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x x x y sec x x θθθθ θθθθ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θθθθ θθθθ θθθθ θθθθ θθθθ ==== ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂∂ ∂ _⇒⇒⇒⇒ ∂ ₌₌₌₌ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂   ====   ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂   ∂ − ∂ − ∂ − ∂ ₌₌₌₌ − ∂∂∂∂ 2 2

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θ θ θ θ θ θ θ θ θ θ θθθθ θθθθ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ₋₋₋₋ ∂ ∂ _{−−−− ∂ −−−−} ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂  ∂ ∂  ∂ ∂ ∂  ∂ = = + = = + = = + =  =   +   ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂∂ ∂∂ ∂∂ ∂∂ ∂∂ ∂∂ ∂∂ ∂∂ ∂ ∂ ∂  ∂ ∂  ∂ ∂ ∂  ∂

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θ θ θ

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∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂  ∂  ∂ ∂  ∂  ∂ ∂       = = − + − − = = − + − − = = − + − − =  =  −  +  −     −  ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂∂ ∂∂ ∂∂ ∂∂ ∂∂ ∂∂ ∂∂ ∂∂ ∂ ∂ ∂  ∂  ∂ ∂  ∂  ∂ ∂      

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θθθθ _{θ θ}

θ θ θ θ θ θ         ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂  ∂ ∂  ∂ ∂ ∂ − = + − −− == ++ −− − =  + −        ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂∂ ∂∂ ∂∂ ∂∂ ∂ ∂∂ ∂ ∂  ∂ ∂  _ ∂ ∂ ∂ ∂ _ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ₊₊₊₊ ∂ ₋₋₋₋ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ 2 2 2

2 2 2 2

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cos sen cos

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θ θ θ θ θ θ θ θ θ θ

θ θ θ θ θ θ θ θ θ θ

θθθθ

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∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

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∂ ∂ ∂ ∂ ∂ ∂

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∂ ∂ ∂ ∂ ∂ ∂ ∂

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cos sen sen2

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r ,

θθθθ

2 2 2

2 2

x 2 2 2 2

1 1 1 1

sen cos sen2

r r r r r r r

φ φ φ φ φ

φφ φφ φφ φφ φφ

φ φ φ φ φ

σ θ θ θ

σ θ θ θ

σ θ θ θ

σ θ θ θ

θ θ θ

θ θ θ

θ θ θ

θ θ θ

   

   

   

   

∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂

= + + − −

= + + − −

= + + − −

= +  + −  − 

∂ ∂ ∂ ∂ ∂ ∂

∂∂ ∂∂ ∂∂ ∂∂ ∂ ∂∂ ∂

'

2 2 2

2 2

y 2 2 2 2

1 1 1 1

cos sen sen2

r r r r r r r

φ φ φ φ φ

φφ φφ φφ φφ φφ

φ φ φ φ φ

σ θ θ θ

σσ θθ θθ θθ

σ θ θ θ

θ θ θ

θ θ θ

θ θ θ

θ θ θ

   

   

   

∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂

= + + − − +

= + + − − +

= + + − − +

= +  + − − + 

∂ ∂ ∂ ∂ ∂ ∂

∂∂ ∂∂ ∂∂ ∂∂ ∂ ∂∂ ∂

∂  ∂ ∂   ∂ ∂ ∂ 

;

,

.

(

.

(

6

?

.

/

B

2A

(

,

2A

7

(

7, ,

/ , .

(

.

, ( /

7.

A

.

'

, .

7

. ,

(

. (

7.

8

(((( ))))

r ,

θθθθ

r

,

θθθθ

,

rθθθθ

'

x

y

x

y

r

A

θθθθ

x

σσσσ

x

σσσσ

y

σσσσ

y

σσσσ

xy

ττττ

>













−

























−

=













θ

θ

θ

θ

σ

τ

τ

σ

θ

θ

θ

θ

σ

τ

τ

σ

θ θ

θ

cos

cos

cos

cos

sen

sen

sen

sen

r

r r

y xy

'

2

2

x

sen

θ

θ

θ

θ

cos

r

sen2

r

θ

θ

θ

θ

σ

θσ

θσ

θτ

σ

θσ

θσ

θτ

σ

θσ

θσ

θτ

σ

=

=

=

=

θσ

+

+

+

+

θσ

−

−

−

−

θτ

.

- <

=, <

=

<

=,

. ( 6

(

< :=, <

=

< ;=,

(

8

2

r 2 2

1

1

r

r

r

φ

φ

φ

φ

φ

φ

φ

φ

σσσσ

θθθθ















_∂

_∂

_∂

_∂

_∂

_∂

_∂

_∂



=

+

=

+

=

+

=









+









∂

∂

∂

∂

∂

∂

∂

∂













2

2

r

θθθθ

φφφφ

σσσσ

====

∂∂∂∂

∂∂∂∂

'

2

r 2

1

1

r

r r

θθθθ

φ

φ

φ

φ

φ

φ

φ

φ

ττττ

θ

θ

θ

θ

θ

θ

θ

θ















_∂

_∂

_∂

_∂

_∂

_∂

_∂

_∂



=

−

=

−

=

−

=









−









∂

∂ ∂

∂

∂ ∂

∂

∂ ∂

∂

∂ ∂

















9

+

F

?

