip48 \bibindent=\dimen138 ) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty Package: amsmath 2021/10/15 v2.17l AMS math features \@mathmargin=\skip49 For additional information on amsmath, use the `?' option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty Package: amstext 2021/08/26 v2.01 AMS text (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty File: amsgen.sty 1999/11/30 v2.0 generic functions \@emptytoks=\toks16 \ex@=\dimen139 )) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty Package: amsbsy 1999/11/29 v1.2d Bold Symbols \pmbraise@=\dimen140 ) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty Package: amsopn 2021/08/26 v2.02 operator names ) \inf@bad=\count194 LaTeX Info: Redefining \frac on input line 234. \uproot@=\count195 \leftroot@=\count196 LaTeX Info: Redefining \overline on input line 399. \classnum@=\count197 \DOTSCASE@=\count198 LaTeX Info: Redefining \ldots on input line 496. LaTeX Info: Redefining \dots on input line 499. LaTeX Info: Redefining \cdots on input line 620. \Mathstrutbox@=\box50 \strutbox@=\box51 \big@size=\dimen141 LaTeX Font Info: Redeclaring font encoding OML on input line 743. LaTeX Font Info: Redeclaring font encoding OMS on input line 744. \macc@depth=\count199 \c@MaxMatrixCols=\count266 \dotsspace@=\muskip16 \c@parentequation=\count267 \dspbrk@lvl=\count268 \tag@help=\toks17 \row@=\count269 \column@=\count270 \maxfields@=\count271 \andhelp@=\toks18 \eqnshift@=\dimen142 \alignsep@=\dimen143 \tagshift@=\dimen144 \tagwidth@=\dimen145 \totwidth@=\dimen146 \lineht@=\dimen147 \@envbody=\toks19 \multlinegap=\skip50 \multlinetaggap=\skip51 \mathdisplay@stack=\toks20 LaTeX Info: Redefining \[ on input line 2938. LaTeX Info: Redefining \] on input line 2939. ) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty Package: amsthm 2020/05/29 v2.20.6 \thm@style=\toks21 \thm@bodyfont=\toks22 \thm@headfont=\toks23 \thm@notefont=\toks24 \thm@headpunct=\toks25 \thm@preskip=\skip52 \thm@postskip=\skip53 \thm@headsep=\skip54 \dth@everypar=\toks26 ) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty Package: amssymb 2013/01/14 v3.01 AMS font symbols (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty Package: amsfonts 2013/01/14 v3.01 Basic AMSFonts support \symAMSa=\mathgroup4 \symAMSb=\mathgroup5 LaTeX Font Info: Redeclaring math symbol \hbar on input line 98. LaTeX Font Info: Overwriting math alphabet `\mathfrak' in version `bold' (Font) U/euf/m/n --> U/euf/b/n on input line 106. )) (/usr/share/texlive/texmf-dist/tex/latex/graphics/graphicx.sty Package: graphicx 2021/09/16 v1.2d Enhanced LaTeX Graphics (DPC,SPQR) (/usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty Package: keyval 2014/10/28 v1.15 key=value parser (DPC) \KV@toks@=\toks27 ) (/usr/share/texlive/texmf-dist/tex/latex/graphics/graphics.sty Package: graphics 2021/03/04 v1.4d Standard LaTeX Graphics (DPC,SPQR) (/usr/share/texlive/texmf-dist/tex/latex/graphics/trig.sty Package: trig 2021/08/11 v1.11 sin cos tan (DPC) ) (/usr/share/texlive/texmf-dist/tex/latex/graphics-cfg/graphics.cfg File: graphics.cfg 2016/06/04 v1.11 sample graphics configuration ) Package graphics Info: Driver file: pdftex.def on input line 107. (/usr/share/texlive/texmf-dist/tex/latex/graphics-def/pdftex.def File: pdftex.def 2020/10/05 v1.2a Graphics/color driver for pdftex )) \Gin@req@height=\dimen148 \Gin@req@width=\dimen149 ) (/usr/share/texlive/texmf-dist/tex/latex/algorithms/algorithm.sty Package: algorithm 2009/08/24 v0.1 Document Style `algorithm' - floating enviro nment (/usr/share/texlive/texmf-dist/tex/latex/float/float.sty Package: float 2001/11/08 v1.3d Float enhancements (AL) \c@float@type=\count272 \float@exts=\toks28 \float@box=\box52 \@float@everytoks=\toks29 \@floatcapt=\box53 ) (/usr/share/texlive/texmf-dist/tex/latex/base/ifthen.sty Package: ifthen 2020/11/24 v1.1c Standard LaTeX ifthen package (DPC) ) \@float@every@algorithm=\toks30 \c@algorithm=\count273 ) (/usr/share/texlive/texmf-dist/tex/latex/algorithms/algorithmic.sty