# Mathematical Methods /Numerical codes
#http://www1.uprh.edu/rbaretti
#http://www1.uprh.edu/rbaretti/methodsoftheoreticalphysics.htm
#http://www1.uprh.edu/rbaretti/MethodsofTheoreticalPhysicsPart2.htm
#http://www1.uprh.edu/rbaretti/MethodsofTheoreticalPhysicsPart3.htm
#http://www1.uprh.edu/rbaretti/MethodsofTheoreticalPhysicsPart4.htm
#http://www1.uprh.edu/rbaretti/MethodsofTheoreticalPhysicsPart5.htm
diff(x^3,x,2)
6*x 6*x |
f=x^2
diff(f,x)
2*x 2*x |
integral(f,x)
x^3/3 x^3/3 |
integral(f,x,0,3)
9 9 |
#x,y=var('x,y')
# plot3d(16-x^2-y^2,(x,-2,2),(y,-2,2))
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#y=plot(sin(x),0,pi)
#show(y)
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y=plot(exp(-x),0,4)
show(y)
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plot ((x^2+1),x,-2,2).show(xmin=-3,xmax=3,ymin=0.,ymax=6)
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integral(exp(-2*x),x,0,20)
1/2 - e^-40/2 1/2 - e^-40/2 |
# two linear equations
x, y = var('x, y')
solve([8*x + y == 3, -x +7*y == 0],x,y)
[[x == 7/19, y == 1/19]] [[x == 7/19, y == 1/19]] |
# example of Kirchoff eqs fro three currents
i1,i2,i3=var('i1,i2,i3')
solve([8*i1 + i2 +i3== 3, -i1 +7*i2 +i3== 0,i1+i2+i3==1],i1,i2,i3)
[[i1 == 2/7, i2 == -1/14, i3 == 11/14]] [[i1 == 2/7, i2 == -1/14, i3 == 11/14]] |
# To solve the equation x''+9*x=0:
t = var('t') # define a variable t
x = function('x',t) # define x to be a function of that variable
DE = lambda y: diff(y,t,2) +9*y
desolve(DE(x(t)), [x,t])
k1*sin(3*t) + k2*cos(3*t) k1*sin(3*t) + k2*cos(3*t) |
# To solve the equation x''+ 9*x=0:
t = var('t') # define the variable t
x=function('x',t)
DE=lambda x:diff(x,t,2)+diff(x,t)+ 9*x
desolve (DE(x(t)),[x,t])
e^(-(t/2))*(k1*sin(sqrt(35)*t/2) + k2*cos(sqrt(35)*t/2)) e^(-(t/2))*(k1*sin(sqrt(35)*t/2) + k2*cos(sqrt(35)*t/2)) |
var('x,y,k ,L')
x=10
f=2*k*L/(x^2+y^2)^(1/2)
integral(f,y,0,100*x)
2*arcsinh(100)*k*L 2*arcsinh(100)*k*L |
# the R declaration defines the decimal result in terms of bits
R=RealField(30)
f=1/(5+x^2)^(1/2)
g=integral(f,x,0,5)
R(integral(f,x,0,5))
1.5444850 1.5444850 |
var('x, y')
y = integral(sqrt(1 + x^2), x, 0, 2)
print y
# n(y,20)
arcsinh(2) + 2 sqrt(5)
--------------------
2
arcsinh(2) + 2 sqrt(5)
--------------------
2
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n((arcsinh(2) + 2*sqrt(5))/2,20)
2.9579 2.9579 |
# 2x2 matrix
A = matrix(2, 2,[1/(3), -1/(3), 1/(3), 1/(3)]);
print(A)
Syntax Error:
A = matrix(2, 2,[1/(3), -1/(3), 1/(3), 1/(3)]);
Syntax Error:
A = matrix(2, 2,[1/(3), -1/(3), 1/(3), 1/(3)]);
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plot (.434*log(x),x,1,10).show(xmin=0,xmax=10,ymin=0.,ymax=1)
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a=pi;p=3;
y=plot((a/pi)^(1/4)*cos(p*x)*exp(-a*x^2/2),-3,3)
show(y)
#cos(px/h') (α/π )1/4 exp ( - αx2 /2)
