Difference between revisions of "Diamond integral"

From timescalewiki
Jump to: navigation, search
Line 1: Line 1:
 +
=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:

References

The Diamond Integral on Time Scales