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A3 2009 gyak 1 - Laptörténet
2024-03-29T10:06:57Z
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Mozo: \
2009-10-28T14:02:53Z
<p>\</p>
<table class='diff diff-contentalign-left'>
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<col class='diff-content' />
<col class='diff-marker' />
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<td colspan='2' style="background-color: white; color:black;">←Régebbi változat</td>
<td colspan='2' style="background-color: white; color:black;">A lap 2009. október 28., 14:02-kori változata</td>
</tr><tr><td colspan="2" class="diff-lineno">97. sor:</td>
<td colspan="2" class="diff-lineno">97. sor:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Az adott pontban:</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Az adott pontban:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>\frac{\partial r}{\partial v}=(0,0,-3),\quad \frac{\partial r}{\partial v}=(0,-1,0),\quad \frac{\partial r}{\partial v}\times\frac{\partial r}{\partial v}=(-3,0,0)</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>\frac{\partial r}{\partial v}=(0,0,-3),\quad \frac{\partial r}{\partial v}=(0,-1,0),\quad \frac{\partial r}{\partial v}\times\frac{\partial r}{\partial v}=(-3,0,0)</math></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">'''4.''' Integráljuk a felső egységsugarú, origóközépponttú félgömbefelületre az</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>v(x,y,z)=(\frac{1}{x},\frac{x}{y^2},\frac{1}{z^3})</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">vektormezőt!</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">''Mo.'' Paraméterezzük gömbi koordinátákkal: ''r''(&phi;,&theta;)=(''x''(&phi;,&theta;),''y''(&phi;,&theta;),''z''(&phi;,&theta;))</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>x=\cos\varphi\cos\vartheta</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>y=\sin\varphi\cos\vartheta</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>z=\sin\vartheta</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">A koordinátavonal irányú vektorok:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>\frac{\partial r}{\partial \varphi}=(-\sin\varphi\cos\vartheta,\cos\varphi\cos\vartheta,0)</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>\frac{\partial r}{\partial \vartheta}=(-\cos\varphi\sin\vartheta,-\sin\varphi\sin\vartheta,\cos\vartheta)</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>\frac{\partial r}{\partial \varphi}\times\frac{\partial r}{\partial \vartheta}=(\cos^2\vartheta\cos\varphi,\cos^2\vartheta\sin\varphi,\cos\vartheta\sin\vartheta)</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Az integrál:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>\int\limits_{\varphi=0}^{2\pi}\int\limits_{\vartheta=0}^{\pi/2}\frac{\cos^2\vartheta\cos\varphi}{\cos\varphi\cos\vartheta}+\frac{\cos^2\vartheta\sin\varphi\cos\varphi\cos\vartheta}{\sin^2\varphi\cos^2\vartheta}+\frac{\cos\vartheta\sin\vartheta}{\sin^3\vartheta}\,\mathrm{d}\varphi\,\mathrm{d}\vartheta=</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"><math>=\int\limits_{\varphi=0}^{2\pi}\int\limits_{\vartheta=0}^{\pi/2}\cos^2\vartheta+\frac{\cos\varphi\cos \vartheta}{\sin\varphi}+\frac{\cos\vartheta}{\sin^2\vartheta}\,\mathrm{d}\varphi\,\mathrm{d}\vartheta=</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Gauss-tétel==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Gauss-tétel==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Ha '''v''' folytonosan differenciálható vektormező és ''V'' az értelmezési tartományába eső kompakt térrész, melynek határa a &part;''V'' felület, akkor  </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Ha '''v''' folytonosan differenciálható vektormező és ''V'' az értelmezési tartományába eső kompakt térrész, melynek határa a &part;''V'' felület, akkor  </div></td></tr>
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Mozo
http://wiki.math.bme.hu/index.php?title=A3_2009_gyak_1&diff=5693&oldid=prev
Mozo: /* Rotációmentes vektormező, potenciál */
2009-10-28T13:01:24Z
<p><span class="autocomment">Rotációmentes vektormező, potenciál</span></p>
<table class='diff diff-contentalign-left'>
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<col class='diff-content' />