Package: algorithmic 2009/08/24 v0.1 Document Style `algorithmic' \c@ALC@unique=\count274 \c@ALC@line=\count275 \c@ALC@rem=\count276 \c@ALC@depth=\count277 \ALC@tlm=\skip55 \algorithmicindent=\skip56 ) \c@proposition=\count278 \c@lemma=\count279 \c@theorem=\count280 \c@definition=\count281 \c@example=\count282 \c@corollary=\count283 \@indexfile=\write3 \openout3 = `vector-analysis.idx'. Writing index file vector-analysis.idx (/usr/share/texlive/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def File: l3backend-pdftex.def 2022-01-12 L3 backend support: PDF output (pdfTeX) \l__color_backend_stack_int=\count284 \l__pdf_internal_box=\box54 ) (./vector-analysis.aux (./1.aux) (./2.aux) (./3.aux) (./4.aux) (./5.aux) (./6.aux)) \openout1 = `vector-analysis.aux'. LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 32. LaTeX Font Info: ... okay on input line 32. LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 32. LaTeX Font Info: ... okay on input line 32. LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 32. LaTeX Font Info: ... okay on input line 32. LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 32. LaTeX Font Info: ... okay on input line 32. LaTeX Font Info: Checking defaults for TS1/cmr/m/n on input line 32. LaTeX Font Info: ... okay on input line 32. LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 32. LaTeX Font Info: ... okay on input line 32. LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 32. LaTeX Font Info: ... okay on input line 32. (/usr/share/texlive/texmf-dist/tex/context/base/mkii/supp-pdf.mkii [Loading MPS to PDF converter (version 2006.09.02).] \scratchcounter=\count285 \scratchdimen=\dimen150 \scratchbox=\box55 \nofMPsegments=\count286 \nofMParguments=\count287 \everyMPshowfont=\toks31 \MPscratchCnt=\count288 \MPscratchDim=\dimen151 \MPnumerator=\count289 \makeMPintoPDFobject=\count290 \everyMPtoPDFconversion=\toks32 ) (/usr/share/texlive/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty Package: epstopdf-base 2020-01-24 v2.11 Base part for package epstopdf Package epstopdf-base Info: Redefining graphics rule for `.eps' on input line 4 85. (/usr/share/texlive/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Liv e )) LaTeX Font Info: Trying to load font information for U+msa on input line 47. (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd File: umsa.fd 2013/01/14 v3.01 AMS symbols A ) LaTeX Font Info: Trying to load font information for U+msb on input line 47. (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd File: umsb.fd 2013/01/14 v3.01 AMS symbols B ) [1 {/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map}] [4 ] (./vector-analysis.toc [5] [6]) \tf@toc=\write4 \openout4 = `vector-analysis.toc'. [7] [8 ] \openout2 = `1.aux'. (./1.tex [1 ] [2] Chapter 1. [3 ] [4] Chapter 2. Overfull \hbox (8.28156pt too wide) detected at line 60 [] \OT1/cmr/bx/n/12 f\OT1/cmr/m/n/12 (\OT1/cmr/bx/n/12 x\OT1/cmr/m/n/12 ) = \OT 1/cmr/bx/n/12 L \OMS/cmsy/m/n/12 ([]) 8\OML/cmm/m/it/12 " \OMS/cmsy/m/n/12 2 \OT1/cmr/m/n/12 (0\OML/cmm/m/it/12 ; \OMS/cmsy/m/n/12 1\OT1/cmr/m/n/12 )(\OMS/c msy/m/n/12 9\OML/cmm/m/it/12 ^^N \OMS/cmsy/m/n/12 2 \OT1/cmr/m/n/12 (0\OML/cmm/ m/it/12 ; \OMS/cmsy/m/n/12 1\OT1/cmr/m/n/12 )[\OMS/cmsy/m/n/12 k\OT1/cmr/bx/n/1 2 x \OMS/cmsy/m/n/12 ^^@ \OT1/cmr/bx/n/12 p\OMS/cmsy/m/n/12 k \OML/cmm/m/it/12 < ^^N \OT1/cmr/m/n/12 =[]\OMS/cmsy/m/n/12 ) k\OT1/cmr/bx/n/12 f\OT1/cmr/m/n/1 2 (\OT1/cmr/bx/n/12 x\OT1/cmr/m/n/12 ) \OMS/cmsy/m/n/12 ^^@ \OT1/cmr/bx/n/12 L\ OMS/cmsy/m/n/12 k \OML/cmm/m/it/12 < "\OT1/cmr/m/n/12 ]) [] [5 ] [6 ] Chapter 3. [7] [8 ] Chapter 4. [9] [10] [11] [12 ] Chapter 5. [13] [14] [15] Overfull \hbox (17.93182pt too wide) in paragraph at lines 275--276 []\OT1/cmr/m/n/12 This the-o-rem has many names \OT1/cmr/m/it/12 Clairaut's the -o-rem, Young's the-o-rem, Schwarz' [] [16] [17] [18 ] Chapter 6. [19] Overfull \hbox (21.3847pt too wide) in paragraph at lines 351--358 \OT1/cmr/m/n/12 the space, to each point in said space $\OML/cmm/m/it/12 X[]$\O T1/cmr/m/n/12

