PREP Quickstart, Multivariable Calculus Sage Quickstart for ...

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2/20/13 PREP Quickstart, Multivariable Calculus -- Sageds=diff(r,t).norm()s=line_integral(ds)centroid_x=line_integral(x*ds)/scentroid_y=line_integral(y*ds)/scentroid_z=line_integral(z*ds)/sdm=density(x,y,z)*dsm=line_integral(dm)avg_density = m/smoment_about_yz_plane=line_integral(x*dm)moment_about_xz_plane=line_integral(y*dm)moment_about_xy_plane=line_integral(z*dm)center_mass_x = moment_about_yz_plane/mcenter_mass_y = moment_about_xz_plane/mcenter_mass_z = moment_about_xy_plane/mIx=line_integral((y^2+z^2)*dm)Iy=line_integral((x^2+z^2)*dm)Iz=line_integral((x^2+y^2)*dm)Rx = sqrt(Ix/m)Ry = sqrt(Iy/m)Rz = sqrt(Iz/m)Let's display everything in a nice table.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) $\left(\frac{1}{m}\intx\,ds,\frac{1}{m}\int y\,ds, \frac{1}{m}\int z\,ds\right)$",(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,test.sagenb.org/home/Carlos_Rodriguez/20/print 4/9
2/20/13 PREP Quickstart, Multivariable Calculus -- Sagecenter_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]])Traceback (click to the left of this block for traceback)...NameError: name 'mp' is not definedWe can also plot vector fields, even in three dimensions.x,y,z=var('x y z')plot_vector_field3d((x*cos(z),-y*cos(z),sin(z)), (x,0,pi),(y,0,pi), (z,0,pi),colors=['red','green','blue'])Toggle Advanced ControlsHelp for Jmol 3-D viewerFunctions of Several VariablesThis 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 betest.sagenb.org/home/Carlos_Rodriguez/20/print 5/9
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