\[\begin{split}\newcommand{\N}{\mathbb N} \newcommand{\Z}{\mathbb Z} \newcommand{\Q}{\mathbb Q} \newcommand{\R}{\mathbb R} \newcommand{\C}{\mathbb C} \newcommand{\ba}{\mathbf{a}} \newcommand{\bb}{\mathbf{b}} \newcommand{\bc}{\mathbf{c}} \newcommand{\bd}{\mathbf{d}} \newcommand{\be}{\mathbf{e}} \newcommand{\bbf}{\mathbf{f}} \newcommand{\bF}{\mathbf{F}} \newcommand{\bh}{\mathbf{h}} \newcommand{\bi}{\mathbf{i}} \newcommand{\bj}{\mathbf{j}} \newcommand{\bk}{\mathbf{k}} \newcommand{\bN}{\mathbf{N}} \newcommand{\bn}{\mathbf{n}} \newcommand{\bo}{\mathbf{0}} \newcommand{\bp}{\mathbf{p}} \newcommand{\bq}{\mathbf{q}} \newcommand{\br}{\mathbf{r}} \newcommand{\bR}{\mathbf{R}} \newcommand{\bs}{\mathbf{s}} \newcommand{\bT}{\mathbf{T}} \newcommand{\bu}{\mathbf{u}} \newcommand{\bv}{\mathbf{v}} \newcommand{\bw}{\mathbf{w}} \newcommand{\bx}{\mathbf{x}} \newcommand{\by}{\mathbf{y}} \newcommand{\bz}{\mathbf{z}} \newcommand{\re}{\operatorname{Re}} \newcommand{\im}{\operatorname{Im}} \newcommand{\bA}{\mathbf{A}} \newcommand{\cE}{\mathcal{E}} \newcommand{\cB}{\mathcal{B}} \newcommand{\cC}{\mathcal{C}} \newcommand{\dist}{\operatorname{d}} \newcommand{\diag}{\operatorname{diag}} \newcommand{\proj}{\operatorname{proj}} \newcommand{\rank}{\operatorname{rank}} \newcommand{\Span}{\operatorname{span}} \newcommand{\row}{\operatorname{row}} \newcommand{\col}{\operatorname{col}} \newcommand{\Null}{\operatorname{null}} \newcommand{\id}{\operatorname{id}} \newcommand{\piste}{\boldsymbol{\cdot}} \newcommand{\kappale}{\newline \hspace{17pt}} \newcommand{\kohta}[1]{\textbf{#1)}\hspace{5px}} \newcommand{\kohtav}[1]{\hspace{7pt}\textbf{#1)}\hspace{5px}} \newcommand{\tilaa}{\vspace{7pt}\\} \newcommand{\bigfrac}[2]{{\displaystyle{\frac{#1}{#2}}}} \newcommand{\smallfrac}[2]{{\textstyle{\frac{#1}{#2}}}} \newcommand{\xn}[2]{(#1_1,#1_2,\ldots,#1_{#2})} \newcommand{\vastaus}[1]{\null\hfill({\footnotesize#1})} \newcommand{\mathvastaus}[1]{\eqno{\mbox{({\footnotesize#1})}}} \newcommand{\ep}[1]{\textnormal{ (\cite[#1]{ep})}} \newcommand{\pysty}[1]{\left[\begin{array}{@{}r@{}}#1\end{array}\right]} \newcommand{\sij}[2]{\bigg/_{\mspace{-15mu}#1}^{\,#2}} \newcommand{\qedhere}{}\end{split}\]

Yleinen muuttujanvaihto avaruusintegraaleilla

Tarkastellaan muuttujanvaihtoa \(\R^3\):ssa muuttujien \(uvw\) ja \(xyz\) välillä. Muuttujanvaihtokuvaus on

(1)\[F\colon\R_{uvw}^3\to\R_{xyz}^3,\ F(u,v,w)=(x(u,v,w),\,y(u,v,w),\,z(u,v,w))\]

ja sen Jacobin determinantti

\[\begin{split}J_F(u,v,w)=\det(F'(u,v,w)) =\frac{\partial(x,y,z)}{\partial(u,v,w)} =\begin{vmatrix} \dfrac{\partial x}{\partial u} & \dfrac{\partial x}{\partial v} & \dfrac{\partial x}{\partial w}\\[10pt] \dfrac{\partial y}{\partial u} & \dfrac{\partial y}{\partial v} & \dfrac{\partial y}{\partial w}\\[10pt] \dfrac{\partial z}{\partial u} & \dfrac{\partial z}{\partial v} & \dfrac{\partial z}{\partial w} \end{vmatrix}.\end{split}\]