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var('x,n '); n=9;
f=n/(pi*(1+(n*x)^2));
integral(f,x,-10,10)
2*arctan(90)/pi 2*arctan(90)/pi |
R=RealField(30)
R(2*arctan(90)/pi)
0.99292674 0.99292674 |
var('x,mu,sigma');sigma=.001;mu=0;eta=20;
gauss=(1/sigma)*(1/(2*pi)^(1/2))*exp(-(x-mu)^2/(2*sigma^2));
y=plot(-eta*gauss,x,-3*sigma,3*sigma)
show(y)
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var('x,mu,sigma');sigma=.001;mu=0;eta=20;
gauss=(1/sigma)*(1/(2*pi)^(1/2))*exp(-(x-mu)^2/(2*sigma^2));
integral(gauss,x,-oo,oo)
0.707106781187*sqrt(2) 0.707106781187*sqrt(2) |
var('t');
y=plot(10*t*exp(-t^2),0,2.5)
show(y)
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#f=2*x^2+1 ;
#integral(f,x,2,5)
K=81, m=0.6 ,
v=(2*K/m)^(1/2)
Traceback (click to the left of this block for traceback) ... ValueError: need more than 1 value to unpack Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "/home/sage/sagenb/sage_notebook/worksheets/reibaretti/11/code/6.py", line 8, in <module>
K=_sage_const_81 , m=_sage_const_0p6 ,
ValueError: need more than 1 value to unpack
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K=81; m=0.6 ;
n(2*K/m)^(1/2)
16.4316767251550 16.4316767251550 |
var('t')
x=5*t^2+3*t;
v=diff(x,t);
10*t + 3 10*t + 3 |
var('x');
y=5*x^2;v=diff(y,x)
x=2;
show(v)
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r = 4
area = pi*r^2
area,area.n()
(16*pi, 50.2654824574367) (16*pi, 50.2654824574367) |
var('x');
y=5*x^2;
v=diff(y,x)
v
10*x 10*x |
v=10*x
x=3
v,v.n()
(30, 30.0000000000000) (30, 30.0000000000000) |
var('r');
f=27- 18*r + 2*r^2 ;
y=plot(f,r,0,10)
show (y)
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f=r^3*exp(-r);
exp(r)*diff(f,r,3)
-(r^3*e^(-r) - 9*r^2*e^(-r) + 18*r*e^(-r) - 6*e^(-r))*e^r -(r^3*e^(-r) - 9*r^2*e^(-r) + 18*r*e^(-r) - 6*e^(-r))*e^r |
#( 27- 18 r + 2 r2 )2 exp( -2r/3) 4π r2
f=(27- 18*r + 2*r^2)^2*4*pi*r^2*exp(-2*r/3);
integral(f,r,0,oo)
19683*pi 19683*pi |
81^2*3
19683 19683 |
#{ 81(3 π) }-1/2 ( 27- 18 r + 2 r2 )2 exp( -2r/3)
f=(81*3*pi)^(-1/2)*( 27- 18*r + 2*r^2)*exp( -r/3)
plot(f,r,0,20)
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u=var('u');
#(1/2)*( 5*x^3 - 3*x)
#f=(1/2)*(5*(2*u-1)^3 -3*(2*u-1) );
f1=5*(2*u-1)^3;
f1
5*(2*u - 1)^3 5*(2*u - 1)^3 |
u=var('u');
p2(u)= 6*u^2 -6*u + 1; p3(u)=20*u^3 -30*u^2 +12*u -1;
integral(p2(u)*p3(u),u,0,1)
0 0 |
var('a');Z=2;En=(5/4)*(a^2/2 -Z*a); E1= 2*(a^2/2 -Z*a);E2= 2*(1/4)*(a^2/2 -Z*a);
e=(5/4)*(a^2 /2 -Z*a) + (17/81)*a + (8.936E-2* a)^2 /(En -E1)+ (8.582e-3*a)^2/(En -E2);
plot(e,a,1.4,2.2)
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integral(1/(1+x^2),x)
arctan(x) arctan(x) |
f(x)=exp(-x^2);
integral(f(x),x,-oo,oo)
sqrt(pi) sqrt(pi) |
n( pi^(1/2) )
1.77245385090552 1.77245385090552 |
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