.. _r7.01.44-jacobien:

Appendix: Expression of the Jacobian matrix
============================================

In the case where neither the plasticity criterion :math:`f` nor :math:`F_m` is verified after the elastic prediction, we recall the expression of the system 
of nonlinear equations :eq:`residu_plastique` and the definition of the Jacobian matrix :eq:`jacobienne`:

.. math::
\ begin {Bmatrix}\ boldsymbol {r} _1\\\ boldsymbol {r} _2\\ r_3\\ r_4\\ r_5\ end {Bmatrix} =
  \ begin {Bmatrix} 
  \ Delta\ boldsymbol {\ varepsilon} ^e -\ Delta\ boldsymbol {\ varepsilon} + 
  \ left (\ mathbb {J} +\ rho\ mathbb {K}\ right):\ Delta\ lambda\ cfrac {\ partial f} {\ partial\ boldsymbol {X}}\ big {|} _ {n+1}} +
  \ left (\ mathbb {J} + (1-\ rho)\ mathbb {K}\ right):\ left (\ Delta\ lambda_m\ cfrac {\ partial F_m} {\ partial F_m} {\ partial\ boldsymbol {A} ^m}\ big {|} _ {n+1}}
  +\ sum_ {i=1} ^ {m-1}\ Delta\ boldsymbol {\ alpha} ^i\ right)\\
  \ Delta\ boldsymbol {\ varepsilon} ^x -\ Delta\ boldsymbol {\ varepsilon} ^e + 
  \ left (1-\ rho\ right)\ mathbb {K}:\ left (\ Delta\ Lambda\ cfrac {\ partial f} {\ partial\ boldsymbol {X}}}\ Big {|} {K}}\ Big {|} _ {n+1}
  -\ Delta\ lambda_m\ cfrac {\ partial F_m} {\ partial\ boldsymbol {A} ^m}\ Big {|} _ {n+1} 
  -\ sum_ {i=1} ^ {m-1}\ Delta\ boldsymbol {\ alpha} ^i\ right)\\
  \ cfrac {f_ {n+1}} {K}\\
  \ cfrac {F_ {m, n+1}} {K}\\
  \ Delta\ xi -\ Delta\ Lambda\ cfrac {\ partial f} {\ partial p_c}\ Big {|} _ {n+1}\\
  \ end {Bmatrix}
  :label: plastic_residure_appendix

.. math::
 J =\ begin {Bmatrix}
  \ cfrac {\ partial\ boldsymbol {r} _1} {\ partial\ Delta\ boldsymbol {\ varepsilon} ^e}} &
  \ cfrac {\ partial\ boldsymbol {r} _1} {\ partial\ Delta\ boldsymbol {\ varepsilon} ^x}} &
  \ cfrac {\ partial\ boldsymbol {r} _1} {\ partial\ Delta\ lambda} &
  \ cfrac {\ partial\ boldsymbol {r} _1} {\ partial\ Delta\ lambda_m} &
  \ cfrac {\ partial\ boldsymbol {r} _1} {\ partial\ Delta\ xi}\\
  \ cfrac {\ partial\ boldsymbol {r} _2} {\ partial\ Delta\ boldsymbol {\ varepsilon} ^e}} &
  \ cfrac {\ partial\ boldsymbol {r} _2} {\ partial\ Delta\ boldsymbol {\ varepsilon} ^x}} &
  \ cfrac {\ partial\ boldsymbol {r} _2} {\ partial\ Delta\ lambda} &
  \ cfrac {\ partial\ boldsymbol {r} _2} {\ partial\ Delta\ lambda_m} &
  \ cfrac {\ partial\ boldsymbol {r} _2} {\ partial\ Delta\ xi}\\
  \ cfrac {\ partial r_3} {\ partial\ Delta\ Delta\ boldsymbol {\ varepsilon} ^e}} &
  \ cfrac {\ partial r_3} {\ partial\ Delta\ Delta\ boldsymbol {\ varepsilon} ^x}} &
  \ cfrac {\ partial r_3} {\ partial\ Delta\ lambda} &
  \ cfrac {\ partial r_3} {\ partial\ Delta\ lambda_m} &
  \ cfrac {\ partial r_3} {\ partial\ Delta\ xi}\\
  \ cfrac {\ partial r_4} {\ partial\ Delta\ Delta\ boldsymbol {\ varepsilon} ^e}} &
  \ cfrac {\ partial r_4} {\ partial\ Delta\ Delta\ boldsymbol {\ varepsilon} ^x}} &
  \ cfrac {\ partial r_4} {\ partial\ Delta\ lambda} &
  \ cfrac {\ partial r_4} {\ partial\ Delta\ lambda_m} &
  \ cfrac {\ partial r_4} {\ partial\ Delta\ xi}\\
  \ cfrac {\ partial r_5} {\ partial\ Delta\ Delta\ boldsymbol {\ varepsilon} ^e}} &
  \ cfrac {\ partial r_5} {\ partial\ Delta\ Delta\ boldsymbol {\ varepsilon} ^x}} &
  \ cfrac {\ partial r_5} {\ partial\ Delta\ lambda} &
  \ cfrac {\ partial r_5} {\ partial\ Delta\ lambda_m} &
  \ cfrac {\ partial r_5} {\ partial\ Delta\ xi}
  \ end {Bmatrix}
 :label: Jacobian_appendix
 
