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Faglige notater: TMA4105 - Sideversjonshistorikk
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Jonathrg på 4. mai 2015 kl. 16:01
2015-05-04T16:01:19Z
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Jonathrg
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Jonathrg på 4. mai 2015 kl. 16:01
2015-05-04T16:01:13Z
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Jonathrg
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Rssaetra på 4. mai 2015 kl. 12:02
2015-05-04T12:02:13Z
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>====Stokes' teorem====</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>====Stokes' teorem====</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>\oint_C \mathbf{F} \cdot \mathbf{T}\,ds = \int \! \! \! \! \int_S \operatorname{curl}\,\mathbf{F} \cdot \mathbf{n}\,d\sigma</math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>\oint_C \mathbf{F} \cdot \mathbf{T}\,ds = \int \! \! \! \! \int_S \operatorname{curl}\,\mathbf{F} \cdot \mathbf{n}\,d\sigma</math></div></td></tr>
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Rssaetra
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Kristsof: Ny side: Faglige notater til [http://www.nanowiki.no/wiki/TMA4105_-_Matematikk_2 Matematikk 2] === Nablavektoren === Nablavektoren: <math>\vec{\nabla}</math> Nablaoperatoren er definert som <ma...
2015-05-04T11:50:56Z
<p>Ny side: Faglige notater til [http://www.nanowiki.no/wiki/TMA4105_-_Matematikk_2 Matematikk 2] === Nablavektoren === Nablavektoren: <math>\vec{\nabla}</math> Nablaoperatoren er definert som <ma...</p>
<p><b>Ny side</b></p><div>Faglige notater til [http://www.nanowiki.no/wiki/TMA4105_-_Matematikk_2 Matematikk 2]<br />
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=== Nablavektoren === <br />
Nablavektoren: <math>\vec{\nabla}</math><br />
<br />
Nablaoperatoren er definert som <math>\nabla f(x,y,z) =\frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} + \frac{\partial f}{\partial z}</math>.<br />
En naturlig utvidelse av dette blir nablavektoren, definert som vektoren <math>\vec{\nabla} f(x,y,z) =[\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z}]</math>.<br />
Retningen til denne vektoren angir den retningen der funksjonen ''f(x,y,z)'' vokser raskest, mens verdien til vektoren angir hvor stor denne veksten er. Ved å følge nablavektoren vil man altså alltid komme til nærmeste ekstrempunkt i funskjonen.<br />
Dersom man setter ''f(x,y,z)=k'', der ''k'' er en konstant, altså man lager nivåkurver i ''f(x,y,z)'' vil nablavektoren stå normal på disse nivåkurvene.<br />
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===Dekomponering av akselerasjonsvektor===<br />
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<math>\mathbf{a}(t) = v'(t)\mathbf{T}(t) + \kappa (t)v^2(t)\mathbf{N}(t)</math><br />
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===Diskriminanten i annenderiverttesten===<br />
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<math>\Delta = AC - B^2\,</math> der <math>A = f_{xx},\,\,\, B = f_{xy},\,\,\, C = f_{yy}</math><br />
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===Sylinderkoordinater=== <br />
Sylinderkoordinater: (<math>r,\,\theta,\,z</math>)<br />
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<math>x = r \cos{\theta}\quad y = r \sin{\theta} \quad z = z</math><br />
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<math>r^2 = x^2 + y^2 \quad dV = r dz \,dr \,d\theta</math><br />
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===Kulekoordinater===<br />
Kulekoordinater: (<math>\rho,\,\phi,\,\theta</math>) <br />
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<math>x = \rho \sin{\phi}\cos{\theta}\quad y = \rho \sin{\phi}\sin{\theta} \quad z = \rho\cos{\phi}</math><br />
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<math>\rho^2 = x^2 + y^2 + z^2 \quad dV = \rho^2 \sin{\phi} \,d\rho\, d\phi \,d\theta</math><br />
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===Flateintegral===<br />
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<math>d\sigma = \left|\mathbf{N}(u,v)\right|\,du\,dv = \left|\frac{\partial\mathbf{r}}{\partial u}\times \frac{\partial \mathbf{r}}{\partial v}\right|\,du\,dv</math><br />
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===Tyngdepunkt og treghetsmoment===<br />
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<math>\bar{x} = \frac{1}{m} \int \!\!\!\! \int \!\!\!\! \int_T x \, dm \quad \bar{y} = \frac{1}{m} \int \!\!\!\! \int \!\!\!\! \int_T y \, dm \quad \bar{z} = \frac{1}{m} \int \!\!\!\! \int \!\!\!\! \int_T z \, dm \quad dm = \delta(x,y,z)\,dV</math><br />
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<math>I_x = \int \!\!\!\! \int \!\!\!\! \int_T \left( y^2 + z^2 \right) \, dm \quad I_y = \int \!\!\!\! \int \!\!\!\! \int_T \left( x^2 + z^2 \right) \, dm \quad I_z = \int \!\!\!\! \int \!\!\!\! \int_T \left( x^2 + y^2 \right) \, dm \quad I_L = \int \!\!\!\! \int \!\!\!\! \int_T R(x,y,z)^2 \, dm \quad</math><br />
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===Vektoranalyse===<br />
====Greens teorem====<br />
<math>\oint_C P \, dx + Q \, dy = \int \! \! \! \! \int_R \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \, dA</math><br />
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====Divergensteoremet====<br />
<math>\int \!\!\!\! \int_S \mathbf{F}\cdot\mathbf{n}\,d\sigma = \int \!\!\!\! \int \!\!\!\! \int_T \operatorname{div}\, \mathbf{F}\,dV</math><br />
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====Stokes' teorem====<br />
<math>\oint_C \mathbf{F} \cdot \mathbf{T}\,ds = \int \! \! \! \! \int_S \operatorname{curl}\,\mathbf{F} \cdot \mathbf{n}\,d\sigma</math></div>
Kristsof