Wolfram Calculus & Algebra | Things to Try

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.
Symbolic & Numeric Mathematical Computation. For everyone from middle-school students to advanced math researchers. Centuries of mathematical development packed into a set of exceptionally powerful functions tightly integrated with advanced visualization and immediately computable built-in data.

Simplify, Factor or Expand Algebraic Expressions

Simplify an expression involving rational functions:
Run
In[]:=
Simplify
1
x+1
-
1
2
(x+1)

Specify assumptions that affect the result of a calculation:
Run
In[]:=
Assumingx<0,Simplify
2
x

Efficiently factor polynomials of high degree:
Run
In[]:=
Factor[
15
x
-1]
Check factorization by multiplying polynomials:
Run
In[]:=
Expand[(-1+x)(1+x+
2
x
)(1+x+
2
x
+
3
x
+
4
x
)(1-x+
3
x
-
4
x
+
5
x
-
7
x
+
8
x
)]

Solve an Equation or System of Equations

Obtain the values that solve equations with symbolic parameters:
Run
In[]:=
SolveValues[a
2
x
+bx+c==0,x]
Obtain replacement rules that solve a system of equations for specified variables:
Run
In[]:=
Solve[{x+y+z==4,x-z==2,x-2y+z==2},{x,y,z}]
Solve the linear system
m.xb
:
Run
In[]:=
LinearSolve
-3
6
-3
7
-1
5
8
8
-3
,
2
9
8


Compute Derivatives of Functions

Compute the derivative of a function with respect to specified variables:
Run
In[]:=
D[Sin[x],x]
Compute the
th
n
derivative of a function with respect to specified variables:
Run
In[]:=
D[
ax

,{x,n}]

Evaluate Integrals

Evaluate symbolic indefinite integrals:
Run
In[]:=
∫xx
Evaluate integrals numerically:
Run
In[]:=
NIntegrate
-
2
x

,{x,0,24}
Evaluate integrals with multiple variables of integration:
Run
In[]:=
∞
∫
0
∞
∫
0
-
2
x
+
2
y


xy

Solve Differential Equations

Solve ordinary differential equations symbolically:
Run
In[]:=
DSolve[u''[x]+
2
k
u[x]==0,u[x],x]
Obtain numerical solutions to differential equations:
Run
In[]:=
lorenzsolution=NDSolve[{
′
x
[t]-3(x[t]-y[t]),
′
y
[t]26.5`x[t]-y[t]-x[t]z[t],
′
z
[t]x[t]y[t]-z[t],x[0]z[0]0,y[0]1},{x,y,z},{t,0,202},MaxSteps∞]
Visualize solutions with 3D graphics:
Run
In[]:=
ParametricPlot3DEvaluate[{x[t],y[t],z[t]}/.lorenzsolution],{t,0,200},
Nice options for Lorenz system plot


Compute Dot Products, Cross Products, etc.

Compute the dot product of two vectors:
Run
In[]:=
{a,b,c}.{x,y,z}
Compute the cross product of two vectors:
Run
In[]:=
{a,b,c}{x,y,z}
Compute the norm of a vector:
Run
In[]:=
Norm[{3,4,12}]

Compute Gradient, Divergence and Curl

Compute the gradient of a function to obtain a vector field:
Run
In[]:=
vfield[x,y,z]=Grad[x
2
y
3
z
,{x,y,z}]
Compute the divergence of a vector field:
Run
In[]:=
Div[vfield[x,y,z],{x,y,z}]
Compute the curl of a vector field:
Run
In[]:=
Curl[vfield[x,y,z],{x,y,z}]

Perform Matrix Arithmetic

Compute the sum or difference of matrices:
Run
In[]:=
a
b
c
d
+
e
f
g
h
//MatrixForm
Compute the product of matrices:
Run
In[]:=
a
b
c
d
.
x
y
//MatrixForm
Efficiently compute powers of matrices:
Run
In[]:=
MatrixPower
1
-2
3
-2
3
-1
3
-1
2
,4//MatrixForm

Compute Matrix Determinant, Inverse, Transpose, etc.

Compute the determinant of a matrix:
Run
In[]:=
Det
a
b
c
d

Compute the inverse of a matrix:
Run
In[]:=
Inverse
a
b
c
d
//MatrixForm
Transpose a matrix:
Run
In[]:=
Transpose
1
2
3
4
//MatrixForm
Find the conjugate transpose of a matrix (also known as the Hermitian transpose):
Run
In[]:=
ConjugateTranspose
1
2
3
4
//MatrixForm

Matrix Decompositions

Find a similarity matrix to transform into the Jordan canonical form of a matrix:
Run
In[]:=
JordanDecomposition
9
-7
3
12
-10
3
16
-16
1
//Map[MatrixForm]
Find the singular value decomposition (SVD) of a matrix:
Run
In[]:=
SingularValueDecomposition
1
2
1
2
//Map[MatrixForm]

Use and Visualize Mathematical Functions

Numerically evaluate functions to specified precision:
Run
In[]:=
N[BesselJ[2,3],50]
Use a variety of special functions as part of inputs or outputs:
Run

Compute Limits of Sequences or Functions

Find Asymptotic Relations

Compute Integral and Summation Transformations