User:Eml5526.s11.team5.srv/hw2

Home Work 2.5
'''Show that $$\overset{b _{i}}{\rightarrow}\centerdot \overset{P}{\rightarrow}\overset{(V)}{\rightarrow}=0$$  where i = 1, 2,. . ., n is the same as $$\overset{W}{\rightarrow}\centerdot \overset{P}{\rightarrow}\overset{(V)}{\rightarrow}=0$$   such that     $$\overset{W}{\rightarrow}=\sum\limits_{i=1}^{n}$$'''

Solution:
For i = 1, 2,. . ., n


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$$\overset{b _{i}}{\rightarrow}\centerdot\overset{P}{\rightarrow}\overset{(V)}{\rightarrow}=0$$ $$
 * style="width:95%" |
 * style="width:95%" |
 *  $$ \displaystyle (Eq. 2.5.1)
 * }

Multiplying the arbitrary constant αi   on both sides ;


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$${{\alpha }_{i}}(\overset{b _{i}}{\rightarrow}\centerdot\overset{P}{\rightarrow}\overset{(V)}{\rightarrow})=0$$ $$ From the properties of Dot product of a vector, $$c\left( \vec{u}\cdot \vec{v} \right)=c(\vec{u})\cdot \vec{v}=\vec{u}\cdot c(\vec{v})$$, where 'c' is an arbitrary constant.
 * style="width:95%" |
 * style="width:95%" |
 *  $$ \displaystyle (Eq. 2.5.2)
 * }

Therefore, equation (2.5.2) becomes,


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$${{\alpha }_{i}}\overset{b _{i}}{\rightarrow}\centerdot \overset{P}{\rightarrow}\overset{(V)}{\rightarrow}=0$$
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 * style="width:95%" |
 *  $$ \displaystyle\!$$
 * }

For i = 1;


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$${{\alpha }_{1}}\overset{b _{1}}{\rightarrow}\centerdot \overset{P}{\rightarrow}\overset{(V)}{\rightarrow}=0$$ $$
 * style="width:95%" |
 * style="width:95%" |
 *  $$ \displaystyle (Eq.1)
 * }

For i = 2;


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 * style="width:95%" |
 * style="width:95%" |

$${{\alpha }_{2}}\overset{b _{2}}{\rightarrow}\centerdot \overset{P}{\rightarrow}\overset{(V)}{\rightarrow}=0$$ $$
 *  $$ \displaystyle (Eq. 2)
 * }

Similarly, For all i = n;


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$${{\alpha }_{n}}\overset{b _{n}}{\rightarrow}\centerdot \overset{P}{\rightarrow}\overset{(V)}{\rightarrow}=0$$ $$
 * style="width:95%" |
 * style="width:95%" |
 *  $$ \displaystyle (Eq. n)
 * }

Adding the equations Eq. 1, Eq. 2 .... upto Eq. n, we get,


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$${{\alpha }_{1}}\overset{b _{1}}{\rightarrow}\centerdot \overset{P}{\rightarrow}\overset{(V)}{\rightarrow}$$ + $${{\alpha }_{2}}\overset{b _{2}}{\rightarrow}\centerdot \overset{P}{\rightarrow}\overset{(V)}{\rightarrow}$$ +. . . . . . + $${{\alpha }_{n}}\overset{b _{n}}{\rightarrow}\centerdot \overset{P}{\rightarrow}\overset{(V)}{\rightarrow}=0$$ $$
 * style="width:95%" |
 * style="width:95%" |
 *  $$ \displaystyle (Eq. 2.5.3)
 * }

Using the Distributive property of dot Product of a vector, Eqn. 2.5.3 can be written as,


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$$({{\alpha }_{1}}\overset{b _{1}}{\rightarrow} + {{\alpha }_{2}}\overset{b _{2}}{\rightarrow} + . . . . . . + {{\alpha }_{n}}\overset{b _{n}}{\rightarrow})\centerdot \overset{P}{\rightarrow}\overset{(V)}{\rightarrow}=0$$ $$
 * style="width:95%" |
 * style="width:95%" |
 *  $$ \displaystyle (Eq. 2.5.4)
 * }

