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Anal3 2017 2. félév - 3. gyakorlat. Segédszámítások

Teljes Matlab script (és live script) kiegészítő függvényekkel.
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Author: Péter Polcz ppolcz@gmail.com

Created on 2017. September 25.

2. feladat, felületintegrál

$s\left(u,v\right)=\left(\begin{array}{c}u\\v\\1-u-v\end{array}\right)$, ahol $u\in \left\lbrack 0,1\right\rbrack$ és $v\in \left\lbrack 0,1-u\right\rbrack$.

F = [ 1 ; 0 ; 0 ];

syms u v real
s = [ u ; v ; 1-u-v ];

dS = cross(diff(s,u),diff(s,v))
Integrand = double(dot(F,dS));
f_Integrand = @(u,v) Integrand + zeros(size(u));
integral2(f_Integrand,0,1,0,@(u) 1-u)

Output:

dS =
 1
 1
 1
ans =
    0.5000

A felület, ami mentén integráltam:

s = matlabFunction(s(3));
[u,v] = meshgrid(0:0.01:1);
z = s(u,v);
z(u+v>1) = NaN;
surf(u,v,z), axis vis3d; shading interp
view([75.300 41.200])
xlabel('x')
ylabel('y')
zlabel('z')

4. feladat

syms u v real
syms x y z real
r = [x;y;z];

f = x + y + z;
s = [
    u+v
    u-v
    u
    ];
fs = subs(f,r,s)
dS = norm(cross(diff(s,u),diff(s,v)))
I = fs*dS;

Output:

fs =
3*u
dS =
6^(1/2)

Symbolical integration:

result = int(int(I,u,[0,1]),v,[0,1]), double(result)

Output:

result =
(3*6^(1/2))/2
ans =
    3.6742

Numerical integration:

result = integral2(matlabFunction(I,'vars',{u v}),0,1,0,1)

Output:

result =
    3.6742

6. feladat

syms u v R real
syms x y z real
r = [x;y;z];

F = [y;x;z];
s = [
    R*cos(u)*sin(v)
    R*sin(u)*sin(v)
    R*cos(v)
    ];

dS = cross(diff(s,u),diff(s,v))
Fs = subs(F,r,s)
I = simplify(dot(Fs,dS))
result = int(int(I,u,0,2*pi),v,0,pi/2)

Output:

dS =
                                      -R^2*cos(u)*sin(v)^2
                                      -R^2*sin(u)*sin(v)^2
 - R^2*cos(u)^2*cos(v)*sin(v) - R^2*cos(v)*sin(u)^2*sin(v)
Fs =
 R*sin(u)*sin(v)
 R*cos(u)*sin(v)
        R*cos(v)
I =
-R^3*sin(v)*(2*cos(u)*sin(u) + cos(v)^2 - 2*cos(u)*cos(v)^2*sin(u))
result =
-(2*pi*R^3)/3

9. feladat (felületintegrál)

syms u v R real
syms x y z real
r = [x;y;z];

F = [x;y;z]/R;
F = [x;2*y;5*z];
s = [
    R*cos(u)*sin(v)
    R*sin(u)*sin(v)
    R*cos(v)
    ];

dS = cross(diff(s,u),diff(s,v))
Fs = subs(F,r,s)
I = simplify(dot(Fs,dS))
result = int(int(I,u,0,2*pi),v,0,pi)

Output:

dS =
                                      -R^2*cos(u)*sin(v)^2
                                      -R^2*sin(u)*sin(v)^2
 - R^2*cos(u)^2*cos(v)*sin(v) - R^2*cos(v)*sin(u)^2*sin(v)
Fs =
   R*cos(u)*sin(v)
 2*R*sin(u)*sin(v)
        5*R*cos(v)
I =
-R^3*sin(v)*(sin(u)^2*sin(v)^2 - 4*sin(v)^2 + 5)
result =
-(32*pi*R^3)/3

9. feladat (térfogati integrál)

syms theta phi rho real
syms x y z real
r = [x;y;z];
R = 2;

F = [x;2*y;5*z];
divF = double(divergence(F,r));

Phi = [
    rho*cos(theta)*sin(phi)
    rho*sin(theta)*sin(phi)
    rho*cos(phi)
    ];

Descartes koordinátákban kiszámolva:

$$\int_{-R}^R \int_{-\sqrt{R^2-x^2}}^{\sqrt{R^2-x^2}} \int_{-\sqrt{R^2-x^2-y^2}}^{\sqrt{R^2-x^2-y^2}} \nabla F(x,y,z) \,\mathrm{d} z \, \mathrm{d} y \, \mathrm{d}x$$

integral3(@(x,y,z) divF + zeros(size(x)),...
    -R,R,...
    @(x) -sqrt(R^2-x.^2),@(x) sqrt(R^2-x.^2),...
    @(x,y) -sqrt(R^2-x.^2-y.^2),@(x,y) sqrt(R^2-x.^2-y.^2))

Output:

ans =
  268.0826

Gömbi koordinátákban kiszámolva:

$$\int_0^R \int_0^\pi \int_0^{2\pi} \nabla F(\Phi(\rho,\vartheta,\varphi)) \cdot \Big|\det J(\rho,\vartheta,\varphi)\Big| \,\mathrm{d} \varphi\, \mathrm{d}\vartheta \, \mathrm{d}\rho$$

ahol $J(\rho,\vartheta,\varphi) = \mathrm{D}\,\Phi(\rho,\vartheta,\varphi)$ a $\Phi:\mathbb{R}^3 \to \mathbb{R}^3$ leképzés Jacobi mátrixa.

J = jacobian(Phi,[rho;theta;phi])
detJ = abs(simplify(det(J)))
Integrand = subs(divF,r,Phi)*detJ;
f_Integrand = matlabFunction(Integrand,'vars',[rho;theta;phi]);
integral3(f_Integrand,0,R,0,2*pi,0,pi)

Output:

J =
[ cos(theta)*sin(phi), -rho*sin(phi)*sin(theta), rho*cos(phi)*cos(theta)]
[ sin(phi)*sin(theta),  rho*cos(theta)*sin(phi), rho*cos(phi)*sin(theta)]
[            cos(phi),                        0,           -rho*sin(phi)]
detJ =
rho^2*abs(sin(phi))
ans =
  268.0826

Ellenőrzésképpen

32*pi*R^3 / 3

Output:

ans =
  268.0826

Published with MATLAB® R2017b, and sanitarized by BeautifulSoup (Python script using BSoup is written by Polcz).