$[]([])$

Position [] ! LaTeX Error: Unicode character 勾 (U+52FE) not set up for use with LaTeX. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.364

Gradient Gradiente 勾 配

? ! LaTeX Error: Unicode character 配 (U+914D) not set up for use with LaTeX. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.364

Gradient Gradiente 勾配

? ! Missing $ inserted. $ l.370

See Mathematics 2

? ! Missing $ inserted. $ l.371 ? Overfull \hbox (40.38226pt too wide) in paragraph at lines 360--371 \OT1/cmr/m/n/12 to-wards their meth-ods of cal-cu-la-tion

$\OMS/cmsy/m/n /12 r \OT1/cmr/m/n/12 = []$

Gradient [] ! LaTeX Error: Unicode character 発 (U+767A) not set up for use with LaTeX. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.372

Divergence Divergenza 発 散

? ! LaTeX Error: Unicode character 散 (U+6563) not set up for use with LaTeX. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.372

Divergence Divergenza 発散

? ! Undefined control sequence. l.382 ... is a sink } \iff \nabla \cdot f(x,y) \lt 0\)

? ! Undefined control sequence. l.385 ...s a source } \iff \nabla \cdot f(x,y) \gt 0\)

? Overfull \hbox (28.61372pt too wide) in paragraph at lines 372--386 []\OT1/cmr/m/n/12

Divergence Di-ver-genza

Vector op-er- a-tor that returns [] Overfull \hbox (0.84431pt too wide) in paragraph at lines 372--386 \OT1/cmr/m/n/12 flow

$(\OML/cmm/m/it/12 x; y\OT1/cmr/m/n/12 )[] \OMS/cm sy/m/n/12 ([]) r ^^A \OML/cmm/m/it/12 f\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 x; y\ OT1/cmr/m/n/12 )0$

Source

[] Overfull \hbox (46.89368pt too wide) in paragraph at lines 372--386 \OT1/cmr/m/n/12

