Cálculo Infinitesimal III. Martes de 7:am a 9:am en el 67-411. Jueves de 9:am a 11:am en el 54-104.
Problemas:
(Texto Guia) Marsden J.: Vector Calculus 3 ◦ edition, W.H. Freeman and Company, 1988.
Cap 1.4 5,7,22.
Cap 2.1 3, 22-26.
Cap 2.2 9, 11, 33.
Cap 2.3 3,4,7,10,13,21,25.
Cap 2.4 2,4,6,17,24.
Cap 2.5 6,7,15,35.
Cap 2.6 10, 17, 21, 31, 32.
Cap 3.1: 3,6,16,26.
Cap 3.2: 1,5,7.
Cap 3.3: 2,7,9,20,25,29.
Cap 3.4: 4,10,16,21,28.
Cap 4.1: 11,20,24.
Cap 4.2: 15,16,17,20.
Cap 4.3: 9,11,13,21,22.
Cap 4.4: 9,13,25,30.
Cap 5.1: 1,3,5,8.
Cap 5.2: 3,5,10,13,14,15.
Cap 5.3: 1,4,5,8,15.
Cap 6.1: 11,14.
Cap 6.2: 2,13,15,16,17,25,30.
Cap 7.1: 5,7,11,25.
Cap 7.2: 4,5,6, 17,18.
Cap 7.3: 7,8,11,19.
Cap 7.4: 4,6,8.
Cap 7.5: 1,4,9,11.
Producto cruz y producto punto:
sage: v = vector(RR, [1.2, 3.5, 4.6])sage: w = vector(RR, [1.7,-2.3,5.2])sage: v*w
sage: v.cross_product(w)
Líneas y planos:
sage: # designed with intersection at t = 2, i.e. (7, -10, 3)sage: var('t, x, y')(t, x, y)sage: line = parametric_plot3d([2*t+3, -3*t-4, t+1], (t, 0, 4),color='red')sage: plane = plot3d((1/5)*(-12+x-2*y), (x, 4, 10), (y, -13,-7), opacity=0.5)sage: intersect=point3d([7,-10,3],color='black',size=30)sage: line+plane+intersect
Gráficas en coordenadas polares:
sage: polar_plot(2 + 2*cos(x), (x, 0, 2*pi), color=hue(0.5), thickness=4)
Gráficas en coordenadas cilíndricas:
sage: T = Cylindrical('height', ['radius', 'azimuth']) sage: r, theta, z = var('r theta z')sage: T.transform(radius=r, azimuth=theta, height=z)) sage: plot3d(1, (r, 0, 3), (theta, 0, pi), transformation=T)
Funciones vector valuadas:
sage: var('t')tsage: r=vector((2*t-4, t^2, (1/4)*t^3))sage: r
sage: velocity = r.diff(t) # velocity(t) = list(r.diff(t)) also would worksage: velocity
sage: T=velocity/velocity.norm()
sage: arc_length = numerical_integral(velocity.norm(), 0,1)sage: arc_length
sage: x,y,z=var('x y z')sage: plot_vector_field3d((x*cos(z),-y*cos(z),sin(z)), (x,0,pi), (y,0,pi), (z,0,pi),colors=['red','green','blue'])
Funciones de varias variables:
sage: g(x,y)=e^-x*sin(y)sage: contour_plot(g, (x, -2, 2), (y, -4*pi, 4*pi), cmap = 'Blues', contours=10, colorbar=True)
Derivadas parciales:
sage: f(x, y) = x^2 + x*y + y^2 - 6*x + 2
sage: fx(x,y)= f.diff(x)sage: fy(x,y) = f.diff(y)sage: fx; fy
sage: f.gradient()
sage: solve([fx==0, fy==0], (x, y))
sage: H = f.hessian()
sage: plot3d(f,(x,-5,5),(y,-5,5))+point((4,-2,f(4,-2)),color='red',size=20)
Campos Vectoriales:
sage: x,y = var('x,y')sage: plot_vector_field((x,-y), (x,-10,10), (y,-10,10))
Integrales múltiples:
sage: f(x,y)=x^2*ysage: # integrate in the order dy dxsage: f(x,y).integrate(y,0,4*x).integrate(x,0,3)
sage: # another way to integrate, and in the opposite order toosage: integrate( integrate(f(x,y), (x, y/4, 3)), (y, 0, 12) )
sage: var('u v')(u, v)sage: surface = plot3d(f(x,y), (x, 0, 3.2), (y, 0, 12.3), color = 'blue', opacity=0.3)sage: domain = parametric_plot3d([3*u, 4*(3*u)*v,0], (u, 0, 1), (v, 0,1), color = 'green', opacity = 0.75)sage: image = parametric_plot3d([3*u, 4*(3*u)*v, f(3*u, 12*u*v)], (u, 0, 1), (v, 0,1), color = 'green', opacity = 1.00)sage: surface+domain+image
Gráficas paramétricas de superficies:
sage: u, v = var('u,v')sage: parametric_plot3d((v*cos(u), v*sin(u), u), (u,0,2*pi), (v,-pi,pi))
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