<col class='diff-marker' />
<col class='diff-content' />
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<td colspan='2' style="background-color: white; color:black;">←Régebbi változat</td>
<td colspan='2' style="background-color: white; color:black;">A lap 2009. október 28., 13:01-kori változata</td>
</tr><tr><td colspan="2" class="diff-lineno">33. sor:</td>
<td colspan="2" class="diff-lineno">33. sor:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>\int\limits_{r_1}^{r_2}\mathrm{grad}\, \Phi\,\mathrm{d}r=\Phi(r_2)-\Phi(r_1)\,</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>\int\limits_{r_1}^{r_2}\mathrm{grad}\, \Phi\,\mathrm{d}r=\Phi(r_2)-\Phi(r_1)\,</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">'''Stokes-tétel''' Ha ''v'' egy folytonosan differenciálható vektormező és ''F'' olyan felület, mely peremével együtt a ''v'' értelmezési tartományában van, akkor </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>\int\limits_{\partial F} v\,\mathrm{d}r=\int\limits_{F}\,\mathrm{rot}\,v\,\mathrm{d}F</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">(megj.: ha ''G'' egy ''egyszeresen összefüggő'' tartománybeli zárt görbe, akkor mindig létezik olyan ''F'' felület ezen belül, melynek pereme ''G''.)</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">  </ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''2.''' Integráljuk a   </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''2.''' Integráljuk a   </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>v(x,y,z)=(2xy-z,x^2+z,y-x)\,</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>v(x,y,z)=(2xy-z,x^2+z,y-x)\,</math></div></td></tr>
<tr><td colspan="2" class="diff-lineno">68. sor:</td>
<td colspan="2" class="diff-lineno">72. sor:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Ezzel a görbére a vonalintegrál:</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Ezzel a görbére a vonalintegrál:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>\Phi(-3,0,2)-\Phi(3,0,2)=12</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>\Phi(-3,0,2)-\Phi(3,0,2)=12</math></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Felületi integrál==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Felületi integrál==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
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Mozo
http://wiki.math.bme.hu/index.php?title=A3_2009_gyak_1&diff=5692&oldid=prev
Mozo: /* Gauss-tétel */
2009-10-28T12:53:38Z
<p><span class="autocomment">Gauss-tétel</span></p>
<table class='diff diff-contentalign-left'>
<col class='diff-marker' />
<col class='diff-content' />
<col class='diff-marker' />
<col class='diff-content' />
<tr valign='top'>
<td colspan='2' style="background-color: white; color:black;">←Régebbi változat</td>
<td colspan='2' style="background-color: white; color:black;">A lap 2009. október 28., 12:53-kori változata</td>
</tr><tr><td colspan="2" class="diff-lineno">93. sor:</td>
<td colspan="2" class="diff-lineno">93. sor:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>\frac{\partial r}{\partial v}=(0,0,-3),\quad \frac{\partial r}{\partial v}=(0,-1,0),\quad \frac{\partial r}{\partial v}\times\frac{\partial r}{\partial v}=(-3,0,0)</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>\frac{\partial r}{\partial v}=(0,0,-3),\quad \frac{\partial r}{\partial v}=(0,-1,0),\quad \frac{\partial r}{\partial v}\times\frac{\partial r}{\partial v}=(-3,0,0)</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Gauss-tétel==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Gauss-tétel==</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Ha '''v''' folytonosan differenciálható vektormező és ''V'' az értelmezési tartományába eső kompakt térrész, melynek határa a &part;''V'' felület, akkor </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>\int\limits_{\partial V} v\;\mathrm{d}F=\int\limits_{V}\,\mathrm{div}\,v\,\mathrm{d}V</math> </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''5.''' Számítsuk ki az</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''5.''' Számítsuk ki az</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>x^2+y^2=z</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>x^2+y^2=z</math></div></td></tr>
</table>
Mozo
http://wiki.math.bme.hu/index.php?title=A3_2009_gyak_1&diff=5691&oldid=prev
