# Quick Maple Primer for Physics Students
# By Miles V. Aronnax
# Updated: 10 June 2008
# Indefinite Integrals
# Say you want to integrate e to the k*x with respect to x...
int( exp(k*x), x ); NiMqJkkia0c2IiEiIi1JJGV4cEc2JEkqcHJvdGVjdGVkR0YqSShfc3lzbGliR0YlNiMqJkYkIiIiSSJ4R0YlRi5GLg== # Definite Integrals
# What if I wanted to integrate e to the k*x from -2 to 73.1?
int( exp(k*x), x=-2..73.1 ); NiMsJComLCYtSSRleHBHNiRJKnByb3RlY3RlZEdGKUkoX3N5c2xpYkc2IjYjLCRJImtHRiskISIjIiIhIiIiLUYnNiMsJEYuJCIrKysrNXQhIikkISIiRjFGMkYuRjpGOQ== # Numeric Answers
# I want my answer to be a decimal number, not a hideous ratio
# or other complicated form. Oh, and k=0.1...
k := 0.1;
evalf( int( exp(k*x), x=-2..73.1 ) ); NiM+SSJrRzYiJCIiIiEiIg== NiMkIitlJWVWXCIhIiY= # Defining Functions
# I'm tired of typing that function, can't I just define it?
SomeStupidFunction := exp(k*x);
evalf( int( SomeStupidFunction, x=-2..73.1 ) );
# Note: k has already been defined above, hense the
# function is now exp(0.1 * x). NiM+STNTb21lU3R1cGlkRnVuY3Rpb25HNiItSSRleHBHNiRJKnByb3RlY3RlZEdGKUkoX3N5c2xpYkdGJTYjLCRJInhHRiUkIiIiISIi NiMkIitlJWVWXCIhIiY= # Derivatives
# Okay, now take the derivative of a function with respect to x.
# Oh, and let's use a new function...
SomewhatEvilFunction := exp( a*x^2 - b*x*y + c*y^2 + 42 );
diff( SomewhatEvilFunction, x ); NiM+STVTb21ld2hhdEV2aWxGdW5jdGlvbkc2Ii1JJGV4cEc2JEkqcHJvdGVjdGVkR0YpSShfc3lzbGliR0YlNiMsKiomSSJhR0YlIiIiSSJ4R0YlIiIjRi8qKEkiYkdGJUYvRjBGL0kieUdGJUYvISIiKiZJImNHRiVGL0Y0RjFGLyIjVUYv NiMqJiwmKiZJImFHNiIiIiJJInhHRidGKCIiIyomSSJiR0YnRihJInlHRidGKCEiIkYoLUkkZXhwRzYkSSpwcm90ZWN0ZWRHRjJJKF9zeXNsaWJHRic2IywqKiZGJkYoRilGKkYoKihGLEYoRilGKEYtRihGLiomSSJjR0YnRihGLUYqRigiI1VGKEYo # Evaluating Derivatives
# Note: Use the eval function to test a value that is
# variable. If we set x:=2 and y:=3.14, then the
# function would be a constant value and
# we can't take the derivative of a constant.
#
# Also, if this function only had one variable,
# then {x=2, y...} could be replaced with x=2
# (ie, the curly braces are needed if defining
# multiple variables in a single eval command).
eval( diff( SomewhatEvilFunction, x ), {x=2, y=Pi, a=1, b=2, c=3} ); NiMqJiwmIiIlIiIiSSNQaUdJKnByb3RlY3RlZEdGKCEiI0YmLUkkZXhwRzYkRihJKF9zeXNsaWJHNiI2IywoIiNZRiZGJyEiJSokRiciIiMiIiRGJg== # Multiple Derivatives
# How do I take multiple derivatives, either of the
# same variable or of different ones?
# Double derivative in x and one derivative in y:
diff( SomewhatEvilFunction, x, x, y );
NiMsKCooSSJhRzYiIiIiLCYqJkkiYkdGJkYnSSJ4R0YmRichIiIqJkkiY0dGJkYnSSJ5R0YmRiciIiNGJy1JJGV4cEc2JEkqcHJvdGVjdGVkR0Y0SShfc3lzbGliR0YmNiMsKiomRiVGJ0YrRjBGJyooRipGJ0YrRidGL0YnRiwqJkYuRidGL0YwRiciI1VGJ0YnRjAqKCwmKiZGJUYnRitGJ0YwKiZGKkYnRi9GJ0YsRidGMUYnRipGJyEiIyooRj1GMEYoRidGMUYnRic= # Differential Equations