2A

8

x xy

r r

r xy y

cos

sen

cos

sen

sen

cos

sen

cos

θθθθ

θ

θ

θ

θ

θ

θ

θ

θ

σ

τ

σ

σ

τ

τ

σ

τ

σ

τ

σ

τ

σ

τ

σ

τ

θ

θ

θ

θ

θ

θ

θ

θ

θ

θ

θ

θ

θ

θ

θ

θ

τ

σ

τ

τ

σ

σ

τ

σ

θ

θ

θ

θ

θ τ

θ τ

θ τ

θ τ

σ

σ

σ

σ

θ

θ

θ

θ

θ

θ

θ

θ

















₋₋₋₋

















































====

































































−−−−

































































_

_

_

_

_

_

_

_

2

2

r

cos

x

sen

y

sen2

xy

σ

θσ

θσ

θτ

σ

θσ

θσ

θτ

σ

θσ

θσ

θτ

σ

=

=

=

=

θσ

+

+

+

+

θσ

+

+

+

+

θτ

2

2

x

y

xy

sen

cos

sen2

θθθθ

σ

θσ

θσ

θτ

σ

θσ

θσ

θτ

σ

θσ

θσ

θτ

σ

=

=

=

=

θσ

+

+

+

+

θσ

−

−

−

−

θτ

'

((((

))))

r

θθθθ

sen cos

y

x

cos 2

xy

τ

θ

θ σ

σ

θτ

τ

θ

θ σ

σ

θτ

τ

θ

θ σ

σ

θτ

τ

=

=

=

=

θ

θ σ

−

−

−

−

σ

+

+

+

+

θτ

*

2A

(

x

y

x

y

r

θθθθ

u

v

u

r

u

_θθθθ

;

r

u

u

cos

sen

v

v

sen

cos

_θθθθ

θ

θ

θ

θ

θ

θ

θ

θ

θ

θ

θ

θ

θ

θ

θ

θ

−−−−

















 





 

 









 





====

 





 

 









 

_

_

_

_

_

_

_

_





 





 

 









 



  













'

xy r

r x

r xy

y

cos

sen

cos

sen

2

2

sen

cos

sen

cos

2

₂

θθθθ θθθθ

θθθθ

γγγγ

γγγγ

ε

ε

ε

ε

ε

ε

ε

_θ

_θ

_θ

_θ

_θ

_θ

_θ

_θ

ε

_θ

_θ

_θ

_θ

_θ

_θ

_θ

_θ

γ

θ

θ γ

θ

θ

γ

θ

θ γ

θ

θ

γ

θ

θ γ

θ

θ

γ

_εεεε

θ

θ γ

θ

θ

εεεε

































































_

_

_

_

_

_

_

_

_

_

_

_

₋₋₋₋

_

_

_

_

















====

















_

_

_

_

_

_

_

_

_

_

_

_

_

_

_

_

−−−−































































































_



_













_

_

_

_

_

_

_

_

xy

2 2

r

cos

x

sen

y

sen2

2

γγγγ

ε

θε

θε

θ

ε

ε

θε

θε

θε

θε

θ

θ

ε

=

=

=

=

θε

+

+

+

+

θε

+

+

+

+

θ

xy

2

2

x

y

sen

cos

sen2

2

θθθθ

γγγγ

ε

θε

θε

θ

ε

ε

θε

θε

θε

θε

θ

θ

ε

=

=

=

=

θε

+

+

+

+

θε

−

−

−

−

θ

((((

))))

xy

r

y

x

sen cos

cos 2

2

2

θθθθ

γγγγ

γγγγ

_θ

_θ

_θ

_θ

_{θ ε}

_{θ ε}

_{θ ε}

_{θ ε}

_ε

_ε

_ε

_ε

_θ

_θ

_θ

_θ

=

−

+

=

−

+

=

−

+

=

−

+

*

*

*

'

#. (

<

=G <

= G < >=

<* =, <* =

<**=,

8

r r

u

r

εεεε

====

∂∂∂∂

∂∂∂∂

r

v

u

1

r

r

θθθθ

θθθθ

εεεε

θθθθ

∂∂∂∂

=

+

=

+

=

+

=

+

∂∂∂∂

r

r

v

v

u

1

r

r

r

θ

θ

θ

θ

θ

θ

θ

θ

θθθθ

γγγγ

θθθθ

∂∂∂∂

∂∂∂∂

=

+

−

=

+

−

=

+

−

=

+

−

∂

∂

∂

∂

∂

∂

∂

∂

*9

*+

*:

?

