Integral transforms:

The new integral transform package inttrans includes: inttrans[laplace], inttrans[invlaplace], inttrans[mellin], inttrans[fourier], inttrans[invfourier], inttrans[fouriersin], inttrans[fouriercos], inttrans[hankel], inttrans[hilbert].

inttrans[fouriersin](3/(t + a),t,w);

$\frac{3\sqrt{2}\left(-\cos \left(aw\right)\mathrm{Ssi}\left(aw\right)+\sin \left(aw\right)\mathrm{Ci}\left(aw\right)\right)}{\sqrt{\mathrm{\pi }}}$

assume(a>0);

inttrans[hilbert](exp(I*a*x),x,y);

$\mathrm{I}{\ⅇ}^{\mathrm{I}y\mathrm{a~}}$

a := 'a':

User defined lookup tables are added to the Fourier and Laplace transforms via routine inttrans[addtable]. Table entries can have an arbitrary number of constant parameters:

with(inttrans):

addtable(mellin,f1(t),F1(s),t,s);

addtable(mellin,f2(t),F2(s),t,s);

inttrans[mellin](t^a*int(x^b*f1(t/x)*f2(x),x=0..infinity),t,s);

$\mathrm{F1}\left(s+a\right)\mathrm{F2}\left(s+a+b+1\right)$

Recognition and transformation of general convolution integrals is added to all three transforms:

addtable(mellin,f(t),F(s),t,s);

inttrans[mellin](diff(f(t),t),t,s);

$-\left(s-1\right)F\left(s-1\right)$

inttrans[mellin](int(f(x),x=t..infinity),t,s);

$\frac{F\left(s+1\right)}{s}$

Functions which are being integrated and differentiated with respect to parameters other than the transform parameter are now simplified using Leibniz and Fubini's theorem.

inttrans[fourier](diff(f(t,u),u),t,s);

$\mathrm{fourier}\left(\mathrm{diff}\left(f\left(t\,u\right)\,u\right)\,t\,s\right)$

inttrans[laplace](int(f(t,u),u=a..b),t,s);

$\mathrm{int}\left(\mathrm{laplace}\left(f\left(t\,u\right)\,t\,s\right)\,u\=a..b\right)$

Transforms can now return multiple integrals, allowing for expressions of the form f(t)/t^n to be transformed for n>1:

inttrans[laplace](f(t)/t^4,t,s);

$\mathrm{laplace}\left(\frac{f\left(t\right)}{t^4}\,t\,s\right)$

In many cases, hypergeometric functions are returned where formerly only definite integrals were returned:

inttrans[fourier]((1+I*t)^(-1/2)*(1+2*I*t)^(-3/2),t,w);

$-\frac{\sqrt{2}\mathrm{\pi }{\ⅇ}^{w}w\mathrm{hypergeom}\left(\left[\frac{3}{2}\right]\,\left[2\right]\,-\frac{w}{2}\right)\mathrm{Heaviside}\left(-w\right)}{2}$

DEtools: Differential Equations Tools package

There are significant changes over release 3 version. Some new functionalities include:

DEtools[PDEchangecoords](D[1,2](w)(x,y,z)-D[1](w)(x,y,z), [x,y,z],
cylindrical,[r,phi,theta]);

$\frac{\sin \left(\mathrm{\phi }\right)\cos \left(\mathrm{\phi }\right){\mathrm{D}}_{1,1}\left(w\right)\left(r\,\mathrm{\phi }\,\mathrm{\theta }\right)r^2-\cos \left(\mathrm{\phi }\right){\mathrm{D}}_1\left(w\right)\left(r\,\mathrm{\phi }\,\mathrm{\theta }\right)r\sin \left(\mathrm{\phi }\right)+2{\cos \left(\mathrm{\phi }\right)}^2{\mathrm{D}}_{1,2}\left(w\right)\left(r\,\mathrm{\phi }\,\mathrm{\theta }\right)r-\cos \left(\mathrm{\phi }\right){\mathrm{D}}_1\left(w\right)\left(r\,\mathrm{\phi }\,\mathrm{\theta }\right)r^2-\cos \left(\mathrm{\phi }\right)\sin \left(\mathrm{\phi }\right){\mathrm{D}}_{2,2}\left(w\right)\left(r\,\mathrm{\phi }\,\mathrm{\theta }\right)+\sin \left(\mathrm{\phi }\right){\mathrm{D}}_2\left(w\right)\left(r\,\mathrm{\phi }\,\mathrm{\theta }\right)r-2{\cos \left(\mathrm{\phi }\right)}^2{\mathrm{D}}_2\left(w\right)\left(r\,\mathrm{\phi }\,\mathrm{\theta }\right)-{\mathrm{D}}_{1,2}\left(w\right)\left(r\,\mathrm{\phi }\,\mathrm{\theta }\right)r+{\mathrm{D}}_2\left(w\right)\left(r\,\mathrm{\phi }\,\mathrm{\theta }\right)}{r^2}$

combstruct: Combinatorial Structures package

The package combstruct is used to define, count and generate combinatorial structures.