Mozo: /* Gauss-tétel */
2009-10-28T10:38:06Z
<p><span class="autocomment">Gauss-tétel</span></p>
<table class='diff diff-contentalign-left'>
<col class='diff-marker' />
<col class='diff-content' />
<col class='diff-marker' />
<col class='diff-content' />
<tr valign='top'>
<td colspan='2' style="background-color: white; color:black;">←Régebbi változat</td>
<td colspan='2' style="background-color: white; color:black;">A lap 2009. október 28., 10:38-kori változata</td>
</tr><tr><td colspan="2" class="diff-lineno">93. sor:</td>
<td colspan="2" class="diff-lineno">93. sor:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>\frac{\partial r}{\partial v}=(0,0,-3),\quad \frac{\partial r}{\partial v}=(0,-1,0),\quad \frac{\partial r}{\partial v}\times\frac{\partial r}{\partial v}=(-3,0,0)</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>\frac{\partial r}{\partial v}=(0,0,-3),\quad \frac{\partial r}{\partial v}=(0,-1,0),\quad \frac{\partial r}{\partial v}\times\frac{\partial r}{\partial v}=(-3,0,0)</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Gauss-tétel==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Gauss-tétel==</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>'''5.''' Számítsuk ki <del class="diffchange diffchange-inline">a</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>'''5.''' Számítsuk ki <ins class="diffchange diffchange-inline">az</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>x^2+y^2=z</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>x^2+y^2=z</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>felület z=1 és z=4 síkok közé eső darabjára a</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>felület z=1 és z=4 síkok közé eső darabjára a</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>:<math>v(x,y,z)=(3x-zx^2+\sin y,\mathrm{sh}\,z^2+5xy+<del class="diffchange diffchange-inline">2y</del>,xz^2-5xz+4x)\,</math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>:<math>v(x,y,z)=(3x-zx^2+\sin y,\mathrm{sh}\,z^2+5xy+<ins class="diffchange diffchange-inline">2006y</ins>,xz^2-5xz+4x)\,</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>vektormező felületmenti integrálját!</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>vektormező felületmenti integrálját!</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">''Mo.'' Vegyük észre, hogy a fedő és alaplapokon a vektormezőnek csak felületirányú komponense van, ezért ezeken a vektormező integrálja 0. A vektormező divergenciájának integrálja a térrészre tehát egyenlő lesz a palást felületi integráljával.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>\mathrm{div}\,v(x,y,z)=(3-2zx)+(5x+2006)+(2xz-5x)=2009\,</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Beparaméterezve a forgástestet (áttérve hengerkoordinátákra)</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>r(\rho,\varphi,z)=\begin{bmatrix}\rho\cos\varphi\\\rho\sin\varphi\\z\end{bmatrix}</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">és a tartomány:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>T_{\rho,\varphi,z}=\{(\rho,\varphi,z)\mid 0\leq\varphi\leq 2\pi,\;1\leq z\leq 4\;0\leq\rho\leq \sqrt{z}\}</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">hiszen </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>\rho=\sqrt{x^2+y^2}=\sqrt{z}</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">tudjuk, hogy a hengerkoordinátáknál:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>\mathrm{det}\,\mathrm{J}(\varphi,\rho,z)=\rho</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">ezért </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"><math>\int\limits_{T_{x,y,z}}\mathrm{div}\,v(x,y,z)\,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z=\int\limits_{T_{\rho,\varphi,z}}\rho\,\mathrm{div}\,v(x(\rho,\varphi,z),y(\rho,\varphi,z),z)\,\mathrm{d}\rho\,\mathrm{d}\varphi\,\mathrm{d}z=</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"><math>=\int\limits_{z=1}^{4}\int\limits_{\varphi=0}^{2\pi}\int\limits_{\rho=0}^{\sqrt{z}}2009\rho\,\mathrm{d}\rho\,\mathrm{d}\varphi\,\mathrm{d}z</math></ins></div></td></tr>
</table>
Mozo
http://wiki.math.bme.hu/index.php?title=A3_2009_gyak_1&diff=5690&oldid=prev
Mozo: /* Felületi integrál */
2009-10-28T10:13:22Z