 
In MFront, this :math:`J` matrix can be obtained by numerical disturbance or analytically, as is the case presented below. Its components are detailed below.

For this purpose, note:
 
* The flow directions of :math:`\Delta\boldsymbol{\varepsilon}^p,\left(\Delta\boldsymbol{\alpha}^i\right)_{1\leq i\leq m}` and :math:`\Delta\xi`:
 
.. math::
\ begin {align}
  &\ boldsymbol {N} _ {n+1} =\ frac {\ partial f} {\ partial\ boldsymbol {X}}\ big {|} _ {n+1} =\ frac {\ cfrac {3} {3} {2M^2} =\ frac {3} {2M^2} {2M^2}}\ boldsymbol {2}}\ boldsymbol {2}}\ boldsymbol {Y} _ {d, n+1} +Y_ {m, n+1} =\ frac {\ cfrac {3} {2M^2} {2M^2}}\ boldsymbol {2}}\ frac {\ boldsymbol {I}} {3}} {T_ {eq} (\ boldsymbol {Y} _ {n+1})}
  ,\ quad\ text {with}\ quad\ boldsymbol {Y} =\ boldsymbol {X} +\ left (P_c-s\ right)\ boldsymbol {I},\ quad T_ {eq} (\ boldsymbol {Y}) (\ boldsymbol {Y}) =\ sqrt {Y}) =\ sqrt {Y}) =\ sqrt {X} +\ left (\ frac {Y_ {eq}} {M}\ right) ^2+y_M^2}\ right) ^2+y_M^2}\\
  &\ boldsymbol {M} ^i_ {n+1} =\ frac {\ partial F_i} {\ partial\ boldsymbol {A} ^i}\ Big {|} _ {n+1} =\ frac {\ cfrac {3} {2} =\ frac {\ cfrac {3} {2} {2}\ boldsymbol {A} ^i_ {d, n+1}} +\ left (\ cfrac {R_i}} {C}\ right) ^2\ left\ langle A^i_ {m, n+1} +C\ right\ rangle\ cfrac {\ boldsymbol {I}} {3}} {t^i_ {eq}} {t^i_ {eq} (\ boldsymbol {A} ^i_ {n+1})}
  ,\ quad\ text {with}\ quad T^i_ {eq} (\ boldsymbol {A} ^i) =\ sqrt {\ left (A^i_ {eq}\ right) ^2+\ left (\ cfrac {R_i} {eq} {eq} {eq}} (\ boldsymbol {A} ^i) =\ sqrt {\ left (A^i_ {eq}\ right) ^2+\ left (\ cfrac {R_i} {eq} {eq}\ left (\ cfrac {R_i}} {C}\ left\ langle A^i_ {m} +C\ right\ rangle\ right) ^2}\\ right
  &N_ {v, n+1} =\ frac {\ partial f} {\ partial p_c}\ Big {|} _ {n+1} =\ boldsymbol {I}:\ boldsymbol {I}:\ boldsymbol {N} _ {n+1}\\
  \ end {align}
  :label: derivee_utile_1
  
* Their following three derivatives:
 
