1 ) For $ mathbf a a x mathbf e x a y mathbf e y a z mathbf e z , mathbf b b x mathbf e x b y mathbf e y b z mathbf e z$ find $ mathbf c mathbf a times mathbf b $ by three methods term by term, resolving the determinant $ mathbf c left begin array lll mathbf e x mathbf e y mathbf e z a x a y a z b x b y b z end array right $ , and using the permutation symbol $ mathbf c mathbf a times mathbf b sum i 1 3 sum m 1 3 sum n 1 3 varepsilon i m n a m b n mathbf e i begin aligned varepsilon 1 2 3 varepsilon 2 3 1 varepsilon 3 1 2 1 varepsilon 3 2 1 varepsilon 2 1 3 varepsilon 1 3 2 1 end aligned $ , all others $ 0 $ 2 ) For a tensor represented in Cartesian matrix form as $ mathbf T left begin array ccc T x x T x y 0 T y x T y y 0 0 0 T z z end array right $ , Find the double dot product $ mathbf T mathbf T $ 3 ) For $ mathbf v v x mathbf e x v y mathbf e y v z mathbf e z$ , and $ boldsymbol tau tau x x ( x , y , z ) mathbf e x mathbf e x tau x y ( x , y , z ) mathbf e x mathbf e y ldots tau z z ( x , y , z ) mathbf e z mathbf e z$ , find $ mathbf v cdot abla boldsymbol tau $ 4 ) 2 D Cartesian coordinate system $ left mathbf e x prime , mathbf e y prime right $ is oriented at angle $ theta 3 5 circ $ , with repect to the horizontal vertical system $ left mathbf e x , mathbf e y right $ We have position vectors $ mathbf r 3 mathbf e x 4 mathbf e y$ and $ mathbf r prime 4 7 5 2 mathbf e x prime 1 5 5 6 mathbf e y prime $ Show that the two vectors are equal, $ mathbf r prime mathbf r $ i e , that they have the same magnitude and direction There is a bit of roundoff error

The Answer is in the image, click to view ...

Question: 1 ) For $ mathbf { a } = a _ x mathbf { e } _ x + a _ y

1)

For $

\

mathbf

{

} =

_

\

mathbf

{

}_

+

_

\

mathbf

{

}_

+

_

\

mathbf

{

}_

, \

mathbf

{

} =

_

\

mathbf

{

}_

+

_

\

mathbf

{

}_

+

_

\

mathbf

{

}_

z$ find $

\

mathbf

{

} = \

mathbf

{

} \

times

\

mathbf

{

}

$ by three methods: term

-

-

term, resolving the determinant $

\

mathbf

{

} = \

left

| \

begin

{

array

} {

lll

} \

mathbf

{

}_

x &

\

mathbf

{

}_

y &

\

mathbf

{

}_

\ \

_

x & a

_

y & a

_

\ \

_

x & b

_

y & b

_

\

end

{

array

} \

right

|

,

and using the permutation symbol: $

\

mathbf

{

} = \

mathbf

{

} \

times

\

mathbf

{

} = \

sum

_{

= 1}^3 \

sum

_{

= 1}^3 \

sum

_{

= 1}^3 \

varepsilon

_{

i m n

}

_

m b

_

\

mathbf

{

}_

i ;

\

begin

{

aligned

}

\

varepsilon

_{123} = \

varepsilon

_{231} = \

varepsilon

_{312} = + 1 \ \

\

varepsilon

_{321} = \

varepsilon

_{213} = \

varepsilon

_{132} = - 1 \

end

{

aligned

}

,

all others $

= 0

2)

For a tensor represented in Cartesian matrix form as $

\

mathbf

{

} = \

left

| \

begin

{

array

} {

ccc

}

_{

x x

}

& T

_{

x y

}

0 \ \

_{

y x

}

& T

_{

y y

}

0 \ \ 0

0

& T

_{

z z

} \

end

{

array

} \

right

|

,

Find the double dot product $

\

mathbf

{

}

\

mathbf

{

}

.

3)

For $

\

mathbf

{

} =

_

\

mathbf

{

}_

+

_

\

mathbf

{

}_

+

_

\

mathbf

{

}_

,

and $

\

boldsymbol

{\

tau

} = \

tau

_{

x x

} (

,

,

) \

mathbf

{

}_

\

mathbf

{

}_

+ \

tau

_{

x y

} (

,

,

) \

mathbf

{

}_

\

mathbf

{

}_

\

ldots

+ \

tau

_{

z z

} (

,

,

) \

mathbf

{

}_

\

mathbf

{

}_

,

find $

\

mathbf

{

} \

cdot

abla

\

boldsymbol

{\

tau

}

.

4) 2 -

D Cartesian coordinate system $

\

left

[\

mathbf

{

}_

^{\

prime

}, \

mathbf

{

}_

^{\

prime

} \

right

]

$ is oriented at angle $

\

theta

= 35^{\

circ

}

,

with repect to the horizontal

/

vertical system $

\

left

[\

mathbf

{

}_

, \

mathbf

{

}_

\

right

]

.

We have position vectors $

\

mathbf

{

} = 3 \

mathbf

{

}_

+ 4 \

mathbf

{

}_

y$ and $

\

mathbf

{

}^{\

prime

} = 4.752 \

mathbf

{

}_

^{\

prime

} + 1.556 \

mathbf

{

}_

^{\

prime

}

.

Show that the two vectors are equal, $

\

mathbf

{

}^{\

prime

} = \

mathbf

{

}

$; i

.

.,

that they have the same magnitude and direction. There is a bit of roundoff error.

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