<p><span class="autocomment">Felületi integrál</span></p>
<table class='diff diff-contentalign-left'>
<col class='diff-marker' />
<col class='diff-content' />
<col class='diff-marker' />
<col class='diff-content' />
<tr valign='top'>
<td colspan='2' style="background-color: white; color:black;">←Régebbi változat</td>
<td colspan='2' style="background-color: white; color:black;">A lap 2009. október 28., 10:13-kori változata</td>
</tr><tr><td colspan="2" class="diff-lineno">92. sor:</td>
<td colspan="2" class="diff-lineno">92. sor:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Az adott pontban:</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Az adott pontban:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>\frac{\partial r}{\partial v}=(0,0,-3),\quad \frac{\partial r}{\partial v}=(0,-1,0),\quad \frac{\partial r}{\partial v}\times\frac{\partial r}{\partial v}=(-3,0,0)</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>\frac{\partial r}{\partial v}=(0,0,-3),\quad \frac{\partial r}{\partial v}=(0,-1,0),\quad \frac{\partial r}{\partial v}\times\frac{\partial r}{\partial v}=(-3,0,0)</math></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">==Gauss-tétel==</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">'''5.''' Számítsuk ki a</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>x^2+y^2=z</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">felület z=1 és z=4 síkok közé eső darabjára a</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>v(x,y,z)=(3x-zx^2+\sin y,\mathrm{sh}\,z^2+5xy+2y,xz^2-5xz+4x)\,</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">vektormező felületmenti integrálját!</ins></div></td></tr>
</table>
Mozo
http://wiki.math.bme.hu/index.php?title=A3_2009_gyak_1&diff=5689&oldid=prev
Mozo: /* Felületi integrál */
2009-10-28T10:01:56Z
<p><span class="autocomment">Felületi integrál</span></p>
<table class='diff diff-contentalign-left'>
<col class='diff-marker' />
<col class='diff-content' />
<col class='diff-marker' />
<col class='diff-content' />
<tr valign='top'>
<td colspan='2' style="background-color: white; color:black;">←Régebbi változat</td>
<td colspan='2' style="background-color: white; color:black;">A lap 2009. október 28., 10:01-kori változata</td>
</tr><tr><td colspan="2" class="diff-lineno">71. sor:</td>
<td colspan="2" class="diff-lineno">71. sor:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''3.''' Határozzuk meg az</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''3.''' Határozzuk meg az</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>:<math>r(u,v)=(\cos u\,\mathrm{ch}\,v,\cos u\mathrm{sh}\,v,3\sin u),\quad\quad (u,v)\in [0,4]\times [0,2]</math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>:<math>r(u,v)=(\cos u\,\mathrm{ch}\,v,\cos u<ins class="diffchange diffchange-inline">\,</ins>\mathrm{sh}\,v,3\sin u),\quad\quad (u,v)\in [0,4]\times [0,2]</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>felület normálvektorát a (u,v)=(&pi;,0)-hoz tartozó pontban!</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>felület normálvektorát a (u,v)=(&pi;,0)-hoz tartozó pontban!</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td colspan="2" class="diff-lineno">77. sor:</td>
<td colspan="2" class="diff-lineno">77. sor:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>x=\cos u\,\mathrm{ch}\,v</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>x=\cos u\,\mathrm{ch}\,v</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>y=\cos u\mathrm{sh}\,v</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>y=\cos u\mathrm{sh}\,v</math></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>:<math>z=3\sin u</math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>:<math>z=3\sin u<ins class="diffchange diffchange-inline">\,</ins></math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>ezért  </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>ezért  </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>x^2-y^2=\cos^2u(\mathrm{ch}^2\,v-\mathrm{sh}^2\,v)=\cos^2 u</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>x^2-y^2=\cos^2u(\mathrm{ch}^2\,v-\mathrm{sh}^2\,v)=\cos^2 u</math></div></td></tr>
<tr><td colspan="2" class="diff-lineno">91. sor:</td>
