Difference between revisions of "Diamond integral"
From timescalewiki
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| + | =Properties= | ||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <strong>Theorem:</strong> The following formula holds: | ||
| + | $$\int_a^a f(t) \Diamond t=0.$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <strong>Theorem:</strong> The following formula holds: | ||
| + | $$\int_a^b f(t) \Diamond t= \int_a^c f(t) \Diamond t + \int_c^b f(t) \Diamond t.$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <strong>Theorem:</strong> The following formula holds: | ||
| + | $$\int_a^b f(t) \Diamond t= -\int_b^a f(t) \Diamond t.$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <strong>Theorem (Sum Rule):</strong> The following formula holds: | ||
| + | $$\int_a^b (f+g)(t) \Diamond t= \int_a^b f(t) \Diamond t + \int_a^b g(t) \Diamond t.$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <strong>Theorem (Constant Multiple):</strong> The following formula holds: | ||
| + | $$\int_a^b \alpha f(t) \Diamond t= \alpha \int_a^b f(t) \Diamond t.$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
=References= | =References= | ||
[http://arxiv.org/pdf/1306.0988.pdf The Diamond Integral on Time Scales] | [http://arxiv.org/pdf/1306.0988.pdf The Diamond Integral on Time Scales] | ||
Revision as of 21:44, 20 October 2014
Properties
Theorem: The following formula holds: $$\int_a^a f(t) \Diamond t=0.$$
Proof: █
Theorem: The following formula holds: $$\int_a^b f(t) \Diamond t= \int_a^c f(t) \Diamond t + \int_c^b f(t) \Diamond t.$$
Proof: █
Theorem: The following formula holds: $$\int_a^b f(t) \Diamond t= -\int_b^a f(t) \Diamond t.$$
Proof: █
Theorem (Sum Rule): The following formula holds: $$\int_a^b (f+g)(t) \Diamond t= \int_a^b f(t) \Diamond t + \int_a^b g(t) \Diamond t.$$
Proof: █
Theorem (Constant Multiple): The following formula holds: $$\int_a^b \alpha f(t) \Diamond t= \alpha \int_a^b f(t) \Diamond t.$$
Proof: █