Lause 8.5.1

Vastaavin oletuksin kuin lauseessa 7.6.2 pätee:

\[\iiint_A f(x,y,z)\,dx\,dy\,dz =\iiint_S f(F(u,v,w)) \left| \frac{\partial(x,y,z)}{\partial(u,v,w)}\right|\,du\,dv\,dw.\]

Lause 8.5.1 ja seuraavissa esimerkeissä laskettavat suurennussuhteet

\[\left|\frac{\partial(x,y,z)}{\partial(u,v,w)}\right|\]

todistavat muuttujanvaihtolauseet 8.2.1 ja 8.2.4 sylinteri- ja pallokoordinaateille.

Esimerkki 8.5.2

Sylinterikoordinaattikuvaukselle

\[F(r,\theta,z) =(x(r,\theta,z),\,y(r,\theta,z),\,z(r,\theta,z)) =(r\cos\theta,\,r\sin\theta,\,z)\]

on

\[\begin{split}\begin{aligned} \frac{\partial(x,y,z)}{\partial(r,\theta,z)} =\begin{vmatrix} \dfrac{\partial x}{\partial r} & \dfrac{\partial x}{\partial\theta} & \dfrac{\partial x}{\partial z}\\[10pt] \dfrac{\partial y}{\partial r} & \dfrac{\partial y}{\partial\theta} & \dfrac{\partial y}{\partial z}\\[10pt] \dfrac{\partial z}{\partial r} & \dfrac{\partial z}{\partial\theta} & \dfrac{\partial z}{\partial z} \end{vmatrix} =\begin{vmatrix} \cos\theta & -r\sin\theta & 0\\ \sin\theta & r\cos\theta & 0\\ 0 & 0 & 1 \end{vmatrix} =r. \end{aligned}\end{split}\]

Laske determinantti kehittämällä viimeisen rivin suhteen ja käytä trigonometrian peruskaavaa \(\sin^2\theta+\cos^2\theta=1\).

Esimerkki 8.5.3

Pallokoordinaattikuvaukselle

\[\begin{split}\begin{aligned} F(\rho,\phi,\theta) &=(x(\rho,\phi,\theta),\,y(\rho,\phi,\theta),\,z(\rho,\phi,\theta))\\ &=(\rho\sin\phi\cos\theta,\,\rho\sin\phi\sin\theta,\,\rho\cos\phi) \end{aligned}\end{split}\]

on

\[\begin{split}\begin{aligned} \frac{\partial(x,y,z)}{\partial(\rho,\phi,\theta)} &=\begin{vmatrix} \dfrac{\partial x}{\partial\rho} & \dfrac{\partial x}{\partial\phi} & \dfrac{\partial x}{\partial\theta}\\[10pt] \dfrac{\partial y}{\partial\rho} & \dfrac{\partial y}{\partial\phi} & \dfrac{\partial y}{\partial\theta}\\[10pt] \dfrac{\partial z}{\partial\rho} & \dfrac{\partial z}{\partial\phi} & \dfrac{\partial z}{\partial\theta} \end{vmatrix} =\begin{vmatrix} \sin\phi\cos\theta & \rho\cos\phi\cos\theta & -\rho\sin\phi\sin\theta\\ \sin\phi\sin\theta & \rho\cos\phi\sin\theta & \rho\sin\phi\cos\theta\\ \cos\phi & -\rho\sin\phi & 0 \end{vmatrix}\\ &=\rho^2\sin\phi. \end{aligned}\end{split}\]

Laske determinantti kehittämällä viimeisen rivin suhteen ja käytä toistuvasti trigonometrian peruskaavaa. Välillä \(0\le\phi\le\pi\) on \(\sin\phi\ge0\), joten

\[\begin{aligned} \left|\frac{\partial(x,y,z)}{\partial(\rho,\phi,\theta)}\right| =\rho^2|\sin\phi|=\rho^2\sin\phi. \end{aligned}\]
Palautusta lähetetään...