<td colspan="2" class="diff-lineno">91. sor:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>\frac{\partial r}{\partial v}=(\cos u\,\mathrm{sh}\,v,\cos u\,\mathrm{ch}\,v,0)\,</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>\frac{\partial r}{\partial v}=(\cos u\,\mathrm{sh}\,v,\cos u\,\mathrm{ch}\,v,0)\,</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Az adott pontban:</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Az adott pontban:</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>:<math>\frac{\partial r}{\partial v}=(0,0,-3)<del class="diffchange diffchange-inline"></math></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>:<math>\frac{\partial r}{\partial v}=(0,0,-3)<ins class="diffchange diffchange-inline">,\quad </ins>\frac{\partial r}{\partial v}=(0,-1<ins class="diffchange diffchange-inline">,0),\quad \frac{\partial r}{\partial v}\times\frac{\partial r}{\partial v}=(-3,0</ins>,0)</math></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">:<math></del>\frac{\partial r}{\partial v}=(0,-1,0)</math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
</table>
Mozo
http://wiki.math.bme.hu/index.php?title=A3_2009_gyak_1&diff=5688&oldid=prev
Mozo: /* Rotációmentes vektormező, potenciál */
2009-10-27T10:21:40Z
<p><span class="autocomment">Rotációmentes vektormező, potenciál</span></p>
<table class='diff diff-contentalign-left'>
<col class='diff-marker' />
<col class='diff-content' />
<col class='diff-marker' />
<col class='diff-content' />
<tr valign='top'>
<td colspan='2' style="background-color: white; color:black;">←Régebbi változat</td>
<td colspan='2' style="background-color: white; color:black;">A lap 2009. október 27., 10:21-kori változata</td>
</tr><tr><td colspan="2" class="diff-lineno">68. sor:</td>
<td colspan="2" class="diff-lineno">68. sor:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Ezzel a görbére a vonalintegrál:</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Ezzel a görbére a vonalintegrál:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>\Phi(-3,0,2)-\Phi(3,0,2)=12</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>\Phi(-3,0,2)-\Phi(3,0,2)=12</math></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">==Felületi integrál==</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">'''3.''' Határozzuk meg az</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>r(u,v)=(\cos u\,\mathrm{ch}\,v,\cos u\mathrm{sh}\,v,3\sin u),\quad\quad (u,v)\in [0,4]\times [0,2]</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">felület normálvektorát a (u,v)=(&pi;,0)-hoz tartozó pontban!</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">''Mo.'' (1) Felírhatjuk például a görbét implicit alakban. Mivel</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>x=\cos u\,\mathrm{ch}\,v</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>y=\cos u\mathrm{sh}\,v</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>z=3\sin u</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">ezért </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>x^2-y^2=\cos^2u(\mathrm{ch}^2\,v-\mathrm{sh}^2\,v)=\cos^2 u</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>x^2-y^2+\frac{1}{9}z^2=1</math> ill. <math>x^2-y^2+\frac{1}{9}z^2-1=0</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Ekkor az </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>F(x,y,z)=x^2-y^2+\frac{1}{9}z^2-1\,</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">gradiense a normálvektort adja:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>n(x,y,z)=\mathrm{grad}\,F=(2x,-2y,\frac{2}{9}z)\,</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">azaz az adott pontban</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>n(x(u,v),y(u,v),z(u,v))=(-2,0,0)\,</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">(2) A felületi normálist a koordinátavonal irányú vektorok vektoriális szorzata is kiadja.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>\frac{\partial r}{\partial u}=(-\sin u\,\mathrm{ch}\,v,-\sin u\,\mathrm{sh}\,v,3\cos u)\,</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>\frac{\partial r}{\partial v}=(\cos u\,\mathrm{sh}\,v,\cos u\,\mathrm{ch}\,v,0)\,</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Az adott pontban:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>\frac{\partial r}{\partial v}=(0,0,-3)</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>\frac{\partial r}{\partial v}=(0,-1,0)</math></ins></div></td></tr>
