Class Notes 1-5-2010 - Revision history

Rgratias at 19:37, 20 January 2010

2010-01-20T19:37:09Z

← Older revision		Revision as of 12:37, 20 January 2010
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	==Reviewed By==		==Reviewed By==
			*[[Gratias, Ryan\|Ryan Gratias]]

Brian.Roath at 16:27, 20 January 2010

2010-01-20T16:27:45Z

← Older revision		Revision as of 09:27, 20 January 2010
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			The individual component representation of vector <math> \vec{v} </math> in the x-direction.
	:<math>v_\mathrm{x} = \vec{v} \cdot \mathbf{\hat{i}}</math>		:<math>v_\mathrm{x} = \vec{v} \cdot \mathbf{\hat{i}}</math>
			The two-dimensional example of a vector in its components with the vector designations.
	:<math> \vec{v} = v_\mathrm{x} \mathbf{\hat{i}} + v_\mathrm{y} \mathbf{\hat{j}} </math>		:<math> \vec{v} = v_\mathrm{x} \mathbf{\hat{i}} + v_\mathrm{y} \mathbf{\hat{j}} </math>
			This is the summation used to represent the vector <math> \vec{v} </math> as having as many dimensions as needed to express its full value:
	:<math> \vec{v} = \sum_{i} v_\mathrm{i} \mathbf{\hat{a}}_\mathrm{i} </math>		:<math> \vec{v} = \sum_{i} v_\mathrm{i} \mathbf{\hat{a}}_\mathrm{i} </math>
	:<math> \langle v_x, v_y\rangle</math>		:<math> \langle v_x, v_y\rangle</math>
			This is the equation for finding the distance between the two vectors <math> \vec{u} </math> and <math> \vec{v} </math> who are separated by angle <math> \theta </math>.
	:<math> \vec{u} \cdot \vec{v} = \|\vec{u}\| \|\vec{v}\| \cos\theta </math>		:<math> \vec{u} \cdot \vec{v} = \|\vec{u}\| \|\vec{v}\| \cos\theta </math>
	:<math> \vec{v} \cdot \mathbf{\hat{i}} = v_\mathrm{x} (\mathbf{\hat{i}} \cdot \mathbf{\hat{i}}) + v_\mathrm{y} \mathbf{\hat{j}} \cdot \mathbf{\hat{i}} </math>		:<math> \vec{v} \cdot \mathbf{\hat{i}} = v_\mathrm{x} (\mathbf{\hat{i}} \cdot \mathbf{\hat{i}}) + v_\mathrm{y} \mathbf{\hat{j}} \cdot \mathbf{\hat{i}} </math>

Brian.Roath: /* Subjects Covered */

2010-01-17T23:23:57Z

Subjects Covered

← Older revision		Revision as of 16:23, 17 January 2010
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	:<math> \vec{v} \cdot \mathbf{\hat{i}} = v_\mathrm{x} </math>		:<math> \vec{v} \cdot \mathbf{\hat{i}} = v_\mathrm{x} </math>
	:<math> \vec{v} \cdot \mathbf{\hat{a}}_\mathrm{m} = \sum_{i} v_\mathrm{i} \mathbf{\hat{a}}_\mathrm{i} \cdot \mathbf{\hat{a}}_\mathrm{m} = v_\mathrm{m} </math>		:<math> \vec{v} \cdot \mathbf{\hat{a}}_\mathrm{m} = \sum_{i} v_\mathrm{i} \mathbf{\hat{a}}_\mathrm{i} \cdot \mathbf{\hat{a}}_\mathrm{m} = v_\mathrm{m} </math>
	:<math> \delta_\mathrm{i,m} \equiv \begin{cases} 1 & \mbox{if } i = m, \\ 0 & \mbox{else} \end{cases}</math>		:<math> \delta_\mathrm{i,m} \equiv \begin{cases} 1, & \mbox{if } i = m \\ 0, & \mbox{else} \end{cases}</math>

	==Example==		==Example==

Colby.fullerton at 23:23, 17 January 2010

2010-01-17T23:23:51Z

← Older revision		Revision as of 16:23, 17 January 2010
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	Brian Roath		Brian Roath

