this tutorial is a brief slide-show introduction to
Post on 05-May-2022
11 Views
Preview:
TRANSCRIPT
« ≈ 1 of 23
This tutorial is a brief slide-show introduction to elementary mathematics by Mathemat-ica. It covers the following subjects:
Basic Operations; Solving Equations; Sequences & Series
Basic Plotting; Curves and Surfaces
May 26, 2010
Hsin-Yun Hu, Department of Mathematics, Tunghai University
東海大學數學系 胡馨云
Printed from the Mathematica Help Browser 1
©1988-2002 Wolfram Research, Inc. All rights reserved.
« ≈ 2 of 23
Basic OperationsDo arithmetic
In[1]:= 2.3 + 5.36 ∗ 2 − 8.9^2 + 4.5ê1.2
Out[1]= −62.44
Handle numbers of any size
2 Printed from the Mathematica Help Browser
©1988-2002 Wolfram Research, Inc. All rights reserved.
In[2]:= 100!
N@Pi, 100D
Out[2]= 93326215443944152681699238856266700490715968264381621468592963895217599993229915608941463976156518286253697920827223758251185210916864000000000000000000000000
Out[3]= 3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068
Find the prime factors of an integer
In[4]:= FactorInteger@45DFactorInteger@242D
Out[4]= 883, 2<, 85, 1<<
Out[5]= 882, 1<, 811, 2<<
Find the greatest common divisor & least common multiple
In[6]:= GCD@12, 16DLCM@12, 16D
Out[6]= 4
Out[7]= 48
« ≈ 3 of 23
Basic OperationsFactor the polynomials
Printed from the Mathematica Help Browser 3
©1988-2002 Wolfram Research, Inc. All rights reserved.
In[8]:= Factor@x2 − y2DFactor@x9 − y9D
Out[8]= Hx − yL Hx + yL
Out[9]= Hx − yL Hx2 + x y + y2L Hx6 + x3 y3 + y6L
Multiply out products and power
In[10]:= Expand@H1 + xL5DExpandAHx2 − y2 + 10L3E
Out[10]= 1 + 5 x + 10 x2 + 10 x3 + 5 x4 + x5
Out[11]= 1000 + 300 x2 + 30 x4 + x6 − 300 y2 − 60 x2 y2 − 3 x4 y2 + 30 y4 + 3 x2 y4 − y6
Simplify factors the polynomials
In[12]:= Simplify@x2 + 2 x + 1DSimplify@−108 + 108 x + 45 x2 − 40 x3 − 10 x4 + 4 x5 + x6D
Out[12]= H1 + xL2
Out[13]= H−2 + xL2 H−1 + xL H3 + xL3
« ≈ 4 of 23
Basic OperationsPut all terms over a common denominator
In[14]:= TogetherA 5
x − 3−
2
x + 1E
Out[14]=11 + 3 x
H−3 + xL H1 + xL
4 Printed from the Mathematica Help Browser
©1988-2002 Wolfram Research, Inc. All rights reserved.
Separate into terms with simple denominators
In[15]:= ApartA x3 − 3 x + 2
Hx + 1L Hx − 3L3E
Out[15]=5
H−3 + xL3 +19
4 H−3 + xL2 +17
16 H−3 + xL −1
16 H1 + xL
Use some mathematical functions
In[16]:= a = 10.0;Sqrt@aDExp@aDLog@aDSin@aDClear@aDb = 1;4 ∗ ArcTan@bDClear@bD
Out[17]= 3.16228
Out[18]= 22026.5
Out[19]= 2.30259
Out[20]= −0.544021
Out[23]= π
« ≈ 5 of 23
Advance OperationsLimit
Printed from the Mathematica Help Browser 5
©1988-2002 Wolfram Research, Inc. All rights reserved.
