Difference between revisions of "Relationship between delta derivative and nabla derivative"

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(Created page with "<div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> <strong>Theorem:</strong> Let $\mathbb{...")
 
 
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==Theorem==
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<strong>[[Relationship between delta derivative and nabla derivative|Theorem]]:</strong> Let $\mathbb{T}$ be a [[time scale]] and let $f \colon \mathbb{T} \rightarrow \mathbb{R}$. If $f$ is [[nabla derivative|$\nabla$-differentiable]] on $\mathbb{T}_{\kappa}$ and $g^{\nabla}$ is [[ld continuous]] on $\mathbb{T}_{\kappa}$, then $f$ is [[delta derivative|$\Delta$-differentiable]] on $\mathbb{T}^{\kappa}$ and  
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Let $\mathbb{T}$ be a [[time scale]] and let $f \colon \mathbb{T} \rightarrow \mathbb{R}$. If $f$ is [[nabla derivative|$\nabla$-differentiable]] on $\mathbb{T}_{\kappa}$ and $g^{\nabla}$ is [[ld continuous]] on $\mathbb{T}_{\kappa}$, then $f$ is [[delta derivative|$\Delta$-differentiable]] on $\mathbb{T}^{\kappa}$ and  
 
$$g^{\Delta}(t) = g^{\nabla}(\sigma(t)).$$
 
$$g^{\Delta}(t) = g^{\nabla}(\sigma(t)).$$
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<strong>Proof:</strong> █
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==Proof==
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==References==
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[[Category:Theorem]]
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[[Category:Unproven]]

Latest revision as of 00:31, 23 August 2016

Theorem

Let $\mathbb{T}$ be a time scale and let $f \colon \mathbb{T} \rightarrow \mathbb{R}$. If $f$ is $\nabla$-differentiable on $\mathbb{T}_{\kappa}$ and $g^{\nabla}$ is ld continuous on $\mathbb{T}_{\kappa}$, then $f$ is $\Delta$-differentiable on $\mathbb{T}^{\kappa}$ and $$g^{\Delta}(t) = g^{\nabla}(\sigma(t)).$$

Proof

References

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