.. -*- coding: utf-8 -*- .. linkall Sage Quickstart for Multivariable Calculus ========================================== This Sage _ quickstart tutorial was developed for the MAA PREP Workshop "Sage: Using Open\-Source Mathematics Software with Undergraduates" (funding provided by NSF DUE 0817071). It is licensed under the Creative Commons Attribution\-ShareAlike 3.0 license (CC BY\-SA _). Because much related material was covered in the calculus tutorial, this quickstart is designed to: - Give concrete examples of some things not covered there, without huge amounts of commentary, and - Remind the reader of a few of the things in that tutorial. Vector Calculus ---------------- In Sage, vectors are primarily linear algebra objects, but they are slowly becoming simultaneously analytic continuous functions. Dot Product, Cross Product ~~~~~~~~~~~~~~~~~~~~~~~~~~ :: sage: v = vector(RR, [1.2, 3.5, 4.6]) sage: w = vector(RR, [1.7,-2.3,5.2]) sage: v*w 17.9100000000000 :: sage: v.cross_product(w) (28.7800000000000, 1.58000000000000, -8.71000000000000) Lines and Planes ~~~~~~~~~~~~~~~~ Intersect :math:x-2y-7z=6 with :math:\frac{x-3}{2}=\frac{y+4}{-3}=\frac{z-1}{1}. One way to plot the equation of a line in three dimensions is with parametric_plot3d. :: 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 Graphics3d Object Vector\-Valued Functions ~~~~~~~~~~~~~~~~~~~~~~~~ We can make vector-valued functions and do the usual analysis with them. :: sage: var('t') t sage: r=vector((2*t-4, t^2, (1/4)*t^3)) sage: r (2*t - 4, t^2, 1/4*t^3) :: sage: r(t=5) (6, 25, 125/4) The following makes the derivative also a vector\-valued expression. :: sage: velocity = r.diff(t) # velocity(t) = list(r.diff(t)) also would work sage: velocity (2, 2*t, 3/4*t^2) Currently, this expression does *not* function as a function, so we need to substitute explicitly. :: sage: velocity(t=1) (2, 2, 3/4) :: sage: T=velocity/velocity.norm() :: sage: T(t=1).n() (0.683486126173409, 0.683486126173409, 0.256307297315028) Here we compute the arclength between :math:t=0 and :math:t=1 by integrating the normalized derivative. As pointed out in the :doc:calculus tutorial <../Calculus>, the syntax for numerical_integral is slightly nonstandard -- we just put in the endpoints, not the variable. :: sage: arc_length = numerical_integral(velocity.norm(), 0,1) sage: arc_length (2.3169847271197814, 2.572369791753217e-14) We can also plot vector fields, even in three dimensions. :: 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']) Graphics3d Object If we know a little vector calculus, we can also do line integrals. Here, based on an example by Ben Woodruff of BYU-Idaho, we compute a number of quantities in a physical three-dimensional setting. Try to read through the entire code. We make an auxiliary function that will calculate :math:\int (\text{integrand})\, dt. We'll use our formulas relating various differentials to :math:dt to easily specify the integrands. :: sage: density(x,y,z)=x^2+z sage: r=vector([3*cos(t), 3*sin(t),4*t]) sage: tstart=0 sage: tend=2*pi sage: t = var('t') sage: t_range=(t,tstart,tend) sage: def line_integral(integrand): ....: return RR(numerical_integral((integrand).subs(x=r[0], y=r[1],z=r[2]), tstart, tend)[0]) sage: _ = var('x,y,z,t') sage: r_prime = diff(r,t) sage: ds=diff(r,t).norm() sage: s=line_integral(ds) sage: centroid_x=line_integral(x*ds)/s sage: centroid_y=line_integral(y*ds)/s sage: centroid_z=line_integral(z*ds)/s sage: dm=density(x,y,z)*ds sage: m=line_integral(dm) sage: avg_density = m/s sage: moment_about_yz_plane=line_integral(x*dm) sage: moment_about_xz_plane=line_integral(y*dm) sage: moment_about_xy_plane=line_integral(z*dm) sage: center_mass_x = moment_about_yz_plane/m sage: center_mass_y = moment_about_xz_plane/m sage: center_mass_z = moment_about_xy_plane/m sage: Ix=line_integral((y^2+z^2)*dm) sage: Iy=line_integral((x^2+z^2)*dm) sage: Iz=line_integral((x^2+y^2)*dm) sage: Rx = sqrt(Ix/m) sage: Ry = sqrt(Iy/m) sage: Rz = sqrt(Iz/m) Finally, we can display everything in a nice :ref:table . Recall that we use the r"stuff" syntax to indicate "raw" strings so that backslashes from LaTeX won't cause trouble. .. skip :: sage: html.table([ ....: [r"Density $\delta(x,y)$", density], ....: [r"Curve $\vec r(t)$",r], ....: [r"$t$ range", t_range], ....: [r"$\vec r'(t)$", r_prime], ....: [r"$ds$, a little bit of arclength", ds], ....: [r"$s$ - arclength", s], ....: [r"Centroid (constant density) $\left(\frac{1}{m}\int x\,ds,\frac{1}{m}\int y\,ds, \frac{1}{m}\int z\,ds\right)$", (centroid_x,centroid_y,centroid_z)], ....: [r"$dm=\delta ds$ - a little bit of mass", dm], ....: [r"$m=\int \delta ds$ - mass", m], ....: [r"average density $\frac{1}{m}\int ds$" , avg_density.n()], ....: [r"$M_{yz}=\int x dm$ - moment about $yz$ plane", moment_about_yz_plane], ....: [r"$M_{xz}=\int y dm$ - moment about $xz$ plane", moment_about_xz_plane], ....: [r"$M_{xy}=\int z dm$ - moment about $xy$ plane", moment_about_xy_plane], ....: [r"Center of mass $\left(\frac1m \int xdm, \frac1m \int ydm, \frac1m \int z dm\right)$", (center_mass_x, center_mass_y, center_mass_z)], ....: [r"$I_x = \int (y^2+z^2) dm$", Ix],[r"$I_y=\int (x^2+z^2) dm$", Iy],[mp(r"$I_z=\int (x^2+y^2)dm$"), Iz], ....: [r"$R_x=\sqrt{I_x/m}$", Rx],[mp(r"$R_y=\sqrt{I_y/m}"), Ry],[mp(r"$R_z=\sqrt{I_z/m}"),Rz] ....: ]) Functions of Several Variables ------------------------------- This connects directly to other issues of multivariable functions. How to view these was mostly addressed in the various plotting tutorials. Here is a reminder of what can be done. :: sage: # import matplotlib.cm; matplotlib.cm.datad.keys() sage: # 'Spectral', 'summer', 'blues' 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) Graphics object consisting of 1 graphics primitive Partial Differentiation ~~~~~~~~~~~~~~~~~~~~~~~ The following exercise is from Hass, Weir, and Thomas, University Calculus, Exercise 12.7.35. This function has a local minimum at :math:(4,-2). :: sage: f(x, y) = x^2 + x*y + y^2 - 6*x + 2 Quiz: Why did we *not* need to declare the variables in this case? :: sage: fx(x,y)= f.diff(x) sage: fy(x,y) = f.diff(y) sage: fx; fy (x, y) |--> 2*x + y - 6 (x, y) |--> x + 2*y :: sage: f.gradient() (x, y) |--> (2*x + y - 6, x + 2*y) :: sage: solve([fx==0, fy==0], (x, y)) [[x == 4, y == -2]] :: sage: H = f.hessian() sage: H(x,y) [2 1] [1 2] And of course if the Hessian has positive determinant and :math:f_{xx} is positive, we have a local minimum. .. skip :: sage: html("$f_{xx}=%s$"%H(4,-2)[0,0]) sage: html("$D=%s$"%H(4,-2).det()) Notice how we were able to use many things we've done up to now to solve this. - Matrices - Symbolic functions - Solving - Differential calculus - Special formatting commands - And, below, plotting! :: sage: plot3d(f,(x,-5,5),(y,-5,5))+point((4,-2,f(4,-2)),color='red',size=20) Graphics3d Object Multiple Integrals and More ~~~~~~~~~~~~~~~~~~~~~~~~~~~ Naturally, there is lots more that one can do. :: sage: f(x,y)=x^2*y sage: # integrate in the order dy dx sage: f(x,y).integrate(y,0,4*x).integrate(x,0,3) 1944/5 sage: # another way to integrate, and in the opposite order too sage: integrate( integrate(f(x,y), (x, y/4, 3)), (y, 0, 12) ) 1944/5 :: 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 Graphics3d Object Quiz: why did we need to declare variables this time?