</table>
Mozo
http://wiki.math.bme.hu/index.php?title=A3_2009_gyak_1&diff=5687&oldid=prev
Mozo: /* Rotációmentes vektormező, potenciál */
2009-10-27T09:36:28Z
<p><span class="autocomment">Rotációmentes vektormező, potenciál</span></p>
<table class='diff diff-contentalign-left'>
<col class='diff-marker' />
<col class='diff-content' />
<col class='diff-marker' />
<col class='diff-content' />
<tr valign='top'>
<td colspan='2' style="background-color: white; color:black;">←Régebbi változat</td>
<td colspan='2' style="background-color: white; color:black;">A lap 2009. október 27., 09:36-kori változata</td>
</tr><tr><td colspan="2" class="diff-lineno">65. sor:</td>
<td colspan="2" class="diff-lineno">65. sor:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>=\frac{2}{3}x_0^2y_0-\frac{1}{2}x_0z_0+\frac{1}{3}y_0x_0^2+\frac{1}{2}y_0z_0+\frac{1}{2}z_0y_0-\frac{1}{2}z_0x_0=x_0^2y_0+y_0z_0-x_0z_0</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>=\frac{2}{3}x_0^2y_0-\frac{1}{2}x_0z_0+\frac{1}{3}y_0x_0^2+\frac{1}{2}y_0z_0+\frac{1}{2}z_0y_0-\frac{1}{2}z_0x_0=x_0^2y_0+y_0z_0-x_0z_0</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Ellenőrizzük!</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Ellenőrizzük!</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>:<math>\frac{\partial\Phi}{\partial x}=2xy+z,\quad\quad\frac{\partial\Phi}{\partial y}=x^2+z,\quad\quad\frac{\partial\Phi}{\partial z}=y-x<del class="diffchange diffchange-inline">=</del></math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>:<math>\frac{\partial\Phi}{\partial x}=2xy+z,\quad\quad\frac{\partial\Phi}{\partial y}=x^2+z,\quad\quad\frac{\partial\Phi}{\partial z}=y-x</math></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Ezzel:</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Ezzel <ins class="diffchange diffchange-inline">a görbére a vonalintegrál</ins>:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>\Phi(-3,0,2)-\Phi(3,0,2)=12</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>\Phi(-3,0,2)-\Phi(3,0,2)=12</math></div></td></tr>
</table>
Mozo
http://wiki.math.bme.hu/index.php?title=A3_2009_gyak_1&diff=5686&oldid=prev
Mozo: /* Rotációmentes vektormező, potenciál */
2009-10-27T09:33:36Z
<p><span class="autocomment">Rotációmentes vektormező, potenciál</span></p>
<table class='diff diff-contentalign-left'>
<col class='diff-marker' />
<col class='diff-content' />
<col class='diff-marker' />
<col class='diff-content' />
<tr valign='top'>
<td colspan='2' style="background-color: white; color:black;">←Régebbi változat</td>
<td colspan='2' style="background-color: white; color:black;">A lap 2009. október 27., 09:33-kori változata</td>
</tr><tr><td colspan="2" class="diff-lineno">57. sor:</td>
<td colspan="2" class="diff-lineno">57. sor:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Ezért legyen  </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Ezért legyen  </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>\Phi(r):=\int\limits_{0}^{r}\mathrm{grad}\, \Phi\,\mathrm{d}r=\int\limits_{0}^{r}v\,\mathrm{d}r</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>\Phi(r):=\int\limits_{0}^{r}\mathrm{grad}\, \Phi\,\mathrm{d}r=\int\limits_{0}^{r}v\,\mathrm{d}r</math></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>v(x,y,z)=(2xy-z,x^2+z,y-x)\,</math></ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Legyen a kezdőpont (0,0,0), a görbe:</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Legyen a kezdőpont (0,0,0), a görbe:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>s(t)=(x_0t,y_0t,z_0t),\quad\quad 0\leq t\leq 1,\quad\quad \dot{s}(t)=(x_0,y_0,z_0)</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>s(t)=(x_0t,y_0t,z_0t),\quad\quad 0\leq t\leq 1,\quad\quad \dot{s}(t)=(x_0,y_0,z_0)</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Ezzel</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Ezzel</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>:<math>\int\limits_{0}^{(x_0,y_0,z_0)}v\,\mathrm{d}r=\int\limits_{t=0}^{1} 2x_0(x_0ty_0t)-x_0z_0t+y_0(x_0t)^2+y_0z_0t+z_0y_0t-<del class="diffchange diffchange-inline">z_0^2t</del>\;\mathrm{d}t=</math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>:<math>\int\limits_{0}^{(x_0,y_0,z_0)}v\,\mathrm{d}r=\int\limits_{t=0}^{1} 