The above equation can be written as,

<span id="(1)">
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$$\sum\limits_{i=1}^{n}\centerdot \overset{P}{\rightarrow}\overset{(V)}{\rightarrow} = 0$$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. 2.5.5)
 * }

Since $$\sum\limits_{i=1}^{n} = \overset{W}{\rightarrow},$$ Eqn(2.5.5) becomes

<span id="(1)">
 * {| style="width:100%" border="0"

$$\overset{W}{\rightarrow}\centerdot \overset{P}{\rightarrow}\overset{(V)}{\rightarrow}=0$$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. 2.5.6)
 * }

<span id="(1)">
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 * style="width:95%" |
 * style="width:95%" |

$$\overset{W}{\rightarrow}\centerdot \overset{P}{\rightarrow}\overset{(V)}{\rightarrow}=0$$ $$
 * <p style="text-align:right"> $$ \displaystyle\!
 * }

Vignesh Solai Rameshbabu--Eml5526.s11.team5.srv 21:09, 30 January 2011 (UTC)

Home Work 2.8
Show that $$\int_{\Omega}w^{h}(x)P(u^{h}(x))dx = 0 \;\; \forall\; w^{h}(x)$$ where $$\;\;w^{h}(x) = \sum_{i=1}^{n}c_{i}b_{i}(x)$$is the same as  $$\int_{\Omega}b_{i}(x)P(u^{h}(x))dx = 0 \;\; for\;\; i = 1,2,...,n$$   where i = 1 to n

$$\int_{\Omega}w^{h}(x)P(u^{h}(x))dx = 0  \;\Leftrightarrow\;  \int_{\Omega}b_{i}(x)P(u^{h}(x))dx = 0 $$

Solution
'Let '

<span id="(1)">
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$$\int_{\Omega}w^{h}(x)P(u^{h}(x))dx = 0 \quad $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. 2.8.1)
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$\int_{\Omega}b_{i}(x)P(u^{h}(x))dx = 0 \quad $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. 2.8.2)
 * }

1)To proove<span id="(1)"> 
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$$\int_{\Omega}w^{h}(x)P(u^{h}(x))dx = 0  \;\Rightarrow\;  \int_{\Omega}b_{i}(x)P(u^{h}(x))dx = 0 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle\!
 * }

Choice 1:
 * $$\;\; \big[c_{1}, c_{2}, c_{3},.........., c_{n}\big] = \big[ 1,0,0,...,0 \big] $$

<span id="(1)">
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$$\therefore\;\; w^{h}(x) = b_{1}(x) $$ $$
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 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle\!
 * }

Therefore Eq. 2.8.1 becomes,

<span id="(1)">
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$$\displaystyle \;\;\int_{\Omega}b_{1}(x)P(u^{h}(x))dx = 0$$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. 2.8.3)
 * }

Choice 2:
 * $$\;\; \big[c_{1}, c_{2}, c_{3},............, c_{n}\big] = \big[ 0,1,0,...,0 \big] $$

<span id="(1)">
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$$\therefore\;\; w^{h}(x) = b_{2}(x) $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle\!
 * }

Therefore Eq. 2.8.1 becomes,

<span id="(1)">
 * {| style="width:100%" border="0"

$$\displaystyle \;\;\int_{\Omega}b_{2}(x)P(u^{h}(x))dx = 0$$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. 2.8.4)
 * }

Choice n:
 * $$\;\; \big[c_{1}, c_{2}, c_{3},............, c_{n}\big] = \big[ 0,0,0,...,1 \big] $$

<span id="(1)">
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$$\therefore\;\; w^{h}(x) = b_{n}(x) $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle\!
 * }