			==Read By==

			==Reviewed By==

Brian.Roath: /* Subjects Covered */

2010-01-17T23:23:02Z

Subjects Covered

← Older revision		Revision as of 16:23, 17 January 2010
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	:<math> \vec{v} = \sum_{i} v_\mathrm{i} \mathbf{\hat{a}}_\mathrm{i} </math>		:<math> \vec{v} = \sum_{i} v_\mathrm{i} \mathbf{\hat{a}}_\mathrm{i} </math>
	:<math> \langle v_x, v_y\rangle</math>		:<math> \langle v_x, v_y\rangle</math>
	:<math> \~~mathbf{\hat~~{u}} \cdot \~~mathbf{\hat~~{v}} = \|\~~mathbf{\hat~~{u}}\| \|\~~mathbf{\hat~~{v}}\| \cos\theta </math>		:<math> \vec{u} \cdot \vec{v} = \|\vec{u}\| \|\vec{v}\| \cos\theta </math>
	:<math> \vec{v} \cdot \mathbf{\hat{i}} = v_\mathrm{x} (\mathbf{\hat{i}} \cdot \mathbf{\hat{i}}) + v_\mathrm{y} \mathbf{\hat{j}} \cdot \mathbf{\hat{i}} </math>		:<math> \vec{v} \cdot \mathbf{\hat{i}} = v_\mathrm{x} (\mathbf{\hat{i}} \cdot \mathbf{\hat{i}}) + v_\mathrm{y} \mathbf{\hat{j}} \cdot \mathbf{\hat{i}} </math>
	:<math> \vec{v} \cdot \mathbf{\hat{i}} = v_\mathrm{x} </math>		:<math> \vec{v} \cdot \mathbf{\hat{i}} = v_\mathrm{x} </math>

Colby.fullerton: /* External Links */

2010-01-17T23:22:02Z

External Links

← Older revision		Revision as of 16:22, 17 January 2010
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	==External Links==		==External Links==
	*[[http://people.wallawalla.edu/~Rob.Frohne/ClassNotes/ENGR351/2010w/Keystone/index.php Class Notes].].		*[http://people.wallawalla.edu/~Rob.Frohne/ClassNotes/ENGR351/2010w/Keystone/index.php Class Notes].

	==Authors==		==Authors==

Brian.Roath: /* Subjects Covered */

2010-01-17T23:21:43Z

Subjects Covered

← Older revision		Revision as of 16:21, 17 January 2010
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	:<math>v_\mathrm{x} = \vec{v} \cdot \mathbf{\hat{i}}</math>		:<math>v_\mathrm{x} = \vec{v} \cdot \mathbf{\hat{i}}</math>
	:<math> \~~mathbf{\hat~~{v}} = v_\mathrm{x} \mathbf{\hat{i}} + v_\mathrm{y} \mathbf{\hat{j}} </math>		:<math> \vec{v} = v_\mathrm{x} \mathbf{\hat{i}} + v_\mathrm{y} \mathbf{\hat{j}} </math>
	:<math> \~~mathbf{\hat~~{v}} = \sum_{i} v_\mathrm{i} \mathbf{\hat{a}}_\mathrm{i} </math>		:<math> \vec{v} = \sum_{i} v_\mathrm{i} \mathbf{\hat{a}}_\mathrm{i} </math>
	:<math> \langle v_x, v_y\rangle</math>		:<math> \langle v_x, v_y\rangle</math>
	:<math> \mathbf{\hat{u}} \cdot \mathbf{\hat{v}} = \|\mathbf{\hat{u}}\| \|\mathbf{\hat{v}}\| \cos\theta </math>		:<math> \mathbf{\hat{u}} \cdot \mathbf{\hat{v}} = \|\mathbf{\hat{u}}\| \|\mathbf{\hat{v}}\| \cos\theta </math>
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	:<math> \vec{v} \cdot \mathbf{\hat{a}}_\mathrm{m} = \sum_{i} v_\mathrm{i} \mathbf{\hat{a}}_\mathrm{i} \cdot \mathbf{\hat{a}}_\mathrm{m} = v_\mathrm{m} </math>		:<math> \vec{v} \cdot \mathbf{\hat{a}}_\mathrm{m} = \sum_{i} v_\mathrm{i} \mathbf{\hat{a}}_\mathrm{i} \cdot \mathbf{\hat{a}}_\mathrm{m} = v_\mathrm{m} </math>
	:<math> \delta_\mathrm{i,m} \equiv \begin{cases} 1 & \mbox{if } i = m, \\ 0 & \mbox{else} \end{cases}</math>		:<math> \delta_\mathrm{i,m} \equiv \begin{cases} 1 & \mbox{if } i = m, \\ 0 & \mbox{else} \end{cases}</math>