.. math::
\ begin {align}  
  &\ boldsymbol {N} _ {n+1,\ boldsymbol {Y} _ {n+1}} =\ frac {\ partial\ boldsymbol {N}} {\ partial\ boldsymbol {Y}}\ Big {|}\ big {|} _ {n+1} =\ frac {\ n+1} =\ frac {\ cfrac {3} {2M^2}}\ mathbb {K} +\ cfrac {\ mathbb} +\ cfrac {\ mathbb {J}} {3} -\ boldsymbol {N} _ {n+1} _ {n+1}\ otimes\ boldsymbol {N} _ {n+1}} {T_ {eq}} (\ boldsymbol {Y} _ {n+1})}\\
  &\ boldsymbol {M} ^i_ {n+1,\ boldsymbol {A} ^i_ {n+1}} =\ frac {\ partial\ boldsymbol {M} ^i} {\ partial\ boldsymbol {A} ^i}\ big {|} _ {n+1}} = 
  \ frac {\ cfrac {3} {2}\ mathbb {K} +\ left (\ cfrac {R_i} {C}\ right) ^2\ mathbf {H}\ left (A^i_ {m, n+1} {m, n+1} +C\ right} +C\ right)\ cfrac {\ mathbb {J}} {3} -\ boldsymbol {M} ^i_ {n+1}\ otimes\ boldsymbol {M} ^i_ {n+1}} {t^i_ {eq} (\ boldsymbol {A} ^i_ {n+1})}\\
  &N_ {v, n+1,\ boldsymbol {Y} _ {n+1}} =\ frac {\ partial N_ {v}} {\ partial\ boldsymbol {Y}}\ big {|} _ {n+1} =\ frac {\ cfrac {\ cfrac {\ boldsymbol {I}}} {3} -N_ {v, n+1}}\ big {|} _ {n+1} =\ frac {\ cfrac {\ boldsymbol {I}}} {3} -N_ {v, n+1}}\ big {|} _ {n+1} =\ frac {\ cfrac {\ boldsymbol {I}}} {3} -N_ {v, n+1}}\ big {|}} _ {n+1} =\ frac {\ n+1}} {T_ {eq} (\ boldsymbol {Y} _ {n+1})} 
  \ end {align}
  :label: derivee_utile_2

* Derivatives of forces :math:`\boldsymbol{Y}=\boldsymbol{Y}(\boldsymbol{\varepsilon}^x,\lambda,\xi)` and 
  :math:`\left(\boldsymbol{A}^i\right)_{1\leq i \leq m}=\left(\boldsymbol{A}^i(\boldsymbol{\varepsilon}^e,\boldsymbol{\varepsilon}^x)\right)_{1\leq i \leq m}`:

.. math::
\ begin {align}  
  &\ boldsymbol {Y} _ {n+1,\ boldsymbol {\ varepsilon} ^x} =\ frac {\ partial\ boldsymbol {Y}} {\ partial\ Delta\ boldsymbol {\ boldsymbol {\ varepsilon} ^x} ^x}\ big {|} ^x}\ big {|} _ {n+1} =\ left (\ mathbb {J}} +\ rho\ mathbb {K}\ right):\ mathbb {C}\\
  &\ boldsymbol {Y} _ {n+1,\ lambda} =\ frac {\ partial\ boldsymbol {Y}} {\ partial\ Delta\ lambda}\ Big {|} _ {n+1} =\ omega S_ {n+1} =\ omega S_ {n+1}\\
  &\ boldsymbol {Y} _ {n+1,\ xi} =\ frac {\ partial\ boldsymbol {Y}} {\ partial\ Delta\ xi}\ Big {|} _ {n+1} = -\ left (\ left (\ omega S_ {n+1}} +\ beta p_ {n+1}}}\ right)\ boldsymbol {I}\\
  &\ boldsymbol {A} ^i_ {n+1,\ boldsymbol {\ varepsilon} ^e} =\ frac {\ partial\ boldsymbol {A} ^i} {\ partial\ Delta\ Delta\ delta\ boldsymbol {\ boldsymbol {\ varepsilon} ^e}\\\
  &\ boldsymbol {A} ^i_ {n+1,\ boldsymbol {\ varepsilon} ^x} =\ frac {\ partial\ boldsymbol {A} ^i} {\ partial\ Delta\ Delta\ boldsymbol {\ varepsilon} boldsymbol {\ boldsymbol {\ varepsilon} ^x}}\ big {|} |} _ {n+1} = -\ rho\ mathbb {K}:\ mathbb {K}:\ mathbb {C}
  \ end {align}
  :label: derivee_utile_3