In[27]:= Clear@nDLimit@xn, x → 2D
Out[28]= 2n
Derivatives
In[29]:= Clear@nDD@xn, xD
Out[30]= n x−1+n
Anti-derivatives
In[31]:= Clear@nD
‡ xn x
Out[32]=x1+n
1 + n
« ≈ 6 of 23
Solving EquationsSolve a polynomial equation in one variable
In[33]:= Solve@a x2 + b x + c == 0, xD
Out[33]= 99x →−b −
è!!!!!!!!!!!!!!!!!!!!b2 − 4 a c2 a
=, 9x →−b +
è!!!!!!!!!!!!!!!!!!!!b2 − 4 a c2 a
==
In[34]:= Solve@x2 + 5 x + 3 == 0, xD
Out[34]= 99x →12I−5 −è!!!!!!13 M=, 9x →
12I−5 +è!!!!!!13 M==
6 Printed from the Mathematica Help Browser
©1988-2002 Wolfram Research, Inc. All rights reserved.
In[35]:= Solve@x2 + x + 3 == 0, xD
Out[35]= 99x →12I−1 − è!!!!!!11 M=, 9x →
12I−1 + è!!!!!!11 M==
Solve a simple equation of higher degree
In[36]:= Solve@3 x3 + 2 x2 − 2 x − 1 0, xD
Out[36]= 98x → −1<, 9x →16I1 −è!!!!!!13 M=, 9x →
16I1 +è!!!!!!13 M==
In[37]:= Solve@x6 1, xD
Out[37]= 88x → −1<, 8x → 1<, 8x → −H−1L1ê3<, 8x → H−1L1ê3<, 8x → −H−1L2ê3<, 8x → H−1L2ê3<<
« ≈ 7 of 23
Solving EquationsSolve two simultaneous algebraic equations
In[38]:= sys1 = 8y 1 + 2 x, x 5 − y<;Solve@sys1, 8x, y<D
Out[39]= 99x →43
, y →113
==
In[40]:= sys2 = 9 1 + x
−1 + y== 0, 2 − 3 y + y2 == 0=;
Solve@sys2, 8x, y<D
Out[41]= 88x → −1, y → 2<<
Solve an equations involving exponentials, logarithms and trigonometric functions
Printed from the Mathematica Help Browser 7
©1988-2002 Wolfram Research, Inc. All rights reserved.
In[44]:= eqn1 = x−2 8;Solve@eqn1, xD
Out[45]= 88x → 2 + Log@8D<<
In[46]:= eqn2 = Log@x + 2D 5;Solve@eqn2, xD
Out[47]= 88x → −2 + 5<<
In[48]:= eqn3 = Sin@xD 1ê2;Solve@eqn3, xD
Out[49]= 99x →π6==
« ≈ 8 of 23
Sequences & SeriesList the terms of a sequence
In[50]:= Table@i2, 8i, 10<DTable@j3, 8j, −10, 10<D
TableA 1
k, 8k, 10<E
TableA H−1Lk−1
k2, 8k, 10<E
Out[50]= 81, 4, 9, 16, 25, 36, 49, 64, 81, 100<
Out[51]= 8−1000, −729, −512, −343, −216, −125, −64, −27, −8, −1, 0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000<
Out[52]= 91,12
,13
,14
,15
,16
,17
,18
,19
,1
10=
Out[53]= 91, −14
, 19
, −1
16, 1
25, −
136
, 149
, −1
64, 1
81, −
1100
=
8 Printed from the Mathematica Help Browser
©1988-2002 Wolfram Research, Inc. All rights reserved.
The telescoping sequence
In[54]:= TableA 1
n−
1
n + 1, 8n, 1, 10<E
SumA 1
n−
1
n + 1, 8n, 1, 10<E
Out[54]= 9 12
,16
,1
12,
120
,1
30,
142
,1
56,
172
,1
90,
1110
=
Out[55]=1011
« ≈ 9 of 23
Sequences & SeriesThe arithmetic sequence
In[56]:= a = 2; d = 2;Table@a + Hn − 1L ∗ d, 8n, 1, 10<DSum@a + Hn − 1L∗ d, 8n, 1, 10<D
Out[57]= 82, 4, 6, 8, 10, 12, 14, 16, 18, 20<
Out[58]= 110
by formula
In[59]:= n = 10; dsum =Ha + a + Hn − 1L∗ dL
2∗ n
Out[59]= 110
The geometric sequence
Printed from the Mathematica Help Browser 9
©1988-2002 Wolfram Research, Inc. All rights reserved.
