The following are examples of some important commands in Mathematica.
Sin[x], Exp[2+3I], Cos[x], Log[x]
Pi, I, E, Integrate[x^2, x], D[Sin[x^2], {x,2}],
Plot[Sin[x]/x, {x, 0, 10}], Plot3D[ Exp[x y], {x,-5,5},{y,-2,2}]
Plot[Cos[x], {x, -4, 4}], Sum[ n^2, {n, 1, 100}], Plot3D[ x y, {x, -2,1},{y,-3,3}]
a={{1,2,3},{4,5,6},{7,8,9}}; Inverse[a]
Solve[a x^2 - b x - c ==0, x]
Solve[ x^3-x -1 ==0, x]//N
NSolve[x^3-x+2==0, x]
Solve[{ 2 x- y + z ==4, 4x+y+z==-1, x-y-z==-4},{x,y,z}]
NRoots[x^6-x-2==0,x]
FindRoot[Exp[-x^2]-Sin[x]==0, {x,2}]
D[x^100 Sin[x], x]
D[Exp[-x^2] x, {x, 3}]
Integrate[ x*Sin[x],x]
Integrate[ x*Sin[x], {x, 0, Pi}]
Integrate[ Exp[-x^2] Cos[x], {x, 0, Infinity}]
NIntegrate[ 1/(Cos[x]^3+ 21), {x, 0, 20}]
mat1={{a11, a12, a13}, {a21, a22, a23}, {a31, a32, a33}}
vec={v1, v2, v3}
mat1.vec
Inverse[mat1]
Eigenvalues[mat1]
DSolve[ y'[x]- a y[x]==1, y[x], x]
DSolve[ {y'[x]- a y[x]==1, y[0]==0},y[x],x]
DSolve[ { x'[t]==y[t], y'[t]==x[t]}, {x[t], y[t]}, t]
f[x_]:= x*Sin[x]; f[x_, y_]:= (x-y)^4; delta[i_, j_]:= If [i==j, 1, 0];
Plot[ Sin[x], {x, -Pi, Pi}]
Plot[ { Sin[x], Cos[x]}, {x, -2Pi, 2Pi}]
Plot3D[ Exp[-x*y], {x, -4, 4}, {y, -1, 10}]
N[Pi, 1000]
I^200
Exp[E]
x^3/.x-> 1+y
(x^2+y^3)/.{x->2, y-> 16}
Series[Sin[x], {x, 0, 100}]
Apart[1/(x^2-4)]
Sum[ 1/i!, {i, 1, 100}]
Sum[ 1/i!, {i, 1, 100}]//N
NSum[1/i!, {i, 1, 100}]
| 2^100//N | Converts symbolic values to numeric values. |
| 1/3+2/7 | Symbolic values |
| 1/3+3/7//N | Numeric values |
| Sqrt[x] | Note the use of []. |
| Exp[x] | Exponential function |
| Log[x] | Natural logarithm |
| Log[b, x] | Logarithm with base b |
| Sin[x], Cos[x], Tan[x] | Trignometric functions |
| ArcSin[x], ArcCos[x], ArcTan[x] | Hyperbolic trignometric functions |
| n! | Factorial |
| Abs[x] | Absolute value |
| Random[] | Randome number |
| Max[x,y,...], Min[x, y, ...] | Maximum, minimum |
| Pi | p=3.141592.... |
| E | e=2.71828.... |
| Degree | 1/100 radian |
| I | imaginary number |
| Infinity | oo |
| expression//N or N[expression] | Numeric values |
| N[expression, n] | Numerica value with n-digit precision |
| x + I y | complex number, x + i y |
| Re[z] | Real part |
| Im[z] | Imaginary part |
| Abs[z] | Absolute value |z| |
| Conjugate[z] | Complex conjugate |
| Arg[z] | Argument of angle |
| % | last result generated |
| %% | next-to-last resutl |
| %n | result on output line Out[n] |
| x=value | assign a value to x |
| x=y=value | assign a value to both x and y |
| Clear[x] | remove any value assigned to x |
| x y | means x times y |
| xy with no space | is the variable with name xy |
| 5x | meanx 5 times x |
{a, b, c} a list
v={1, 2, 3,4}
v[[1]]=4