* Derivatives of all :math:`\left(\Delta\boldsymbol{\alpha}^i=\Delta\lambda_i\boldsymbol{M}_{n+1}^i\right)_{1\leq i\leq m-1}`:

.. math::
\ begin {align} 
  \ frac {\ partial\ Delta\ boldsymbol {\ alpha} ^i} {\ partial\ Delta\ boldsymbol {\ varepsilon} ^e} =\ chi_i\ left (\ Delta\ lambda_i\ boldsymbol {\ alpha} {alpha} ^i_ {i} ^i} ^i} ^i_ {n+1}} =\ chi_i\ left (\ Delta\ lambda_i\ i\ i\ boldsymbol {\ alpha} ^i} ^i_ {n+1}} =\ frac {2} {3H_d_}} +\ frac {2} {3H_d_d+ ^i}\ boldsymbol {M} ^i_ {n+1}\ otimes\ boldsymbol {M} ^i_ {n+1}\ right):\ boldsymbol {A} ^i_ {n+1,\ boldsymbol {\ varepsilon} ^e}\\
  \ frac {\ partial\ Delta\ boldsymbol {\ alpha} ^i} {\ partial\ Delta\ boldsymbol {\ varepsilon} ^x} =\ chi_i\ left (\ Delta\ lambda_i\ boldsymbol {\ alpha} {alpha} ^i_ {i} ^i} ^i_ {n+1}}} =\ chi_i\ left (\ Delta\ Lambda_i\ i\ i\ i\ i\ i\ i\ i\ i\ i\ i\ i\ i\ i\ i\ boldsymbol {\ alpha} ^i} ^i} ^i_ {n+1}} =\ frac {2} {3H_d_d+1}} +\ frac {2} {3H_d_d+ ^i}\ boldsymbol {M} ^i_ {n+1}\ otimes\ boldsymbol {M} ^i_ {n+1}\ right):\ boldsymbol {A} ^i_ {n+1,\ boldsymbol {\ varepsilon} ^x}
  \ end {align}
  :label: derivee_utile_4

where :math:`\chi_i=1` if :math:`\left \langle F_i\left(\boldsymbol{\sigma}_{n+1}-\boldsymbol{X}_{d,n+1} -H_d^i\boldsymbol{\alpha}_{d,n}^i-H_v^i\alpha_{v,n}^i\boldsymbol{I}\right)\right \rangle>0`, zero otherwise.

First line
--------------

The derivation of each term in the first line of the system shown :eq:`jacobienne_annexe` provides:
 
.. math::
\ begin {align}
 &\ frac {\ partial\ boldsymbol {r} _1} {\ partial\ Delta\ boldsymbol {\ varepsilon} ^e} = 
 \ mathbb {I} +\ left (\ mathbb {J} + (1-\ rho)\ mathbb {K}\ right):\ left (\ Delta\ lambda_m\ boldsymbol {M} ^m_ {n+1,\ boldsymbol {N+1} + (1-\ rho) + (1-\ rho)\ mathbb {K}\ right):\ left (\ Delta\ Lambda_m\ boldsymbol {M} ^m_ {n+1,\ m_ {n+1,\),\ boldsymbol {\ v+1, arepsilon} ^e}
 +\ sum_ {i=1} ^ {m-1}\ frac {\ partial\ Delta\ boldsymbol {\ alpha} ^i} {\ partial\ Delta\ boldsymbol {\ varepsilon} ^e}\ right)\\
 &\ frac {\ partial\ boldsymbol {r} _1} {\ partial\ Delta\ boldsymbol {\ varepsilon} ^x} =
 \ left (\ mathbb {J} +\ rho\ mathbb {K}\ right):\ Delta\ lambda\ boldsymbol {N} _ {n+1,\ boldsymbol {Y} _ {n+1}}}:\ boldsymbol {n+1}}}:\ boldsymbol {N} _ {n+1,\ boldsymbol {\ varepsilon} ^x}}
 +\ left (\ mathbb {J} + (1-\ rho)\ mathbb {K}\ right):\ left (\ Delta\ lambda_m\ boldsymbol {M} ^m_ {n+1,\ boldsymbol {A} {n+1),\ boldsymbol {A} {n+1}}:\ boldsymbol {A} ^m_ {n+1,\ boldsymbol {\ varepsilon} ^n+1,\ boldsymbol {\ varepsilon} ^m+ x}
 +\ sum_ {i=1} ^ {m-1}\ frac {\ partial\ Delta\ boldsymbol {\ alpha} ^i} {\ partial\ Delta\ boldsymbol {\ varepsilon} {\ varepsilon} ^x}\ right)\\
 &\ frac {\ partial\ boldsymbol {r} _1} {\ partial\ Delta\ lambda} =\ left (\ mathbb {J} +\ rho\ mathbb {K}\ right):
 \ left (\ boldsymbol {N} _ {n+1}} +\ Delta\ lambda\ boldsymbol {N} _ {n+1,\ boldsymbol {Y} _ {n+1}}}:\ boldsymbol {Y}}} +\ delta\ lambda}\ right)\\
 &\ frac {\ partial\ boldsymbol {r} _1} {\ partial\ Delta\ lambda_m} =\ left (\ mathbb {J} + (1-\ rho)\ mathbb {K}\ right):\ boldsymbol {M} ^m_ {n+1}\\
 &\ frac {\ partial\ boldsymbol {r} _1} {\ partial\ Delta\ xi} =\ left (\ mathbb {J} +\ rho\ mathbb {K}\ right):
 \ Delta\ lambda\ boldsymbol {N} _ {n+1,\ boldsymbol {Y} _ {n+1}}:\ boldsymbol {Y} _ {n+1,\ xi}}:\ boldsymbol {Y} _ {n+1,\ xi}
 \ end {align}
 :label: jacobienne_line1