Points in vec-tor fields with more outward flow

< p>$(\OML/cmm/m/it/12 x; y\OT1/cmr/m/n/12 )[] \OMS/cmsy/m/n/12 ([]) [] ! LaTeX Error: Unicode character 回 (U+56DE) not set up for use with LaTeX. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.387

Curl Rotore 回 転

? ! LaTeX Error: Unicode character 転 (U+8EE2) not set up for use with LaTeX. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.387

Curl Rotore 回転

? [20] ! LaTeX Error: Unicode character ラ (U+30E9) not set up for use with LaTeX. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.396

Laplacian Laplaciano ラ プラス作用素

? ! LaTeX Error: Unicode character プ (U+30D7) not set up for use with LaTeX. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.396

Laplacian Laplaciano ラプ ラス作用素

? ! LaTeX Error: Unicode character ラ (U+30E9) not set up for use with LaTeX. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.396 ...class=cyan>Laplacian Laplaciano ラプラ ス作用素

? ! LaTeX Error: Unicode character ス (U+30B9) not set up for use with LaTeX. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.396 ...ss=cyan>Laplacian Laplaciano ラプラス 作用素

? ! LaTeX Error: Unicode character 作 (U+4F5C) not set up for use with LaTeX. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.396 ...cyan>Laplacian Laplaciano ラプラス作 用素 ? ! LaTeX Error: Unicode character 用 (U+7528) not set up for use with LaTeX. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.396 ...n>Laplacian Laplaciano ラプラス作用 素 ? ! LaTeX Error: Unicode character 素 (U+7D20) not set up for use with LaTeX. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.396 ...aplacian Laplaciano ラプラス作用素 ? Overfull \hbox (21.43324pt too wide) in paragraph at lines 396--400 []\OT1/cmr/m/n/12

Laplacian Lapla-ciano

Vector op-er-a- tor that returns [] ! Missing $ inserted. $ l.407 ... that point, e see Linear Algebra<... ? ! LaTeX Error: Bad math environment delimiter. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.408 ...ormal basises may be dependent on some \( \theta\) or \(\phi\) relat... ? Overfull \hbox (3.3515pt too wide) in paragraph at lines 402--423 \OT1/cmr/m/n/12

  • Cartesian
  • Polar
  • Parabolic
  • Bipola r
  • Elliptic
    • [] Overfull \hbox (48.89288pt too wide) in paragraph at lines 402--423 \OT1/cmr/m/n/12

    $\U/msb/m/n/12 R[]$\OT1/cmr/m/n/12

    < ul>
  • Cartesian
  • Cylindrical
  • Spherical
    • [] Overfull \hbox (12.08464pt too wide) in paragraph at lines 425--441 \OT1/cmr/m/n/12 trans-form

    $[]\OMS/cmsy/m/n/12 f[]\OML/cmm/m/it/12 ; [] ; []\OMS/cmsy/m/n/12 g \OT1/cmr/m/n/12 = \U/msb/m/n/12 R[]$\OT1/cmr/m/n/12

    • $[] = []$
    • $[] = []$
      • [] Overfull \hbox (20.93282pt too wide) in paragraph at lines 425--441 \OT1/cmr/m/n/12
    • $[] = []$

    Volume el-e-ment

    < p>$\OML/cmm/m/it/12 dV \OT1/cmr/m/n/12 = \OML/cmm/m/it/12 dxdydz$\OT1/cmr/m/n/1 2

    [] Underfull \vbox (badness 2503) has occurred while \output is active [] [21] Overfull \hbox (37.2561pt too wide) in paragraph at lines 476--512 \OT1/cmr/m/n/12 class=cyan>Vector trans-form

