## Definitions

**Moderator:** Æron

<span style='font-size:14pt;line-height:100%'><b>FIELD</b></span><br><br>A <i>field</i> is a set with two binary operations (S,+,*) satisfying the following:<br><br>i) SxS->S:=a+b for all a,b in S. (S is closed under addition)<br>ii) a+b=b+a for all a,b in S. (S is commutative under addition)<br>iii) (a+<!--emo&B)--><img src='http://definecynical.mancubus.net/forum/html/emoticons/cool.gif' border='0' style='vertical-align:middle' alt='cool.gif' /><!--endemo-->+c=a+(b+c) for all a,b,c in S. (S is associative under addition)<br>iv) There exists an additive identity 0 such that a+0=a for all a in S.<br>v) For every element a in S there exists an additive invese -a such that a+(-a)=0.<br>S*:=S\{0} (S* is the set S without the additive identity)<br>vi) S*xS*->S*:=a*b for all a,b in S*. (S* is closed under multiplication)<br>vii) a*b=b*a for all a,b in S*. (S* is commutative under multiplication)<br>viii) (a*<!--emo&B)--><img src='http://definecynical.mancubus.net/forum/html/emoticons/cool.gif' border='0' style='vertical-align:middle' alt='cool.gif' /><!--endemo-->*c=a*(b*c) for all a,b,c in S*. (S* is associative under multiplycation)<br>ix) There exists a multiplicative identity 1 such that a*1=a for all a in S*.<br>x) For every element a in S* there exists an a multiplicative inverse a^-1 such that a*(a^-1)=1.<br>xi) a*(b+c)=a*b+a*c for all a,b,c in S. (S has distributivity)

Llewellyn for President 2008 <br><br><img><br><img>

- Salad Man
**Posts:**415**Joined:**Mon Oct 13, 2003 7:17 pm**Location:**Somewhere you just wish you were.-
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No, I don't think you can on this type of board... a mod should be able to lock or delete them for you.<br><br>EDIT: Yup, you can't delete, even if nobody else has replied.

<center><span style='font-family:impact'><b><span style='font-size:26pt;line-height:100%'>afk brb bbq lol</b></span><br><br><!--QuoteBegin-Pokchu+--></div><table border='0' align='center' width='95%' cellpadding='3' cellspacing='1'><tr><td><b>QUOTE</b> (Pokchu)</td></tr><tr><td id='QUOTE'><!--QuoteEBegin-->Tell her she drew the Shiznit. =O<!--QuoteEnd--></td></tr></table><div class='signature'><!--QuoteEEnd--><br><br><span style='font-size:19pt;line-height:100%'><a href='http://www.livejournal.com/users/saladman/' target='_blank'>(whisper whisper whisper)</a></span></span></center>

<span style='font-size:14pt;line-height:100%'><b>VECTOR SPACE</b></span><br><br>A <i>vector space</i> over a field F is a set V that satisfied the folowing properties:<br><br>i) There exists vector addition VxV->V such that:<br> i) a+b=b+a for all a,b in V<br> ii) a+(b+c)=(a+b )+c for all a,b,c in V<br> iii) There exists a vector 0 in V such that a+0=a for all a in V.<br> iv) For all a in V there exists a vector -a such that a+(-a)=0.<br>ii) There exists scalar multiplication FxV->V such that:<br> i) (a*b )*v=a*(b*v) for all a,b in F and v in V. (scalar multiplication is associative)<br> ii) 1*v=v for all v in V. (not 1 is in the field NOT the vector space)<br> iii) (a+b )*v=a*v+b*v for all a,b in F and v in V. (this one and the next one are distributive laws)<br> iv) a*(v+w)=a*v+a*w for all a in F and v,w in V.

Llewellyn for President 2008 <br><br><img><br><img>

<span style='font-size:14pt;line-height:100%'><b>LINEARLY INDEPENDENT</b></span><br><br>A sub set S:={v_1,v_2,...,v_n} of a vector space V over Fis <i>linearly independent</i> iff for all v_i, v_i cannot be written as a linear combination of the others (i.e. v_i does not equal sum(a_j*v_j) for all j not equal to i and any a_j in F).<br>

Llewellyn for President 2008 <br><br><img><br><img>

<span style='font-size:14pt;line-height:100%'><b>SPANNING SET</b></span><br><br>A set S:={v_1,v_2,...,v_n} is a <i>spanning set</i> of a vector space V over a field F iff all v in V can be written as a linear combination of v_i (i.e. there exists a_i's in F such that v=sum(a_i*v_i) over all i=<n)

Llewellyn for President 2008 <br><br><img><br><img>

<span style='font-size:14pt;line-height:100%'><b>CURVATURE</b></span><br><br>Let r:R^1->R^m be differentiable at every point.<br>r(t)<br>T(t):=(dr/dt)/(norm(dr/dt)<br>r(s):=r(t) a distance s along the curve away from a reference point in a reference direction.<br><br>The <i>curvature</i> or r is kapa:=norm(dT/ds)<br><br>There is an analog to functions from R^n into R^m but I'm not sure what it is.

Llewellyn for President 2008 <br><br><img><br><img>

<b><span style='font-size:14pt;line-height:100%'>ISOTROPIC</span></b><br><br>Something is <i>isotropic at a point</i> if every property changes the same is any direction from the point.<br><br>Something is <i>isotropic</i> if it's isotropic at every point.<br><br>This definition is not entirely rigorous and I apologize.

Llewellyn for President 2008 <br><br><img><br><img>

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