A combinatorial class is defined by writing a grammar specification that describes it. In this way, an extensive collection of different classes may be defined. For example, the system applies to all regular and context-free grammars, grammars to define binary trees, plane general trees, necklaces, functional graphs, expression trees etc.

with(combstruct):

Model alcohol molecules, C_n H_{2n+1} OH.

Define the grammar for an alkyl, C_n H_{2n+1}, which is a ternary rooted tree that moves freely in space.

molecule := {alkyl = Union(H, Prod(C, Set(alkyl, card=3))), H=Atom, C=Atom}:

count([alkyl, molecule], size=6+2*6+1);

$17$

Thus we see that there are 17 different alcohol molecules with 6 carbon atoms. Here is one of them.

draw([alkyl, molecule], size=6+6*2+1);

$\mathrm{Prod}\left(C\,\mathrm{Set}\left(\mathrm{Prod}\left(C\,\mathrm{Set}\left(\mathrm{Prod}\left(C\,\mathrm{Set}\left(H\,H\,H\right)\right)\,H\,H\right)\right)\,\mathrm{Prod}\left(C\,\mathrm{Set}\left(\mathrm{Prod}\left(C\,\mathrm{Set}\left(H\,H\,H\right)\right)\,\mathrm{Prod}\left(C\,\mathrm{Set}\left(H\,H\,H\right)\right)\,H\right)\right)\,H\right)\right)$

tensor: Tensor Analysis and General Relativity Computations

The tensor package deals with tensors, their operations, and their use in General relativity both in the natural basis and in a rigid moving frame. Some utilities to manipulate tensors are also provided, as well as a mechanism for customizing the way simplification is done in multi-step calculations.

Compute the curvature in the natural basis for the Schwarzschild solution:

with(tensor):

g := create([-1,-1], array(1..4, 1..4, symmetric, sparse, [(1,1) = 1-2*M/r,
(2,2) = -1/(1-2*M/r), (3,3) = -r^2, (4,4) = -r^2*sin(theta)^2])):

coord := [t, r, theta, phi]:

`tensor/simp` := proc(a) simplify(a); end:

con_g := invert(g, 'det_g'):

d1_g := d1metric(g, coord):

d2_g := d2metric(d1_g, coord):

Cf1 := Christoffel1(d1_g):

Cf2 := Christoffel2(con_g, Cf1):

Rm := Riemann(con_g, d2_g, Cf1):

Rc := Ricci(con_g, Rm):

Display the non-zero independent components of the covariant Riemann and Ricci tensors:

displayGR('Riemann', Rm);

$\mathrm{The\; Riemann\; Tensor}$

$\mathrm{non-zero\; components\; :}$

$\mathrm{R1212}=\frac{2M}{r^3}$

$\mathrm{R1313}=\frac{\left(-r+2M\right)M}{r^2}$

$\mathrm{R1414}=\frac{\left(-r+2M\right)M{\sin \left(\mathrm{\theta }\right)}^2}{r^2}$

$\mathrm{R2323}=-\frac{M}{-r+2M}$

$\mathrm{R2424}=-\frac{M{\sin \left(\mathrm{\theta }\right)}^2}{-r+2M}$

$\mathrm{R3434}=-2rM{\sin \left(\mathrm{\theta }\right)}^2$

$\mathrm{character\; :\; [-1,\; -1,\; -1,\; -1]}$

displayGR('Ricci', Rc);

$\mathrm{The\; Ricci\; tensor}$

$\mathrm{non-zero\; components\; :}$

$\mathrm{None}$

$\mathrm{character\; :\; [-1,\; -1]}$

Notice that the last result confirms the fact that the Schwarzschild solution is indeed a vacuum solution.

Verify the Bianchi identity by computing the covariant derivatives of the Riemann components and anti-symmetrizing them on the last three indices:

D_Rm := cov_diff(Rm, coord, Cf2):

asymm_D_Rm := antisymmetrize(D_Rm, [3,4,5]):

act('display', asymm_D_Rm);

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{No\; non-zero\; independent\; components.}$

  CHARACTER :

$\left[\mathrm{-1}\,\mathrm{-1}\,\mathrm{-1}\,\mathrm{-1}\,\mathrm{-1}\right]$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

Since all of the components are zero, the Bianchi identity is verified.

geometry: 2D Euclidean Geometry Package

The geometry package is re-designed, and has more functionalities: the conic section is completely implemented, more transformations, graphical visualization is also provided.