http://timescalewiki.org/index.php?title=Relationship_between_nabla_derivative_and_delta_derivative&feed=atom&action=history
Relationship between nabla derivative and delta derivative - Revision history
2024-03-28T21:09:31Z
Revision history for this page on the wiki
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http://timescalewiki.org/index.php?title=Relationship_between_nabla_derivative_and_delta_derivative&diff=1308&oldid=prev
Tom at 06:02, 10 June 2016
2016-06-10T06:02:45Z
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<td colspan='2' style="background-color: white; color:black; text-align: center;">← Older revision</td>
<td colspan='2' style="background-color: white; color:black; text-align: center;">Revision as of 06:02, 10 June 2016</td>
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<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline"><div class</del>=<del class="diffchange diffchange-inline">"toccolours mw-collapsible mw-collapsed" style</del>=<del class="diffchange diffchange-inline">"width:800px"></del></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>==<ins class="diffchange diffchange-inline">Theorem==</ins></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline"><strong>[[Relationship between nabla derivative and delta derivative|Theorem]]:</strong> </del>Let $\mathbb{T}$ be a [[time scale]] and let $f \colon \mathbb{T} \rightarrow \mathbb{R}$. If $f$ is [[Delta derivative|$\Delta$-differentiable]] and $f^{\Delta}$ is [[rd continous]] on $\mathbb{T}^{\kappa}$, then $f$ is [[nabla derivative|$\nabla$-differentiable]] on $\mathbb{T}_{\kappa}$ and  </div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Let $\mathbb{T}$ be a [[time scale]] and let $f \colon \mathbb{T} \rightarrow \mathbb{R}$. If $f$ is [[Delta derivative|$\Delta$-differentiable]] and $f^{\Delta}$ is [[rd continous]] on $\mathbb{T}^{\kappa}$, then $f$ is [[nabla derivative|$\nabla$-differentiable]] on $\mathbb{T}_{\kappa}$ and  </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$$f^{\nabla}(t) = f^{\Delta}(\rho(t)).$$</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$$f^{\nabla}(t) = f^{\Delta}(\rho(t)).$$</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline"><div class</del>=<del class="diffchange diffchange-inline">"mw-collapsible-content"></del></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline"><strong>Proof</del>:<del class="diffchange diffchange-inline"></strong> █ </del></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">==Proof=</ins>=</div></td></tr>
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<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline"></div></del></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">==References==</ins></div></td></tr>
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<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">[[Category</ins>:<ins class="diffchange diffchange-inline">Theorem]]</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">[[Category:Unproven]]</ins></div></td></tr>
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Tom
http://timescalewiki.org/index.php?title=Relationship_between_nabla_derivative_and_delta_derivative&diff=721&oldid=prev
Tom at 09:05, 12 April 2015
2015-04-12T09:05:37Z
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<td colspan='2' style="background-color: white; color:black; text-align: center;">Revision as of 09:05, 12 April 2015</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div><div class="toccolours mw-collapsible mw-collapsed" style="width:800px"></div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div><div class="toccolours mw-collapsible mw-collapsed" style="width:800px"></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><strong>Theorem:</strong> Let $\mathbb{T}$ be a [[time scale]] and let $f \colon \mathbb{T} \rightarrow \mathbb{R}$. If $f$ is [[Delta derivative|$\Delta$-differentiable]] and $f^{\Delta}$ is [[rd continous]] on $\mathbb{T}^{\kappa}$, then $f$ is [[nabla derivative|$\nabla$-differentiable]] on $\mathbb{T}_{\kappa}$ and  </div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><strong><ins class="diffchange diffchange-inline">[[Relationship between nabla derivative and delta derivative|</ins>Theorem<ins class="diffchange diffchange-inline">]]</ins>:</strong> Let $\mathbb{T}$ be a [[time scale]] and let $f \colon \mathbb{T} \rightarrow \mathbb{R}$. If $f$ is [[Delta derivative|$\Delta$-differentiable]] and $f^{\Delta}$ is [[rd continous]] on $\mathbb{T}^{\kappa}$, then $f$ is [[nabla derivative|$\nabla$-differentiable]] on $\mathbb{T}_{\kappa}$ and  </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$$f^{\nabla}(t) = f^{\Delta}(\rho(t)).$$</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$$f^{\nabla}(t) = f^{\Delta}(\rho(t)).$$</div></td></tr>
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Tom
http://timescalewiki.org/index.php?title=Relationship_between_nabla_derivative_and_delta_derivative&diff=720&oldid=prev
Tom: Created page with "<div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> <strong>Theorem:</strong> Let $\mathbb{T}$ be a time scale and let $f \colon \mathbb{T} \rightarro..."
2015-04-12T09:05:11Z
<p>Created page with "<div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> <strong>Theorem:</strong> Let $\mathbb{T}$ be a <a href="/index.php/Time_scale" title="Time scale">time scale</a> and let $f \colon \mathbb{T} \rightarro..."</p>
<p><b>New page</b></p><div><div class="toccolours mw-collapsible mw-collapsed" style="width:800px"><br />
<strong>Theorem:</strong> Let $\mathbb{T}$ be a [[time scale]] and let $f \colon \mathbb{T} \rightarrow \mathbb{R}$. If $f$ is [[Delta derivative|$\Delta$-differentiable]] and $f^{\Delta}$ is [[rd continous]] on $\mathbb{T}^{\kappa}$, then $f$ is [[nabla derivative|$\nabla$-differentiable]] on $\mathbb{T}_{\kappa}$ and <br />
$$f^{\nabla}(t) = f^{\Delta}(\rho(t)).$$<br />
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<strong>Proof:</strong> █ <br />
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Tom