2A

<*9=, <*+=

<*:=,

( 6

8

r

u

v

_θθθθ

2 2 2

r r

r r

2 2 2 2

1

2

1

1

1

r

r

r

r

r

r

r

r

r

θ θ θ θ

θθ θθ θθ θθ

θ θ θ θ

ε

ε

γ

γ

ε

ε

γ

γ

ε

ε

γ

γ

ε

ε

ε

ε

ε

ε

ε

ε

ε

ε

γ

γ

θ

θ

θ

θ

θ

θ

θ

θ

θ

θ

θ

θ

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

₊

₊

₊

₊

∂

∂

∂

∂

₊

₊

₊

₊

∂

₋

₋

₋

₋

∂

∂

∂

∂

₌

₌

₌

₌

∂

₊

₊

₊

₊

∂

∂

∂

∂

∂

∂ ∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂ ∂

∂ ∂

∂

∂

'

#

H

I , .

(

.

, .

(

(

.

<

J=

'

.

7 .

.

?

2A

(

(

(

8

((((

))))

r r θθθθ

2G

r

σ

λ ε ε

ε

σ

λ ε ε

ε

σ

λ ε ε

ε

σ

=

=

=

=

λ ε ε

+

+

+

+

+

+

+

+

ε

((((

r

))))

2G

θ θ θ

θ θ θ

θ θ θ

θ θ θ

σ

λ ε ε

ε

σ

λ ε ε

ε

σ

λ ε ε

ε

σ

=

=

=

=

λ ε ε

+

+

+

+

+

+

+

+

ε

rθθθθ

G

rθθθθ

τ

γ

τ

γ

τ

γ

τ

====

γ

((((

1

))))((((

E

1 2

))))

νννν

λλλλ

_ν

_ν

ν

ν

ν

ν

ν

ν

====

+

−

+

−

+

−

+

−

((((

))))

2

r r

1

1

E

E

θθθθ

ν

ν

ν

ν

ν

ν

ν

ν

νννν

ε

σ

σ

ε

σ

σ

ε

σ

σ

ε

=

=

=

=

−−−−

σ

−

−

−

−

++++

σ

((((

))))

2

r

1

1

E

E

θ θ θ θ θ θ θ θ

ν

ν

ν

ν

ν

ν

ν

ν

νννν

ε

σ

σ

ε

σ

σ

ε

σ

σ

ε

=

=

=

=

−−−−

σ

−

−

−

−

++++

σ

r r

G

θθθθ θθθθ

ττττ

γγγγ

====

((((

E

))))

G

2 1

νννν

====

++++

1

((((

))))

r 2 r

E

1

θθθθ

σ

ε νε

σ

ε νε

σ

ε νε

σ

ε νε

νννν

=

+

=

+

=

+

=

+

−−−−

((((

r

))))

2

E

1

θ θ

θ θ

θ θ

θ θ

σ

νε ε

σ

νε ε

σ

νε ε

σ

νε ε

νννν

=

+

=

=

+

+

=

+

−−−−

rθθθθ

G

rθθθθ

τ

γ

τ

γ

τ

γ

τ

====

γ

((((

))))

r r

1

E

θθθθ

ε

σ νσ

ε

σ νσ

ε

σ νσ

ε

=

=

=

=

σ νσ

−

−

−

−

((((

r

))))

1

E

θ θ θθ θθ θ θ

ε

σ νσ

ε

σ νσ

ε

σ νσ

ε

=

=

=

=

σ νσ

−

−

−

−

r r

G

θθθθ θθθθ

ττττ

γγγγ

====

'

'

9 ?

2A

? C'

#. (

<

=

<

=

.

?

2-

? C'

(

(

,

2

6 K

(

, /

(

?

2A

<

=

<

=,

8

r r r r

r

1

_cos

1

₂

_sen

₀

r

r

r

r

r

r

θ θ θ θ θ

θ θ θ θ θ

θ θ θ θ θ

θ θ θ θ θ

τ

σ σ

σ

τ

τ

τ

σ σ

σ

τ

τ

τ

σ σ

σ

τ

τ

τ

σ σ

σ

τ

τ

σσσσ

_θ

_θ

_θ

_θ

_θ

_θ

_θ

_θ

θ

θ

θ

θ

θ

θ

θ

θ

∂

−

∂

∂

∂

−

∂

∂

∂

−

∂

∂

∂

−

∂

∂

∂∂∂∂

































+

+

−

+

+

=

+

+

−

+

+

=

+

+

−

+

+

=

+

+

−

+

+

=



























_∂

_∂

_∂

_∂

_∂

_∂

_∂

_∂





_∂

_∂

_∂

_∂

_∂

_∂

_∂

_∂



































!

6 E ?

?

2-

6

.

6

0

,

6

8

r r

r

1

0 , em

0

r

r

r

θ θ θ θ θ θ θ θ

τ

σ σ

τ

σ σ

τ

σ σ

τ

σ σ

σσσσ

_θθθθ

θθθθ

∂

−

∂

∂

−

−

∂

−

∂∂∂∂

















+

+

=

=

+

+

+

+

=

=

=

=

+

+

=

=











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