	==Example==		==Example==
	[[Image:January_5_graph_2.jpg\|200px\|thumb\|left\|Function waves with varying periods based on the function x(t) = x(t+T)]]		[[Image:January_5_graph_2.jpg\|200px\|thumb\|left\|Function waves with varying periods based on the function x(t) = x(t+T)]]

Brian.Roath: /* Example */

2010-01-17T23:19:52Z

Example

← Older revision		Revision as of 16:19, 17 January 2010
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	==Example==		==Example==
	[[Image:January_5_graph_2.jpg\|200px\|thumb\|left\|Function waves with varying periods based on the function x(t) = x(t+T)]]		[[Image:January_5_graph_2.jpg\|200px\|thumb\|left\|Function waves with varying periods based on the function x(t) = x(t+T)]]
	Given function:		Given function: <math> x(t) = x(t+T) \,</math>
	:<math> x(t) = x(t+T) \,</math>
	:<math> x(t) = \sum^\infty_{n=1} \left[ b_n \sin \left( \left( \frac {2\pi n} {T} \right) t \right) \right] </math>		:<math> x(t) = \sum^\infty_{n=1} \left[ b_n \sin \left( \left( \frac {2\pi n} {T} \right) t \right) \right] </math>
	1) Use vector analogy		1) Use vector analogy

Brian.Roath: /* Example */

2010-01-17T23:19:11Z

Example

← Older revision		Revision as of 16:19, 17 January 2010
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	==Example==		==Example==
	[[Image:January_5_graph_2.jpg\|200px\|thumb\|left\|Function waves with varying periods based on the function x(t) = x(t+T)]]		[[Image:January_5_graph_2.jpg\|200px\|thumb\|left\|Function waves with varying periods based on the function x(t) = x(t+T)]]
	Given function: :<math> x(t) = x(t+T) </math>		Given function:
			:<math> x(t) = x(t+T) \,</math>
	:<math> x(t) = \sum^\infty_{n=1} \left[ b_n \sin \left( \left( \frac {2\pi n} {T} \right) t \right) \right] </math>		:<math> x(t) = \sum^\infty_{n=1} \left[ b_n \sin \left( \left( \frac {2\pi n} {T} \right) t \right) \right] </math>
	1) Use vector analogy		1) Use vector analogy

Brian.Roath: /* Example */

2010-01-17T23:18:00Z

Example

← Older revision		Revision as of 16:18, 17 January 2010
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	==Example==		==Example==
	[[Image:January_5_graph_2.jpg\|200px\|thumb\|left\|Function waves with varying periods based on the function x(t) = x(t+T)]]		[[Image:January_5_graph_2.jpg\|200px\|thumb\|left\|Function waves with varying periods based on the function x(t) = x(t+T)]]
			Given function: :<math> x(t) = x(t+T) </math>
	:<math> x(t) = \sum^\infty_{n=1} \left[ b_n \sin \left( \left( \frac {2\pi n} {T} \right) t \right) \right] </math>		:<math> x(t) = \sum^\infty_{n=1} \left[ b_n \sin \left( \left( \frac {2\pi n} {T} \right) t \right) \right] </math>
	1) Use vector analogy		1) Use vector analogy
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	:<math> \int_{\frac {-T} {2}}^{\frac {T} {2}} x(t) \sin \left( \frac {2\pi mt} {T} \right) \,dt = v_m</math>		:<math> \int_{\frac {-T} {2}}^{\frac {T} {2}} x(t) \sin \left( \frac {2\pi mt} {T} \right) \,dt = v_m</math>


	==External Links==		==External Links==