file: anal3_2het.m
author: Polcz Péter <ppolcz@gmail.com>
Created on 2016.09.30. Friday, 12:57:31
Contents
2. het HF1
syms R u v x y z real
r = [x;y;z];
C = [
R*cos(v)*cos(u)
R*cos(v)*sin(u)
R*sin(v)
];
J = jacobian(C,[R;u;v])
simplify(det(J))
C = [
R*sin(v)*cos(u)
R*sin(v)*sin(u)
R*cos(v)
];
J = jacobian(C,[R;u;v])
simplify(det(J))
assume([R > 0, in(R, 'real')])
assume([0 <= v < pi, in(v, 'real')])
dS_vec = simplify(-vekanal_cross(diff(C,u), diff(C,v)))
dS = simplify(norm(dS_vec))
fprintf('%s = %s %s \n= %s %s\n', '\mathrm{d}\vec{S}', ...
pcz_latex(R*sin(v)), ...
pcz_latex(simplify(dS_vec / R / sin(v))), ...
pcz_latex(R*sin(v)), ...
pcz_latex(r))
F = [y;x;z];
Integrand = simplify(subs(F,r,C)' * dS_vec);
pretty(Integrand)
expand(Integrand)
Integrand_u = int(Integrand, u, 0, 2*pi)
Integrand_uv = int(Integrand_u, v, 0, pi/2)
disp 'Tehat a fluxus: '
pretty(Integrand_uv)
J =
[ cos(u)*cos(v), -R*cos(v)*sin(u), -R*cos(u)*sin(v)]
[ cos(v)*sin(u), R*cos(u)*cos(v), -R*sin(u)*sin(v)]
[ sin(v), 0, R*cos(v)]
ans =
R^2*cos(v)
J =
[ cos(u)*sin(v), -R*sin(u)*sin(v), R*cos(u)*cos(v)]
[ sin(u)*sin(v), R*cos(u)*sin(v), R*cos(v)*sin(u)]
[ cos(v), 0, -R*sin(v)]
ans =
-R^2*sin(v)
dS_vec =
R^2*cos(u)*sin(v)^2
R^2*sin(u)*sin(v)^2
(R^2*sin(2*v))/2
dS =
R^2*sin(v)
\mathrm{d}\vec{S} = R \sin(v) \left[\begin{array}{c} R \cos(u) \sin(v) \\ R \sin(u) \sin(v) \\ R \cos(v) \end{array}\right]
= R \sin(v) r = \left[\begin{array}{c} x \\ y \\ z \end{array}\right]
3 2 2
R sin(v) (cos(v) + 2 cos(u) sin(u) sin(v) )
ans =
R^3*cos(v)^2*sin(v) + 2*R^3*cos(u)*sin(u)*sin(v)^3
Integrand_u =
-2*pi*R^3*sin(v)*(sin(v)^2 - 1)
Integrand_uv =
(2*pi*R^3)/3
Tehat a fluxus:
3
2 pi R
-------
3
2. het HF2
syms R u v x y z real
r = [x;y;z];
C = [
R*cos(u)
R*sin(u)
v
];
J = jacobian(C,[R;u;v])
simplify(det(J))
assume([R > 0, in(R, 'real')])
assume([0 <= v < 4, in(v, 'real')])
dS_vec = simplify(vekanal_cross(diff(C,u), diff(C,v)))
dS = simplify(norm(dS_vec))
dS23_vec = simplify(vekanal_cross(diff(C,u), diff(C,R)))
dS23 = simplify(norm(dS23_vec))
fprintf('%s = %s %s \n= %s %s\n', '\mathrm{d}\vec{S}', ...
pcz_latex(R*sin(v)), ...
pcz_latex(simplify(dS_vec / R / sin(v))), ...
pcz_latex(R*sin(v)), ...
pcz_latex(r))
F = [x^2;y^2;0];
Integrand = simplify(subs(F,r,C)' * dS_vec);
pretty(Integrand)
expand(Integrand)
Integrand_u = int(Integrand, u, 0, 2*pi)
Integrand_uv = int(Integrand_u, v, 0, 4)
disp 'Tehat a fluxus: '
pretty(Integrand_uv)
J =
[ cos(u), -R*sin(u), 0]
[ sin(u), R*cos(u), 0]
[ 0, 0, 1]
ans =
R
dS_vec =
R*cos(u)
R*sin(u)
0
dS =
R
dS23_vec =
0
0
-R
dS23 =
R
\mathrm{d}\vec{S} = R \sin(v) \left[\begin{array}{c} \frac{\cos(u)}{\sin(v)} \\ \frac{\sin(u)}{\sin(v)} \\ 0 \end{array}\right]
= R \sin(v) r = \left[\begin{array}{c} x \\ y \\ z \end{array}\right]
3 3 3
R (cos(u) + sin(u) )
ans =
R^3*cos(u)^3 + R^3*sin(u)^3
Integrand_u =
0
Integrand_uv =
0
Tehat a fluxus:
0
2. het D3*
syms x y z real
r = [x;y;z];
f = 1 / norm(r);
F = simplify(jacobian(f,r))'
pretty(F)
vekanal_div(F)
simplify(vekanal_div(F))
C = [
R*sin(v)*cos(u)
R*sin(v)*sin(u)
R*cos(v)
];
assume([R > 0, in(R, 'real')])
assume([0 <= v < pi, in(v, 'real')])
dS_vec = simplify(-vekanal_cross(diff(C,u), diff(C,v)))
dS = simplify(norm(dS_vec))
fprintf('%s = %s %s \n= %s %s\n', '\mathrm{d}\vec{S}', ...
pcz_latex(R*sin(v)), ...
pcz_latex(simplify(dS_vec / R / sin(v))), ...
pcz_latex(R*sin(v)), ...
pcz_latex(r))
Integrand = simplify(subs(F,r,C)' * dS_vec);
disp 'Integrand:'
pretty(Integrand)
Integrand_u = int(Integrand, u, 0, 2*pi)
Integrand_uv = int(Integrand_u, v, 0, pi)
F =
-x/(x^2 + y^2 + z^2)^(3/2)
-y/(x^2 + y^2 + z^2)^(3/2)
-z/(x^2 + y^2 + z^2)^(3/2)
/ x \
| - ----------------- |
| 2 2 2 3/2 |
| (x + y + z ) |
| |
| y |
| - ----------------- |
| 2 2 2 3/2 |
| (x + y + z ) |
| |
| z |
| - ----------------- |
| 2 2 2 3/2 |
\ (x + y + z ) /
ans =
(3*x^2)/(x^2 + y^2 + z^2)^(5/2) - 3/(x^2 + y^2 + z^2)^(3/2) + (3*y^2)/(x^2 + y^2 + z^2)^(5/2) + (3*z^2)/(x^2 + y^2 + z^2)^(5/2)
ans =
0
dS_vec =
R^2*cos(u)*sin(v)^2
R^2*sin(u)*sin(v)^2
(R^2*sin(2*v))/2
dS =
R^2*sin(v)
\mathrm{d}\vec{S} = R \sin(v) \left[\begin{array}{c} R \cos(u) \sin(v) \\ R \sin(u) \sin(v) \\ R \cos(v) \end{array}\right]
= R \sin(v) r = \left[\begin{array}{c} x \\ y \\ z \end{array}\right]
Integrand:
-sin(v)
Integrand_u =
-2*pi*sin(v)
Integrand_uv =
-4*pi