b_vector.sagews

%typeset_mode True

var('x, y, xi, eta')
var('x_0, x_1, x_2, x_3, x_4, x_5, x_6, x_7,'
    'y_0, y_1, y_2, y_3, y_4, y_5, y_6, y_7')

assume(x_0, 'constant')
assume(x_1, 'constant')
assume(x_2, 'constant')
assume(x_3, 'constant')
assume(x_4, 'constant')
assume(x_5, 'constant')
assume(x_6, 'constant')
assume(x_7, 'constant')

assume(y_0, 'constant')
assume(y_1, 'constant')
assume(y_2, 'constant')
assume(y_3, 'constant')
assume(y_4, 'constant')
assume(y_5, 'constant')
assume(y_6, 'constant')
assume(y_7, 'constant')

($\displaystyle x$, $\displaystyle y$, $\displaystyle \xi$, $\displaystyle \eta$)

($\displaystyle x_{0}$, $\displaystyle x_{1}$, $\displaystyle x_{2}$, $\displaystyle x_{3}$, $\displaystyle x_{4}$, $\displaystyle x_{5}$, $\displaystyle x_{6}$, $\displaystyle x_{7}$, $\displaystyle y_{0}$, $\displaystyle y_{1}$, $\displaystyle y_{2}$, $\displaystyle y_{3}$, $\displaystyle y_{4}$, $\displaystyle y_{5}$, $\displaystyle y_{6}$, $\displaystyle y_{7}$)

N_0 = (-1.0 / 4.0) * (1 - xi)  * (1 - eta) * (1 + xi + eta)
N_1 = (1.0 / 2.0)  * (1 - eta) * (1 - xi**2)
N_2 = (-1.0 / 4.0) * (1 + xi)  * (1 - eta) * (1 - xi + eta)
N_3 = (1.0 / 2.0)  * (1 + xi)  * (1 - eta**2)
N_4 = (-1.0 / 4.0) * (1 + xi)  * (1 + eta) * (1 - xi - eta)
N_5 = (1.0 / 2.0)  * (1 + eta) * (1 - xi**2)
N_6 = (-1.0 / 4.0) * (1 - xi)  * (1 + eta) * (1 + xi - eta)
N_7 = (1.0 / 2.0)  * (1 - xi)  * (1 - eta**2)

x = N_0 * x_0 \
  + N_1 * x_1 \
  + N_2 * x_2 \
  + N_3 * x_3 \
  + N_4 * x_4 \
  + N_5 * x_5 \
  + N_6 * x_6 \
  + N_7 * x_7

y = N_0 * y_0 \
  + N_1 * y_1 \
  + N_2 * y_2 \
  + N_3 * y_3 \
  + N_4 * y_4 \
  + N_5 * y_5 \
  + N_6 * y_6 \
  + N_7 * y_7

y

print('*******************************************************************')
x

$\displaystyle -{\left(\eta + \xi + 1\right)} {\left(\eta - 1\right)} {\left(0.250000000000000 \, \xi - 0.250000000000000\right)} y_{0} - {\left(\eta - \xi + 1\right)} {\left(\eta - 1\right)} {\left(-0.250000000000000 \, \xi - 0.250000000000000\right)} y_{2} - {\left(\eta + \xi - 1\right)} {\left(\eta + 1\right)} {\left(-0.250000000000000 \, \xi - 0.250000000000000\right)} y_{4} - {\left(\eta - \xi - 1\right)} {\left(\eta + 1\right)} {\left(0.250000000000000 \, \xi - 0.250000000000000\right)} y_{6} - {\left(\xi^{2} - 1\right)} {\left(-0.500000000000000 \, \eta + 0.500000000000000\right)} y_{1} - {\left(\eta^{2} - 1\right)} {\left(0.500000000000000 \, \xi + 0.500000000000000\right)} y_{3} - {\left(\xi^{2} - 1\right)} {\left(0.500000000000000 \, \eta + 0.500000000000000\right)} y_{5} - {\left(\eta^{2} - 1\right)} {\left(-0.500000000000000 \, \xi + 0.500000000000000\right)} y_{7}$

*******************************************************************

$\displaystyle -{\left(\eta + \xi + 1\right)} {\left(\eta - 1\right)} x_{0} {\left(0.250000000000000 \, \xi - 0.250000000000000\right)} - {\left(\eta - \xi - 1\right)} {\left(\eta + 1\right)} x_{6} {\left(0.250000000000000 \, \xi - 0.250000000000000\right)} - {\left(\eta - \xi + 1\right)} {\left(\eta - 1\right)} x_{2} {\left(-0.250000000000000 \, \xi - 0.250000000000000\right)} - {\left(\eta + \xi - 1\right)} {\left(\eta + 1\right)} x_{4} {\left(-0.250000000000000 \, \xi - 0.250000000000000\right)} - {\left(\xi^{2} - 1\right)} {\left(-0.500000000000000 \, \eta + 0.500000000000000\right)} x_{1} - {\left(\xi^{2} - 1\right)} {\left(0.500000000000000 \, \eta + 0.500000000000000\right)} x_{5} - {\left(\eta^{2} - 1\right)} x_{3} {\left(0.500000000000000 \, \xi + 0.500000000000000\right)} - {\left(\eta^{2} - 1\right)} x_{7} {\left(-0.500000000000000 \, \xi + 0.500000000000000\right)}$