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expression/. x -> value replace x by value in the expression
expr/.{x-> xval, y-> yval}
t=1+x^2
t/.x->2
t/.x->5a
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Expand[ (1+x)^2] Factor[%] Simplify[ x^2+2x+1] Integrate[1/(x^4-1),x] D[%,x] Simplify[%] |
a=(x-1)^2 (2+x)/ ( (1+x) (x-3)^2) Expand[a] Together[%] Apart[%] Factor[%] |
e=Expand[(1+3x+4y^2)^2] Coefficient[e,x] Exponent[e,y] r= (1+x)/( 2 (2-y)) Denominator[r] Numerator[r] |
D[x^n, x]
D[ArcTan[x], x]
D[x^n, {x, 3}]
Integrate[x^n, x]
Integrate[1/(x^4-a^4), x]
Integrate[Sin[x^2], x]
Integrate[ x^2, {x, 0, 1}]
N[%]
Integrate[ x^2+y^2, {x, 0, 1}, {y, 0, 1}]
Sum[i^2, {i, 1, n}]
Sum[1/i^2, {i, 1, Infinity}]
Solve[x^2+2x-7==0, x]
Solve[ x^4-5 x^2-3==0, x]
DSolve[ y''[x]+y[x]==0, y[x], x]
Series[ Exp[x], {x, 0, 5}]
Limit[ Sin[x]/x, x->0]
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N[(3+Sqrt[2])]
NSum[1/i^3, {i, 1, Infinity}]
NIntegrate[ 1/Sqrt[ x (1-x)], {x, 0, 1}]
NIntegrate[ Exp[-x^2], {x, -Infinity, Infinity}]
NSolve[ x^5+x+1==0, x]
NSolve[ {x+y==2, x-3y==3, x-y+z==0}, {x, y,z}]
FindRoot[ 3 Cos[x] == Log[x], {x, 1}]
data=Table[Exp[x/5.0], {x, 7}]
Fit[ data, {1, x, x^2}, x]
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f[x_]:=x^2
Clear[f]
Do[ Print[ i], {i, 5}]
Do[ Print[i], {i, 2, 10, 2}]
Print[expression]
Table[ expression, {i, imax}]
Table[i!, {i, 4}]
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{a,b,c} ... vector(a,b,c)
{{a,b},{c,d}} ... matrix
m={{ 1, 4}, {2, 5}}
m[[1,2]]
v={4,5}
m.v .... Product between matrix and vector
MatrixForm[m]
Inverse[m]
Det[m]
Eigenvalues[m]
Eigenvectors[m]
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Plot[ Sin[x], {x, 0, 2Pi}]
Plot[Tan[x], {x, -3, 3}]
Plot[{Sin[3x], Sin[2x]}, {x, 0, 2Pi}]
Plot[ Sin[1/x], {x, -1, 1}]
ContourPlot[ Sin[x] Sin[y], {x, -2,2},{y,-2,2}]
DensitPlot[ Sin[x] Sin[y], {x, -2,2},{y,-2,2}]
Plot3D[ Sin[ x y], {x, 0,3},{y, 0, 3}]
t=Table[ i^2, {i, 10}]
ListPlot[t]
ListPlot[t, PlotJoined->True]
ParametricPlot[ {Sin[t], Sin[2t]}, {t, 0, 2Pi}]
ParametricPlot3D[ { Sin[t], Cos[t], u}, {t, 0, 2Pi}, {u, 0, 4}]
ParametricPlot3D[ { Cos[t] (3+Cos[u]), Sin[t](3+Cos[u]), Sin[u]}, {t, 0, 2Pi}, {u, 0, 2Pi}]
sol=NDSolve[ {x'[t]==-3 (x[t]-y[t]),
y'[t]==-x[t] z[t]+26.5 x[t]-y[t],
z'[t]==x[t] y[t]-z[t],
x[0]==z[0]==0, y[0]==1},
{x,y,z}, {t, 0, 20}, MaxSteps->3000];
ParametricPlot[ Evaluate[{x[t], z[t]} /.sol], {t, 0, 20}, PlotPoints -> 1000]
ParametricPlot3D[ Evaluate[ {x[t],y[t],z[t]}/.sol], {t,0,20},PlotPoints->1000]
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Expand[ (x+y)^3] >> tmp.m <<tmp.m Directory[] SetDirectory["/tmp"] |