    For some cho-sen an-gle $\ OML/cmm/m/it/12 ^^R$\OT1/cmr/m/n/12

    $[]\OMS/cmsy/m/n/12 f[]\OML/cmm/m/i t/12 ; []; []\OMS/cmsy/m/n/12 g \OT1/cmr/m/n/12 = [] Overfull \hbox (75.4431pt too wide) in paragraph at lines 514--548 \OT1/cmr/m/n/12 [0\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 2\OML/cmm/m/it/12 ^^Y\OT1/ cmr/m/n/12 ]\OML/cmm/m/it/12 ; []\OMS/cmsy/m/n/12 f[]\OML/cmm/m/it/12 ; []; []\ OMS/cmsy/m/n/12 g \OT1/cmr/m/n/12 = \U/msb/m/n/12 R[]$\OT1/cmr/m/n/12

    • $[] = []$
    • $[] = []$
      • [] Overfull \hbox (87.6689pt too wide) in paragraph at lines 514--548 \OT1/cmr/m/n/12
    • $[] = []$

    $\OML/cmm/m/it/12 P[] \OT1/cmr/m/n/ 12 = []$

    [] Overfull \hbox (18.18806pt too wide) in paragraph at lines 514--548 \OT1/cmr/m/n/12
  • $\OMS/cmsy/m/n/12 r ^^A [] \OT1/cmr/m/n/12 = [][] + [][] + [][]$
  • $\OMS/cmsy/m/n/12 r ^^B [] \OT1/cmr/m/n/12 = []([] \OMS/cmsy/m/n /12 ^^@ [] Overfull \hbox (8.04431pt too wide) in paragraph at lines 514--548 [][]\OT1/cmr/m/n/12 ([](\OML/cmm/m/it/12 ^^R\OT1/cmr/m/n/12 )[]) + [][]$
    • < /ul>

    Volume el-e-ment

    [] Underfull \vbox (badness 10000) has occurred while \output is active [] [22]) [23] \openout2 = `2.aux'. (./2.tex [24 ] [25] [26] Chapter 7. [27 ] [28] [29] [30 ] Chapter 8. ) [31] \openout2 = `3.aux'. (./3.tex [32 ] [33] [34] Chapter 9. [35 ] Overfull \hbox (0.2838pt too wide) in paragraph at lines 54--54 []\OT1/cmr/bx/n/17.28 Applications for sur-face and flux in-te-grals [] [36] Chapter 10. [37 ] Overfull \hbox (46.93019pt too wide) in paragraph at lines 103--123 \OT1/cmr/m/n/12 the point (dif-fer-en-ti-a-tion of tan-gent with re-spect to ar -clength)
  • Binormal, [] Overfull \hbox (59.96666pt too wide) in paragraph at lines 125--131 \OT1/cmr/m/n/12 tan-gen-tial plane

    $\OML/cmm/m/it/12 ^^T \OT1/cmr/m/n/12 = \OMS/cmsy/m/n/12 k\OT1/cmr/bx/n/12 T[]\OMS/cmsy/m/n/12 k$\OT1/cmr/m/n/12

    $\OML/cmm/m/it/12 ^^T \OT1/cmr/m/n/12 = \OT1/cmr/bx/n/12 T[] \OMS/cmsy/m/n /12 ^^A \OT1/cmr/bx/n/12 N$\OT1/cmr/m/n/12

    Propositions

    [] Overfull \hbox (69.0751pt too wide) in paragraph at lines 133--139 \OT1/cmr/m/n/12 tan-gen-tial plane

    $\OML/cmm/m/it/12 ^^\ \OT1/cmr/m/n/12 = \OMS/cmsy/m/n/12 k\OT1/cmr/bx/n/12 B[]\OMS/cmsy/m/n/12 k$\OT1/cmr/m/n/12

    $\OML/cmm/m/it/12 ^^\ \OT1/cmr/m/n/12 = \OMS/cmsy/m/n/12 ^^@\OT1/cmr/bx/n/ 12 B[] \OMS/cmsy/m/n/12 ^^A \OT1/cmr/bx/n/12 N$\OT1/cmr/m/n/12

    Propositions

    [] Overfull \hbox (45.23334pt too wide) in paragraph at lines 149--155 []\OT1/cmr/m/n/12

    Fundamental planes