Therefore Eq. 2.8.1 becomes,

<span id="(1)">
 * {| style="width:100%" border="0"

$$\displaystyle \;\;\int_{\Omega}b_{n}(x)P(u^{h}(x))dx = 0$$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. 2.8.5)
 * }

In General, Eq. 2.8.3, Eq. 2.8.4, Eq. 2.8.5 can be written as,


 * {| style="width:100%" border="0" align="left"

|$$\displaystyle \int_{\Omega}b_{i}(x)P(u^{h}(x))dx = 0 \;\; where\;\; i = 1,2,...,n  \;( which \;is \;the\; same \;as\; Eq. 2.8.2) $$ $$
 * <p style="text-align:right;">$$\displaystyle\!


 * }
 * }

2)To proove <span id="(1)">
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$$  \int_{\Omega}b_{i}(x)P(u^{h}(x))dx = 0     \;\Rightarrow\;  \int_{\Omega}w^{h}(x)P(u^{h}(x))dx = 0 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle\!
 * }

For i = 1, 2,. . ., n

<span id="(1)">
 * {| style="width:100%" border="0"

$$\int_{\Omega}b_{i}(x)P(u^{h}(x))dx = 0 \quad $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. 2.8.6)
 * }

Multiplying the arbitrary constant $$\;\;c_{i}\;\;$$   on both sides ;

<span id="(1)">
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$$\;\int_{\Omega}c_{i}b_{i}(x)P(u^{h}(x))dx = 0 \;\; for\;\; i = 1,2,...,n$$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. 2.8.7)
 * }

For i = 1;

<span id="(1)">
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$$\;\int_{\Omega}c_{1}b_{1}(x)P(u^{h}(x))dx = 0 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq.1)
 * }

For i = 2;

<span id="(1)">
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$$\;\int_{\Omega}c_{2}b_{2}(x)P(u^{h}(x))dx = 0 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq.2)
 * }

Similarly, For all i = n;

<span id="(1)">
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$$\;\int_{\Omega}c_{n}b_{n}(x)P(u^{h}(x))dx = 0 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq.n)
 * }

Adding the equations Eq. 1, Eq. 2 .... upto Eq. n, we get,

<span id="(1)">
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$$\int_{\Omega}c_{1}b_{1}(x)P(u^{h}(x))dx + \int_{\Omega}c_{2}b_{2}(x)P(u^{h}(x))dx + .\; .\;. + \int_{\Omega}c_{n}b_{n}(x)P(u^{h}(x))dx = 0$$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. 2.8.8)
 * }

<span id="(1)">
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$$\;\;\int_{\Omega}\left [ c_{1}b_{1}(x) + c_{2}b_{2}(x)+ .\; .\;. + c_{n}b_{n}(x)\right ]P(u^{h}(x))dx = 0$$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. 2.8.9)
 * }

The above equation can be written as,

<span id="(1)">
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$$\displaystyle \Rightarrow\;\;\int_{\Omega}\sum_{i=1}^{n}c_{i}b_{i}(x)P(u^{h}(x))dx = 0$$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. 2.8.10)
 * }

Since $$\sum\limits_{i=1}^{n} = \;w^{h}(x)$$ Eqn(2.8.10) becomes

<span id="(1)">
 * {| style="width:100%" border="0"

$$\int_{\Omega}w^{h}(x)P(u^{h}(x))dx = 0 \quad $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. 2.8.11)
 * }

<span id="(1)">
 * {| style="width:100%" border="0"


 * style="width:95%" |
 * style="width:95%" |

$$\int_{\Omega}w^{h}(x)P(u^{h}(x))dx = 0 \;\; \forall\; w^{h}(x)\;\;\; (which \;is\; the\; same\; as\; Eq. 2.8.1)$$ $$
 * <p style="text-align:right"> $$ \displaystyle\!
 * }

Vignesh Solai Rameshbabu--Eml5526.s11.team5.srv 21:09, 30 January 2011 (UTC)