

In [1]:

    
import numpy as np



In [42]:

    
x = np.linspace(1,8,5)



In [43]:

    
x.shape









    Out[43]:





(5,)



In [44]:

    
y = np.sin(x)



In [45]:

    
y.shape









    Out[45]:





(5,)



In [46]:

    
for i in range(y.shape[0]-1):
    print( (y[i+1]-y[i]),(y[i+1]-y[i])/(x[i+1]-x[i]))









    



-0.459809992756 -0.262748567289
-1.35919110972 -0.776680634124
0.944350901118 0.539629086353
1.02253746317 0.584307121812



In [47]:

    
y[1:]-y[:-1]









    Out[47]:





array([-0.45980999, -1.35919111,  0.9443509 ,  1.02253746])



In [48]:

    
y[1:]









    Out[48]:





array([ 0.38166099, -0.97753012, -0.03317922,  0.98935825])



In [ ]:



In [ ]:

    
(y[1:]-y[:-1])/(x[1:]-x[:-1])



In [ ]:

    
np.diff(y)



In [50]:

    
np.diff(x)









    Out[50]:





array([ 1.75,  1.75,  1.75,  1.75])



In [31]:

    
np.roll(y,-1)









    Out[31]:





array([ 0.38166099, -0.97753012, -0.03317922,  0.98935825,  0.84147098])



In [30]:

    
y









    Out[30]:





array([ 0.84147098,  0.38166099, -0.97753012, -0.03317922,  0.98935825])



In [32]:

    
np.gradient(y)









    Out[32]:





array([-0.45980999, -0.90950055, -0.2074201 ,  0.98344418,  1.02253746])



In [66]:

    
import sympy



In [67]:

    
X = sympy.Symbol('X')



In [68]:

    
expr  =  (sympy.sin(X**2+1*sympy.cos(sympy.exp(X)))).diff(X)



In [73]:

    
expr









    Out[73]:





(2*X - exp(X)*sin(exp(X)))*cos(X**2 + cos(exp(X)))



In [71]:

    
f = sympy.lambdify(X,expr,"numpy")



In [72]:

    
f( np.array([1,2,3]))









    Out[72]:





array([  0.87994175,   0.67977486,  12.91025851])



In [77]:

    
import ipywidgets as widgets
from ipywidgets import interact



In [79]:

    
widgets.IntSlider?



In [82]:

    
@interact(x=widgets.IntSlider(1,2,10,1))
def g(x=1):
    print(x)

Całka oznaczona

$$\int_a^b f(x) dx = \lim_{n\to\infty} \sum_{i=1}^{n} f(\hat x_i) \Delta x_i$$



In [68]:

    
import numpy as np
N = 10
x = np.linspace( 0,np.pi*1.23, N)



In [69]:

    
f = np.sin(x)



In [70]:

    
x,f









    Out[70]:





(array([ 0.        ,  0.429351  ,  0.85870199,  1.28805299,  1.71740398,
         2.14675498,  2.57610598,  3.00545697,  3.43480797,  3.86415896]),
 array([ 0.        ,  0.41628079,  0.75699506,  0.96029369,  0.98927233,
         0.83867057,  0.53582679,  0.13571557, -0.2890318 , -0.66131187]))



In [71]:

    
np.diff(x)









    Out[71]:





array([ 0.429351,  0.429351,  0.429351,  0.429351,  0.429351,  0.429351,
        0.429351,  0.429351,  0.429351])



In [72]:

    
np.sum(f[:-1]*np.diff(x))









    Out[72]:





1.8651106037884004



In [73]:

    
w = np.ones_like(x)
h = np.diff(x)[0]



In [74]:

    
w[-1] = 0



In [75]:

    
h*np.sum(w*f)









    Out[75]:





1.8651106037884



In [76]:

    
w[0] = 0.5 
w[-1] = 0.5 
h*np.sum(w*f)









    Out[76]:





1.7231431497698426



In [77]:

    
import scipy.integrate



In [78]:

    
scipy.integrate.









    



  File "<ipython-input-78-52697efd3bc2>", line 1
    scipy.integrate.
                    ^
SyntaxError: invalid syntax



In [ ]:

Całka nieoznaczona

$$\int_a^x f(y) dy = \lim_{n\to\infty} \sum_{i=1}^{n} f(\hat y_i) \Delta y_i$$



In [79]:

    
np.cumsum(f)*h









    Out[79]:





array([ 0.        ,  0.17873057,  0.50374715,  0.9160502 ,  1.34079527,
        1.70087931,  1.93093708,  1.98920669,  1.8651106 ,  1.5811757 ])



In [80]:

    
np.sum(f)*h









    Out[80]:





1.5811756957512852



In [82]:

    
f.shape,np.cumsum(f).shape









    Out[82]:





((10,), (10,))

Pochodne funkcji wielu zmiennych



In [101]:

    
x = np.linspace(0,2,50)
y = np.linspace(0,2,50)



In [102]:

    
X,Y = np.meshgrid(x,y,indexing='xy')



In [103]:

    
X









    Out[103]:





array([[ 0.        ,  0.04081633,  0.08163265, ...,  1.91836735,
         1.95918367,  2.        ],
       [ 0.        ,  0.04081633,  0.08163265, ...,  1.91836735,
         1.95918367,  2.        ],
       [ 0.        ,  0.04081633,  0.08163265, ...,  1.91836735,
         1.95918367,  2.        ],
       ..., 
       [ 0.        ,  0.04081633,  0.08163265, ...,  1.91836735,
         1.95918367,  2.        ],
       [ 0.        ,  0.04081633,  0.08163265, ...,  1.91836735,
         1.95918367,  2.        ],
       [ 0.        ,  0.04081633,  0.08163265, ...,  1.91836735,
         1.95918367,  2.        ]])



In [104]:

    
F = np.sin(X**2 + Y)



In [105]:

    
F[1,2],X[1,2],Y[1,2]









    Out[105]:





(0.047462378916894261, 0.081632653061224483, 0.040816326530612242)



In [106]:

    
%matplotlib inline
import matplotlib.pyplot as plt



In [108]:

    
plt.contour(X,Y,F)









    Out[108]:





<matplotlib.contour.QuadContourSet at 0x7fb8b91b7748>



In [110]:

    
plt.imshow(F,origin='lower')









    Out[110]:





<matplotlib.image.AxesImage at 0x7fb8b7c68208>



In [117]:

    
np.diff(F,axis=1).shape









    Out[117]:





(50, 49)



In [118]:

    
np.diff(F,2,axis=0).shape









    Out[118]:





(48, 50)



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