2x_0(x_0ty_0t)-x_0z_0t+y_0(x_0t)^2+y_0z_0t+z_0y_0t-<ins class="diffchange diffchange-inline">x_0z_0 t</ins>\;\mathrm{d}t=</math></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>:<math>=\int\limits_{t=0}^{1} 2x_0^2y_0t^2-x_0z_0t+y_0x_0^2t^2+y_0z_0t+z_0y_0t-<del class="diffchange diffchange-inline">z_0^2t</del>\;\mathrm{d}t=</math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>:<math>=\int\limits_{t=0}^{1} 2x_0^2y_0t^2-x_0z_0t+y_0x_0^2t^2+y_0z_0t+z_0y_0t-<ins class="diffchange diffchange-inline">x_0z_0t</ins>\;\mathrm{d}t=</math></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>:<math>=\frac{2}{3}x_0^2y_0-\frac{1}{2}x_0z_0+\frac{1}{3}y_0x_0^2+\frac{1}{2}y_0z_0+\frac{1}{2}z_0y_0-\frac{1}{2}<del class="diffchange diffchange-inline">z_0^2</del>=x_0^2y_0+y_0z_0-\frac{<del class="diffchange diffchange-inline">1</del>}{<del class="diffchange diffchange-inline">2</del>}<del class="diffchange diffchange-inline">x_0z_0-</del>\frac{<del class="diffchange diffchange-inline">1</del>}{<del class="diffchange diffchange-inline">2</del>}<del class="diffchange diffchange-inline">z_0</del>^2</math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>:<math>=\frac{2}{3}x_0^2y_0-\frac{1}{2}x_0z_0+\frac{1}{3}y_0x_0^2+\frac{1}{2}y_0z_0+\frac{1}{2}z_0y_0-\frac{1}{2}<ins class="diffchange diffchange-inline">z_0x_0</ins>=x_0^2y_0+y_0z_0-<ins class="diffchange diffchange-inline">x_0z_0</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">Ellenőrizzük!</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">:<math></ins>\frac{<ins class="diffchange diffchange-inline">\partial\Phi</ins>}{<ins class="diffchange diffchange-inline">\partial x</ins>}<ins class="diffchange diffchange-inline">=2xy+z,\quad\quad</ins>\frac{<ins class="diffchange diffchange-inline">\partial\Phi</ins>}{<ins class="diffchange diffchange-inline">\partial y</ins>}<ins class="diffchange diffchange-inline">=x</ins>^2<ins class="diffchange diffchange-inline">+z,\quad\quad\frac{\partial\Phi}{\partial z}=y-x=</ins></math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Ezzel:</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Ezzel:</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>:<math>\Phi(-3,0,2)-\Phi(3,0,2)=<del class="diffchange diffchange-inline">3+3=6</del></math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>:<math>\Phi(-3,0,2)-\Phi(3,0,2)=<ins class="diffchange diffchange-inline">12</ins></math></div></td></tr>
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Mozo
http://wiki.math.bme.hu/index.php?title=A3_2009_gyak_1&diff=5685&oldid=prev
Mozo: /* Rotációmentes vektormező, potenciál */
2009-10-27T09:27:32Z
<p><span class="autocomment">Rotációmentes vektormező, potenciál</span></p>
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<td colspan='2' style="background-color: white; color:black;">←Régebbi változat</td>
<td colspan='2' style="background-color: white; color:black;">A lap 2009. október 27., 09:27-kori változata</td>
</tr><tr><td colspan="2" class="diff-lineno">64. sor:</td>
<td colspan="2" class="diff-lineno">64. sor:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>=\frac{2}{3}x_0^2y_0-\frac{1}{2}x_0z_0+\frac{1}{3}y_0x_0^2+\frac{1}{2}y_0z_0+\frac{1}{2}z_0y_0-\frac{1}{2}z_0^2=x_0^2y_0+y_0z_0-\frac{1}{2}x_0z_0-\frac{1}{2}z_0^2</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>=\frac{2}{3}x_0^2y_0-\frac{1}{2}x_0z_0+\frac{1}{3}y_0x_0^2+\frac{1}{2}y_0z_0+\frac{1}{2}z_0y_0-\frac{1}{2}z_0^2=x_0^2y_0+y_0z_0-\frac{1}{2}x_0z_0-\frac{1}{2}z_0^2</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Ezzel:</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Ezzel:</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>:<math>\Phi(3,0,2)-\Phi(<del class="diffchange diffchange-inline">-</del>3,0,2)=<del class="diffchange diffchange-inline">-</del>3<del class="diffchange diffchange-inline">-(</del>3<del class="diffchange diffchange-inline">)</del>=<del class="diffchange diffchange-inline">-</del>6</math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>:<math>\Phi(<ins class="diffchange diffchange-inline">-</ins>3,0,2)-\Phi(3,0,2)=3<ins class="diffchange diffchange-inline">+</ins>3=6</math></div></td></tr>
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Mozo