Mathematica-Guide.pdf
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827 KB
)
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583770568 UNPDF
CONTENTS
Loading the Mathematica Kernel
1
Function Definition
1
Differentiation
1
Antidifferentiation
2
Graphing a Function
2
Piecewise-Defined Function
4
Root Finding
5
Direction Field
5
Symbolic Solution of First Order Linear Differential
6
Equations
Limiting Behavior
7
Symbolic Solution of Nonlinear First Order Differential
7
Equations
Checking a Solution
8
Graphing Implicit Solutions
8
Symbolic Solution of Second Order Linear Differential
9
Equations
Higher Order Linear Differential Equations
9
Numerical Solution of an Initial Value Problem
10
Matrices and First Order Systems
11
Eigenpair Calculation
13
Symbolic Solution of First Order Systems
13
Plotting a Phase Plane Trajectory
14
Phase Plane Analysis of Autonomous 2-Dimensional
15
Systems
Laplace Transforms
17
Graphing the Periodic Thrust Velocity
17
Series Expansions
18
Equilibrium Points for Two-Dimensional Autonomous
19
Systems
Linearization About an Equilibrium Point
19
Partial Differential Equations and Fourier Series
20
Two-Point Boundary Value Problems
26
ii
CONTENTS
Loading the Mathematica Kernel
1
Function Definition
1
Differentiation
1
Antidifferentiation
2
Graphing a Function
2
Piecewise-Defined Function
4
Root Finding
5
Direction Field
5
Symbolic Solution of First Order Linear Differential
6
Equations
Limiting Behavior
7
Symbolic Solution of Nonlinear First Order Differential
7
Equations
Checking a Solution
8
Graphing Implicit Solutions
8
Symbolic Solution of Second Order Linear Differential
9
Equations
Higher Order Linear Differential Equations
9
Numerical Solution of an Initial Value Problem
10
Matrices and First Order Systems
11
Eigenpair Calculation
13
Symbolic Solution of First Order Systems
13
Plotting a Phase Plane Trajectory
14
Phase Plane Analysis of Autonomous 2-Dimensional
15
Systems
Laplace Transforms
17
Graphing the Periodic Thrust Velocity
17
Series Expansions
18
Equilibrium Points for Two-Dimensional Autonomous
19
Systems
Linearization About an Equilibrium Point
19
Partial Differential Equations and Fourier Series
20
Two-Point Boundary Value Problems
26
ii
Mathematica Technical Supplement
Loading the Mathematica Kernel
Performing a simple arithmetic operation, such as the one shown, can be used to activate the
Mathematica
Kernel, the
software's computational engine.
1
+
1
2
Function Definition
The function
f
H
t
L
=-
2
te
-
t
+
5 sin H2
p
tL
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
ÅÅÅÅÅÅ
+
1 is defined.
f
@
t_
D
:
=
1
-
2
*
t
*
Exp
@
-
t
D
+
5
*
Sin
@
2
*
Pi
*
t
DêH
1
+
t^2
L
;
The use of "
: =
" (rather than simply "
=
") indicates a "delayed assignment". Using the colon tells
Mathematica
to return to the
original definition each time you use the function.
Function evaluation (
f
H
t
L is evaluated at
t
=
3):
f
@
3
D
1
-
6
ÅÅÅÅÅÅ
‰
3
N
@
f
@
3
DD
0.701278
Differentiation
The function
t
3
+
t
sin
t
is differentiated.
D
@
t
*
Sin
@
t
D
+
t^3, t
D
3t
2
+
t Cos
@
t
D
+
Sin
@
t
D
The function,
f
H
t
L
=-
2
te
-
t
+
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
ÅÅÅÅÅ
+
1, that was previously defined, is differentiated.
t
2
+
1
D
@
f
@
t
D
,t
D
-
2
‰
-
t
+
2
‰
-
t
t
+
10
p
Cos
@
2
p
t
D
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
1
+
t
2
-
10 t Sin
@
2
p
t
D
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
H
1
+
t
2
L
2
t
2
+
1
5 sin H2
p
tL
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