Second line
--------------

The second line of the system reads:

.. math::
\ begin {align}
 &\ frac {\ partial r_2} {\ partial\ Delta\ Delta\ boldsymbol {\ varepsilon} ^e} = -\ mathbb {I} - (1-\ rho)\ mathbb {K}:
 \ left (\ Delta\ lambda_m\ boldsymbol {M} ^m_ {n+1,\ boldsymbol {A} ^m_ {n+1}}:\ boldsymbol {A} ^m_ {n+1,\ boldsymbol {M} {n+1,\ boldsymbol {\ varepsilon} ^e}} +\ sum_ {i=1} ^ {m-1}\ frac {\ partial\ symbol {n+1}\ frac {\ partial\ symbol {n+1}\ frac {\ partial\ symbol {n+1}\ frac {\ partial\ symbol {n+1] +1,\ boldsymbol {\ partial\ Delta\ boldsilon} ^e} +\ sum_ {i=1} ^ {m-1}}\ frac {\ partial\ symbol {n+1}\ {\ alpha} ^i} {\ partial\ Delta\ Delta\ boldsymbol {\ varepsilon} ^e}\ right)\\
 &\ frac {\ partial r_2} {\ partial\ Delta\ Delta\ boldsymbol {\ varepsilon} ^x} =\ mathbb {I} + (1-\ rho)\ mathbb {K}:\ left (
 \ Delta\ lambda\ boldsymbol {N} _ {n+1,\ boldsymbol {Y} _ {n+1}}:\ boldsymbol {Y} _ {n+1,\ boldsymbol {\ varepsilon} ^x}} -
 \ Delta\ lambda_m\ boldsymbol {M} ^m_ {n+1,\ boldsymbol {A} ^m_ {n+1}}:\ boldsymbol {A} ^m_ {n+1,\ boldsymbol {\ varepsilon} ^m_ {m_} {n+1}}:\ boldsymbol {\ partial\ n+1}\ frac {\ partial\ Delta\ boldsymbol {\ alpha}} ^i} {\ partial\ Delta\ Delta\ boldsymbol {\ varepsilon} ^x}\ right)\\
 &\ frac {\ partial r_2} {\ partial\ Delta\ lambda} {\ partial\ delta\ lambda} = (1-\ rho)\ mathbb {K}:\ left (\ boldsymbol {N} _ {n+1}} +\ Delta\ lambda\ lambda\\ boldsymbol\\ lambda\\ boldsymbol {N}} +\ delta\ lambda\\\ lambda\ boldsymbol {N}\ Lambda}\ right)\\
 &\ frac {\ partial r_2} {\ partial\ Delta\ delta\ lambda_m} =- (1-\ rho)\ mathbb {K}:\ boldsymbol {M} ^m_ {n+1}\
 &\ frac {\ partial r_2} {\ partial\ Delta\ xi}} = (1-\ rho)\ mathbb {K}:\ Delta\ lambda\ boldsymbol {N} _ {n+1,\ n+1,\ boldsymbol {Y}} _ {n+1}}:\ boldsymbol {Y} _ {n+1,\ xi}
 \ end {align}
 :label: jacobienne_line2
 