In[60]:= b = 1; r = 1 ê3;Table@b ∗ rn−1, 8n, 1, 10<DSum@b ∗ rn−1, 8n, 1, 10<D
Out[61]= 91, 13
, 19
, 127
, 181
, 1243
, 1729
, 12187
, 16561
, 119683
=
Out[62]=2952419683
by formula
In[63]:= n = 10; rsum =b ∗H1 − rnL
1 − r
Out[63]=2952419683
« ≈ 10 of 23
Basic PlottingThe two-dimensional functions plots
10 Printed from the Mathematica Help Browser
©1988-2002 Wolfram Research, Inc. All rights reserved.
In[64]:= Plot@x2 − 2, 8x, −4, 4<, Frame → TrueD;
-4 -2 0 2 4
0
2.5
5
7.5
10
12.5
In[65]:= Plot@Exp@xD, 8x, −4, 4<, Frame → TrueD;
-4 -2 0 2 40
5
10
15
20
25
30
35
Printed from the Mathematica Help Browser 11
©1988-2002 Wolfram Research, Inc. All rights reserved.
In[66]:= Plot@Log@xD, 8x, 0, 4<, Frame → TrueD;
0 1 2 3 4-10
-8
-6
-4
-2
0
« ≈ 11 of 23
Basic PlottingCombining several plots
12 Printed from the Mathematica Help Browser
©1988-2002 Wolfram Research, Inc. All rights reserved.
In[67]:= Plot@82x, 3x, 5x<, 8x, −5, 5<, Frame → True, PlotStyle → 8RGBColor@1, 0, 0D, RGBColor@0, 1, 0D, RGBColor@0, 0, 1D<D;
-4 -2 0 2 40
20
40
60
80
100
120
In[68]:= g1 = Plot@Sin@xD, 8x, −2 π, 4 π<, PlotRange → 8−2, 2<, Frame → True, PlotStyle → RGBColor@1, 0, 1DD;g2 = Plot@Cos@xD, 8x, −2 π, 4 π<, PlotRange → 8−2, 2<, Frame → True, PlotStyle → RGBColor@0, 1, 1DD;Show@g1, g2D;
Printed from the Mathematica Help Browser 13
©1988-2002 Wolfram Research, Inc. All rights reserved.
-5 -2.5 0 2.5 5 7.5 10 12.5
-1.5
-1
-0.5
0
0.5
1
1.5
2
-5 -2.5 0 2.5 5 7.5 10 12.5
-1.5
-1
-0.5
0
0.5
1
1.5
2
14 Printed from the Mathematica Help Browser
©1988-2002 Wolfram Research, Inc. All rights reserved.
-5 -2.5 0 2.5 5 7.5 10 12.5
-1.5
-1
-0.5
0
0.5
1
1.5
2
« ≈ 12 of 23
Basic PlottingThe three-dimensional surfaces plots
Printed from the Mathematica Help Browser 15
©1988-2002 Wolfram Research, Inc. All rights reserved.
In[71]:= g1 = Plot3D@Sin@x yD, 8x, 0, 3<, 8y, 0, 3<D;
0
1
2
3 0
1
2
3
-1-0.5
00.5
1
0
1
2
16 Printed from the Mathematica Help Browser
©1988-2002 Wolfram Research, Inc. All rights reserved.
In[72]:= g2 = Plot3D@Exp@−Hx2 + y2LD, 8x, −3, 3<, 8y, −3, 3<, PlotRange → AllD;
-2
0
2-2
0
2
00.250.5
0.751
-2
0
2
« ≈ 13 of 23
Basic PlottingConti.
