#you can call your variables anything you like; here I've named them to mostly match the traditional names
rho,theta,phi = var('rho,theta,phi')

#plotting 1: if you have rho as a function of theta and phi, i.e., rho = stuff in theta and phi, you can use the command 
#spherical_plot3d(your function,(theta,theta min, theta max),(phi, phi min, phi max))
#for example, the sphere of radius 3, rho = 3, theta from 0 to 2pi, phi from 0 to pi:
spherical_plot3d(3,(theta,0,2/3*pi),(phi,0,pi))

#or a teardrop
spherical_plot3d(exp(-phi),(theta,0,2*pi),(phi,0,pi))

# or anything else of the form rho = stuff in phi, theta; I think this one looks like a seashell
spherical_plot3d(phi * theta, (theta, 0, 2*pi), (phi, 0, pi))

︠85872ed7-ce4c-45bd-bf70-54ea27e104e2︠
#if you want to plot something other than rho as a function of theta and phi, we have to define the spherical transformation to match this case

#the normal setup is
# T = Spherical ('dependent variable', ['independent variable', 'independent variable'])

︠d950ab49-e682-45ca-bb39-182289cc8c4ds︠
# for instance, if we want to plot the cone in spherical coordinates, we want phi constant, and rho and theta as independent variables
# We use the transformation
T1 = Spherical('inclination',['radius','azimuth'])
# and apply the transformation
T1.transform(radius=rho, azimuth=theta, inclination=phi)
#Sage uses radius for rho, azimuth for theta, and inclination for phi, and we must list use the variables in this order when plotting

#for example, we plot the cone, using the plot3d command, with the transformation:
plot3d(pi/2,(rho,0,1),(theta,0,2*pi),transformation=T1)

#You can do the same thing for theta = stuff in rho and phi (or theta constant) by using a transformation of the form
T2= Spherical('azimuth', ['radius', 'inclination'])
T2.transform(radius=rho, azimuth=theta, inclination=phi)
# and a wedge of constant theta:
plot3d(pi/3,(rho,0,1),(phi,pi/4,3*pi/4),transformation = T2)

# equivalent to spherical_plot3d, we could use plot3d with the transformation 
#T = Spherical('radius', ['azimuth', 'inclination'])
#T.transform(radius=rho, azimuth=theta, inclination=phi)
#plot3d(your function of rho,(theta,theta min, theta max),(phi, phi min, phi max),transformation = T)


︠bc25cf83-5a6d-4139-92cc-829533841a06︠
#setting up plots in cylindrical coordinates is similar, but we use 
#T = Cylindrical('height', ['radius', 'azimuth'])
#instead
︠eadce050-6cda-4d38-9523-e8a87adc02e0︠
# Integration in Spherical Coordinates

#no special transformations required for integration
#just remember that in spherical coordinates we have an extra rho^2 * sin(phi) factor 
#and you can name your variables r,t,and p if you like
︠bc47ace3-f26c-49b2-add3-22d027eb6e87︠
#example: integrating over the unit sphere:
r,t,p = var('r,t,p')
integrate(integrate(integrate(1*r^2*sin(p),r,0,1),t,0,2*pi),p,0,pi)

#or you can write out the traditional variable names:
integrate(integrate(integrate(rho^2*sin(phi),rho,0,1),theta,0,2*pi),phi,0,pi)

︠ba91c01a-feb2-4e0b-b481-75057c34fa77︠

︠da40c870-1997-4e53-9dcf-c3d05cad03d0︠

︠6d6d9343-05c1-46d5-ba09-6cf78c72a903︠

︠c124cf93-0af7-47fd-984e-cebfe07ceb01︠
︠4211f1ed-b577-46a9-8999-6a6f9aca2fe5︠

︠8099d834-5d10-439b-b42c-16f841c07941︠

︠468a9dcd-1a5b-4774-a15a-da33671f7684︠

︠ffb6d9a6-a3a3-450b-a0f5-7ad73daeb166︠