Third line
---------------

The third line in the system is:

.. math::
\ begin {align}
 &\ frac {\ partial r_3} {\ partial\ Delta\ Delta\ boldsymbol {\ varepsilon} ^e} =\ boldsymbol {0}\\
 &\ frac {\ partial r_3} {\ partial\ partial\ Delta\ delta\ boldsymbol {\ varepsilon} ^x} =\ cfrac {\ boldsymbol {N} _ {n+1}:\ boldsymbol {Y} _ {n+1}\\
 &\ frac {\ partial r_3} {\ partial\ Delta\ lambda} =\ cfrac {\ boldsymbol {N} _ {n+1}:\ boldsymbol {Y} _ {n+1,\ n+1,\ lambda} -\ lambda} -\ omega S_ {n+1}} {K}\\
 &\ frac {\ partial r_3} {\ partial\ Delta\ lambda_m} =0\\
 &\ frac {\ partial r_3} {\ partial\ Delta\ xi} =\ cfrac {\ boldsymbol {N} _ {n+1}:\ boldsymbol {Y} _ {n+1,\ xi} +\ omega S_ {n+1} +\ omega S_ {n+1} +\ beta R_ {n+1} +\ beta R_ {n+1}\\\
 \ end {align}
 :label: jacobienne_line3

Fourth line
---------------

The fourth line:

.. math::
\ begin {align}
 &\ frac {\ partial r_4} {\ partial\ Delta\ Delta\ boldsymbol {\ varepsilon} ^e} =\ cfrac {\ boldsymbol {M} ^m_ {n+1}:\ boldsymbol {A} {n+1}:\ boldsymbol {A} ^m_ {n+1}:\ boldsymbol {A} ^m_ {n+1}:\ boldsymbol {A} ^m_ {n+1}:\ boldsymbol {A} ^m_ {n+1}:\ boldsymbol {A} ^m_ {n+1}:\ boldsymbol {A} ^m_ {n+1}
 &\ frac {\ partial r_4} {\ partial\ Delta\ Delta\ boldsymbol {\ varepsilon} ^x} =\ cfrac {\ boldsymbol {M} ^m_ {n+1}:\ boldsymbol {A} {n+1}:\ boldsymbol {A} ^m_ {n+1}:\ boldsymbol {A} ^m_ {n+1}:\ boldsymbol {A} ^m_ {n+1}:\ boldsymbol {A} ^m_ {n+1}:\ boldsymbol {A} ^m_ {n+1}:\ boldsymbol {A} ^m_ {n+1}:\ boldsymbol {A} ^m_ {n+1}
 &\ frac {\ partial r_4} {\ partial\ Delta\ lambda} =0\\
 &\ frac {\ partial r_4} {\ partial\ Delta\ lambda_m} =0\\
 &\ frac {\ partial r_4} {\ partial\ Delta\ xi} =0\\
 \ end {align}
 :label: jacobienne_line4

Fifth line
---------------

The fifth and final line:

.. math::
\ begin {align}
 &\ frac {\ partial r_5} {\ partial\ Delta\ Delta\ boldsymbol {\ varepsilon} ^e} =\ boldsymbol {0}\\
 &\ frac {\ partial r_5} {\ partial\ Delta\ Delta\ boldsymbol {\ varepsilon} ^x} = -\ Delta\ lambda N_ {v, n+1,\ boldsymbol {Y} _ {n+1} _ {n+1}}\ boldsymbol {Y} _ {n+1}}\\
 &\ frac {\ partial r_5} {\ partial\ Delta\ lambda} =-N_ {v, n+1} -\ Delta\ lambda N_ {v, n+1,\ boldsymbol {Y} _ {n+1}}\ boldsymbol {Y} _ {n+1}}\ boldsymbol {Y} _ {n+1,\ lambda}\\
 &\ frac {\ partial r_5} {\ partial\ Delta\ lambda_m} =0\\
 &\ frac {\ partial r_5} {\ partial\ Delta\ xi} =1-\ Delta\ lambda N_ {v, n+1,\ boldsymbol {Y} _ {n+1}}\ boldsymbol {n+1}}\ boldsymbol {Y} _ {n+1,\ xi}\\
 \ end {align}
 :label: jacobienne_line5