Printed from the Mathematica Help Browser 17
©1988-2002 Wolfram Research, Inc. All rights reserved.
In[73]:= Show@g1, Mesh → FalseD;
0
1
2
3 0
1
2
3
-1-0.5
00.5
1
0
1
2
18 Printed from the Mathematica Help Browser
©1988-2002 Wolfram Research, Inc. All rights reserved.
In[74]:= Show@g2, Shading → FalseD;
-2
0
2-2
0
2
00.250.5
0.751
-2
0
2
« ≈ 14 of 23
Basic PlottingConti.
Printed from the Mathematica Help Browser 19
©1988-2002 Wolfram Research, Inc. All rights reserved.
In[75]:= Show@ContourGraphics@g1DD;
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
3
In[76]:= Show@g2, ViewPoint → 8−1, −3, 1<D;
-20
2
-20
2
0
0.25
0.5
0.75
1
02
-20
20 Printed from the Mathematica Help Browser
©1988-2002 Wolfram Research, Inc. All rights reserved.
In[77]:= Show@g2, Boxed → FalseD;
-2
0
2
-2
0
2
00.250.5
0.75
1
-2
0
2
« ≈ 15 of 23
Curves In PlaneTeardrop
Printed from the Mathematica Help Browser 21
©1988-2002 Wolfram Research, Inc. All rights reserved.
In[78]:= a = 1.5; b = 3;ParametricPlot@8a ∗ Cos@tD − aê2 ∗ Sin@2 ∗ tD, b ∗ Sin@tD<, 8t, 0, 2 ∗ Pi<, AspectRatio → AutomaticD;
-2 -1 1 2
-3
-2
-1
1
2
3
Ellipse
In[80]:= a = 4; b = 2;ParametricPlot@8a ∗ Cos@tD, b ∗ Sin@tD<, 8t, 0, 2 ∗ Pi<, AspectRatio → AutomaticD;
-4 -2 2 4
-2
-1
1
2
22 Printed from the Mathematica Help Browser
©1988-2002 Wolfram Research, Inc. All rights reserved.
Epicycloid
In[82]:= a = 32; b = 4;
ParametricPlotA9Ha + bL ∗ Cos@tD − b ∗ CosA a + b
b∗ tE, Ha + bL∗ Sin@tD − b ∗ SinA a + b
b∗ tE=, 8t, 0, 2 ∗ Pi<, AspectRatio → AutomaticE;
-30 -20 -10 10 20 30
-30
-20
-10
10
20
30
« ≈ 16 of 23
Involute of a circle
Printed from the Mathematica Help Browser 23
©1988-2002 Wolfram Research, Inc. All rights reserved.
In[84]:= ParametricPlot@8Sin@θD − θ ∗ Cos@θD, Cos@θD + θ ∗ Sin@θD<, 8θ, 0, 6 ∗ Pi<, AspectRatio → AutomaticD;
-15 -10 -5 5 10 15
-15
-10
-5
5
10
Rose curve (odd petal)
24 Printed from the Mathematica Help Browser
©1988-2002 Wolfram Research, Inc. All rights reserved.
In[85]:= r = 2 ∗ Cos@5 ∗ θD;ParametricPlot@8r ∗ Cos@θD, r ∗ Sin@θD<, 8θ, 0, Pi<, AspectRatio → AutomaticD;
-1.5 -1 -0.5 0.5 1 1.5 2
-2
-1
1
2
Rose curve (even petal)
Printed from the Mathematica Help Browser 25
©1988-2002 Wolfram Research, Inc. All rights reserved.
In[87]:= r = 2 ∗ Cos@2 ∗ θD;ParametricPlot@8r ∗ Cos@θD, r ∗ Sin@θD<, 8θ, 0, 2 ∗ Pi<, AspectRatio → AutomaticD;
-2 -1 1 2
-2
-1
1
2
« ≈ 17 of 23
Cardioid / Heart-shaped
26 Printed from the Mathematica Help Browser
©1988-2002 Wolfram Research, Inc. All rights reserved.
