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Not sure if this is user error: I am trying to evaluate the following integral:

var('t')
f = sqrt((e^(i * t) - 1) * (e^(-i*t) - 1))
show(f.integrate(t, 0, 2 * pi))

Sagemath returns 0, but WolframAlpha returns 8. Have I mis-specified the problem, or is this a bug?

In the following expression:

var('z')

u = function('u')(z)
v = function('v')(z)

f = u + i * v

show(f.diff(z).conjugate().expand().simplify_full())
# conjugate(diff(u(z), z)) - I*conjugate(diff(v(z), z))

I would like Sage to make the simplification conjugate(diff(u(z), z)) = diff(u(z), z) because $u, v$ are intended to be real-valued. But my first attempt:

assume(u, 'real')

didn't work:

TypeError: self (=u(z)) must be a relational expression

What is the right way to write this assumption?

Incorrect result for complex integral Not sure if this is user error: I am trying to evaluate the following integral: v

Assume a function is real-valued In the following expression: var('z') u = function('u')(z) v = function('v')(z) f =

Assume a function is real-valued In the following expression: var('z') u = function('u')(z) v = function('v')(z) f =

Assume a function is real-valued In the following expression: var('z') u = function('u')(z) v = function('v')(z) f =

My About tab says: ┌────────────────────────────────────────────────────────────────────┐ │ SageMath version 9.1, Relea

Incorrect result for integral of (cos z) / z I am confused by the following result: I'd like to integrate cos(z) / z on

I'd like to integrate dx /\ dy over the unit square. Naively, I would expect the following to work:

E.<x,y> = EuclideanSpace(2) 

phi = E.diff_form(2)
phi[1, 2] = 1

show(integrate(integrate(phi, x, 0, 1), y, 0, 1))

but it fails with:

TypeError: unable to convert 2-form on the Euclidean plane E^2 to a symbolic expression

and I can't find anything about integration in the DiffFormFreeModule documentation. What is the right way to do this?

Integrating differential forms I'd like to integrate dx /\ dy over the unit square. Naively, I would expect the followin

Integrating differential forms I'd like to integrate dx /\ dy over the unit square. Naively, I would expect the followin

I'd like to compute an exterior derivative in spherical coordinates. So far I have the following:

E.<x,y,z> = EuclideanSpace(3)
Ec.<r,theta,phi> = E.spherical_coordinates()

EcM = Ec.manifold()

EcM.set_default_frame(EcM.spherical_frame())
EcM.set_default_chart(EcM.spherical_coordinates())

F = function('F')
show(F)
F_1 = function('F_r')(r, theta, phi)
F_2 = function('F_theta')(r, theta, phi)
F_3 = function('F_phi')(r, theta, phi)


psi = EcM.diff_form(2, 'psi')

psi[2, 3] = F_1
psi[1, 3] = -F_2
psi[1, 2] = F_3

show(psi.display())

res = psi.exterior_derivative()
show(res.display())

which works for the spherical coordinate transformation that is the default for Euclidean space. But, I see that Sagemath has a different spherical coordinate map than I do:

> print(E.coord_change(E.spherical_coordinates(), E.cartesian_coordinates()).display())
x = r*cos(phi)*sin(theta)
y = r*sin(phi)*sin(theta)
z = r*cos(theta)

Instead, I'd like:

x = r * cos(theta) * cos(phi)
y = r * sin(theta) * cos(phi)
z = r * sin(phi)

How can I supply a different change of coordinates function for this case?

How do I make sure the differential forms are on the right space? (Apologies if my vocabulary is not correct here). When

How do I make sure the differential forms are on the right space? (Apologies if my vocabulary is not correct here). When

Yes, I do -- thanks! How do I get a Manifold that has the right coordinate system attached?

Differential forms on non-standard spherical coordinates I'd like to compute an exterior derivative in spherical coordin

Differential forms on non-standard spherical coordinates I'd like to compute an exterior derivative in spherical coordin

Plotting a 2d subspace of R^3 I'm trying to plot the 2d subspace of R^3 defined by a pair of vectors. Following some exa

Evaluating a form field at a point on vectors I am having trouble matching up terminology in my textbook (Hubbard's Vect

Evaluating a form field at a point on vectors I am having trouble matching up terminology in my textbook (Hubbard's Vect

I am having trouble matching up terminology in my textbook (Hubbard's Vector Calculus) against SageMath operators. I'd like to understand how to solve the following example problem with Sage:

Let phi = cos(x z) * dx /\ dy be a 2-form on R^3. Evaluate it at the point (1, 2, pi) on the vectors [1, 0, 1], [2, 2, 3].

The expected answer is:

cos (1 * pi) * Matrix([1, 2], [0, 2]).det() = -2

So far I have pieced together the following:

E.<x,y,z> = EuclideanSpace(3, 'E')

f = E.diff_form(2, 'f')
f[1, 2] = cos(x * z)
point = E((1,2,pi), name='point')
anchor = f.at(point)

v1 = vector([1, 0, 1])
v2 = vector([2, 2, 3])

show(anchor(v1, v2))

which fails with the error:

TypeError: the argument no. 1 must be a module element

To construct a vector in E, I tried:

p1 = E(v1.list())
p2 = E(v2.list())
show(anchor(p1, p2))

but that fails with the same error. What's the right way to construct two vectors in E?

Evaluating a form field at a point on vectors I am having trouble matching up terminology in my textbook (Hubbard's Vect