In[89]:= r = 2 ∗H1 − Cos@θDL;ParametricPlot@8r ∗ Cos@θD, r ∗ Sin@θD<, 8θ, 0, 2 ∗ Pi<, AspectRatio → AutomaticD;
-4 -3 -2 -1
-2
-1
1
2
Star-shaped
Printed from the Mathematica Help Browser 27
©1988-2002 Wolfram Research, Inc. All rights reserved.
In[91]:= r = 50 ∗ H1 + 0.2 ∗ Cos@6 ∗ θDL;ParametricPlot@8r ∗ Cos@θD, r ∗ Sin@θD<, 8θ, 0, 2 ∗ Pi<, AspectRatio → AutomaticD;
-60 -40 -20 20 40 60
-40
-20
20
40
« ≈ 18 of 23
Curves In SpaceHelix
28 Printed from the Mathematica Help Browser
©1988-2002 Wolfram Research, Inc. All rights reserved.
In[93]:= ParametricPlot3D@83 ∗ Cos@3 tD, 3 ∗ Sin@3 tD, t<, 8t, 0, 4 ∗ Pi<, AspectRatio → AutomaticD;
-20
2
-20
2
0
5
10
-20
2
Perodic curve
Printed from the Mathematica Help Browser 29
©1988-2002 Wolfram Research, Inc. All rights reserved.
In[94]:= ParametricPlot3D@82 ∗ t, Sin@5 ∗ tD + Sin@tD, t<, 8t, 0, 4 ∗ Pi<, AspectRatio → AutomaticD;
0
10
20
-2-10120
5
10
0
5
« ≈ 19 of 23
Surfaces In Space Cylindrical surface
30 Printed from the Mathematica Help Browser
©1988-2002 Wolfram Research, Inc. All rights reserved.
In[95]:= ParametricPlot3D@8Cos@uD, Sin@uD, v<, 8u, 0, 2 ∗ Pi<, 8v, 0, 1<, AspectRatio → AutomaticD;
-1-0.5
00.5
1 -1
-0.5
0
0.5
1
00.250.5
0.751
-1-0.5
00.5
Riemann surface
Printed from the Mathematica Help Browser 31
©1988-2002 Wolfram Research, Inc. All rights reserved.
In[96]:= ParametricPlot3D@8u ∗ Cos@vD, u ∗ Sin@vD, 2 ∗ v<, 8u, 0, 5<, 8v, 0, 4 ∗ Pi<, BoxRatios → 81.5, 1.5, 2<D;
-5 -2.5 0 2.55
-5-2.5
02.5
5
0
10
20
5-2.5
02.5
Sphere
32 Printed from the Mathematica Help Browser
©1988-2002 Wolfram Research, Inc. All rights reserved.
In[97]:= ρ = 1;ParametricPlot3D@8ρ ∗ Sin@uD∗ Cos@vD, ρ ∗ Sin@uD∗ Sin@vD, ρ ∗ Cos@uD<, 8u, 0, Pi<, 8v, 0, 12 ∗ Piê6<, AspectRatio → AutomaticD;
-1-0.5
00.5
1
-1
-0.50
0.51
-1
-0.5
0
0.5
1
-1-0.5
00.5
1
-0.50
0.5
« ≈ 20 of 23
Ellipsoid
Printed from the Mathematica Help Browser 33
©1988-2002 Wolfram Research, Inc. All rights reserved.
In[99]:= a = 1; b = 1; c = 1.5;ParametricPlot3D@8a ∗ Cos@vD∗ Cos@uD, b ∗ Cos@vD∗ Sin@uD, c ∗ Sin@vD<, 8u, 0, 2 ∗ Pi<,8v, 0, 2 ∗ Pi<, AspectRatio → AutomaticD;
-1 -0.5 0 0.51
-1-0.5
00.5
1
-1
0
1
1-0.5
00.5
Paraboloid
34 Printed from the Mathematica Help Browser
©1988-2002 Wolfram Research, Inc. All rights reserved.
In[101]:=
ParametricPlot3D@82 ∗ u ∗ Cos@vD, 2 ∗ u ∗ Sin@vD, u2<, 8u, 0, 4<, 8v, 0, 2 ∗ Pi<, AspectRatio → AutomaticD;
-50
5
-5
0
5
0
5
10
15
-50
5
-5
0
5
« ≈ 21 of 23
Tours
Printed from the Mathematica Help Browser 35
©1988-2002 Wolfram Research, Inc. All rights reserved.
In[102]:=
a = 2;ParametricPlot3D@8H4 + a ∗ Cos@vDL∗ Cos@uD, H4 + a ∗ Cos@vDL∗ Sin@uD, a ∗ Sin@vD<, 8u, 0, 2 ∗ Pi<, 8v, 0, 2 ∗ Pi<, AspectRatio → AutomaticD;
-5
0
5-5
0
5
-2-1012
-5
0
5
Wrinkled sphere
36 Printed from the Mathematica Help Browser
©1988-2002 Wolfram Research, Inc. All rights reserved.
In[104]:=
ρ = 1 + 0.2 ∗ Sin@uD∗ Sin@5 ∗ vD;ParametricPlot3D@8ρ ∗ Sin@uD∗ Cos@vD, ρ ∗ Sin@uD∗ Sin@vD, ρ ∗ Cos@uD<, 8u, 0, Pi<, 8v, 0, 2 ∗ Pi<, AspectRatio → AutomaticD;
-1-0.5
00.5
1
-1-0.5
00.5
1
-1
-0.5
0
0.5
1
-1-0.5
00.5
1
-1-0.5
00.5
« ≈ 22 of 23
Bumpy sphere
Printed from the Mathematica Help Browser 37
©1988-2002 Wolfram Research, Inc. All rights reserved.
In[106]:=
ρ = 1 + 0.2 ∗ Sin@8 ∗ uD ∗ Sin@8 ∗ vD;ParametricPlot3D@8ρ ∗ Sin@uD∗ Cos@vD, ρ ∗ Sin@uD∗ Sin@vD, ρ ∗ Cos@uD<, 8u, 0, Pi<, 8v, 0, 2 ∗ Pi<, AspectRatio → AutomaticD;
-1-0.5
00.5
1
-1-0.5
00.5
1
-1
-0.5
0
0.5
1
-1-0.5
00.5
1
-1-0.5
00.5
Mobius strip
38 Printed from the Mathematica Help Browser
©1988-2002 Wolfram Research, Inc. All rights reserved.
In[108]:=
ParametricPlot3D@8H4 − v ∗ Sin@uDL∗ Cos@2 ∗ uD, H4 − v ∗ Sin@uDL∗ Sin@2 ∗ uD, v ∗ Cos@uD<,8u, 0, Pi<, 8v, −2, 2<, ViewPoint → 81.2, −2.5, 1.2<, AspectRatio → AutomaticD;
-5-2.5
02.5
-5-2.5
02.5
5
-2
-1
0
1
2
-5-2.5
02 5
-5-2.5
02.5
« ≈ 23 of 23
Use the Help Browser to learn more
You now know enough to start using Mathematica.
But the examples you have seen represent only a small part of its capabilities.
Here are some resources you can draw on to learn more about Mathematica.
Printed from the Mathematica Help Browser 39
©1988-2002 Wolfram Research, Inc. All rights reserved.
àBuilt-in Functions
Click a link to view a list of all built-in functions in that category.
Numerical Computation Algebraic Computation Mathematical Functions
Lists and Matrices Graphics and Sound Programming
Input and Output Notebooks System Interface
àHelp Browser
The Help Browser contains detailed information on all aspects of Mathematica.
Built-in Functions Add-ons & Links The Mathematica Book
Front End Getting Started Tour
Demos Master Index
40 Printed from the Mathematica Help Browser
©1988-2002 Wolfram Research, Inc. All rights reserved.
top related