# I need to solve a differential equation, how do I do it?
MyDiffEquation := m*diff( x(t), t, t ) + b*diff( x(t), t) + K*x(t) = 0;
dsolve( MyDiffEquation );
# Yes, the solution has undefined constants _C1 and _C2 which need
# to be determined using boundary conditions. NiM+SS9NeURpZmZFcXVhdGlvbkc2Ii8sKComSSJtR0YlIiIiLUklZGlmZkdJKnByb3RlY3RlZEdGLTYkLUkieEdGJTYjSSJ0R0YlLUkiJEdGLTYkRjIiIiNGKkYqKiZJImJHRiVGKi1GLDYkRi9GMkYqRioqJkkiS0dGJUYqRi9GKkYqIiIh NiMvLUkieEc2IjYjSSJ0R0YmLCYqJkkkX0MxR0YmIiIiLUkkZXhwRzYkSSpwcm90ZWN0ZWRHRjBJKF9zeXNsaWJHRiY2IywkKigsJkkiYkdGJkYsKiQsJiokRjYiIiNGLComSSJLR0YmRixJIm1HRiZGLCEiJSNGLEY6ISIiRixGPUZARihGLCNGQEY6RixGLComSSRfQzJHRiZGLC1GLjYjLCQqKCwmRjZGLEY3RixGLEY9RkBGKEYsRkFGLEYs # Systems of Differential Equations
# I've got a coupled system, and I just want it solved.
AnnoyingSystem := [ diff( x(t), t ) = y(t), diff( y(t), t ) = -x(t) ];
dsolve( AnnoyingSystem ); NiM+SS9Bbm5veWluZ1N5c3RlbUc2IjckLy1JJWRpZmZHSSpwcm90ZWN0ZWRHRio2JC1JInhHRiU2I0kidEdGJUYvLUkieUdGJUYuLy1GKTYkRjBGLywkRiwhIiI= NiM8JC8tSSJ5RzYiNiNJInRHRicsJiomSSRfQzFHRiciIiItSSRjb3NHNiRJKnByb3RlY3RlZEdGMUkoX3N5c2xpYkdGJ0YoRi1GLSomSSRfQzJHRidGLS1JJHNpbkdGMEYoRi0hIiIvLUkieEdGJ0YoLCYqJkYsRi1GNUYtRi0qJkY0Ri1GLkYtRi0= # Transcendental Equations / Numeric Roots
# Got a hideous equation that you can't get an exact root out of?
HideousEQ := x = cos(x);
fsolve( HideousEQ ); NiM+SSpIaWRlb3VzRVFHNiIvSSJ4R0YlLUkkY29zRzYkSSpwcm90ZWN0ZWRHRitJKF9zeXNsaWJHRiU2I0Yn NiMkIitLOCYzUighIzU= # 2-D Graphing
# Note: Right-clicking on plots will give you various options.
plot( sin(x)/x, x); 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 # 3-D Graphing
# Note: Like 2-D graphs, right-click on the plot for options.
# Additionally, you can rotate 3-D graphs by left-clicking
# on the plot and draging the image arround for a better view.
plot3d( sin(x)*y - cos(y)*x, x=-Pi..Pi, y=-Pi..Pi, axes=FRAME ); 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# Series Summations
# Note: Try summing this equation from 0 to infinity.
sum( x^n/n!, n=0..4 ); NiMsLCIiIkYkSSJ4RzYiRiQqJEYlIiIjI0YkRigqJEYlIiIkI0YkIiInKiRGJSIiJSNGJCIjQw== # Evaluating Series Summations
# Note: If you set x=1, not 1.0, then you get an
# exact ratio--your choice: decimal number
# or ratio. If all else fails, you can force
# a ratio to be translated into a decimal by
# wrapping the whole command in evalf().
eval( sum( x^n/n!, n=0..10 ), x=1.0 ); NiMkIisuPUc9RiEiKg== # Multiple Plots, One Graph
plot( [sin(x), sin(x + Pi/7), sin(x + Pi/4)], x );
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# Infinite Square Well
# Below are plots of the antisymmetric and symmetric wave functions
# for two non-interacting particles in a square well. One particle
# is in the ground state, the other is in the first excited state.
# Symmetric vs antisymmetric can be judged by the symmetry along y=x.
AntiSymmetricWaveFunction := cos(Pi*x)*sin(2*Pi*y) - sin(2*Pi*x)*cos(Pi*y);
plot3d( AntiSymmetricWaveFunction, x=-0.5..0.5, y=-0.5..0.5, axes=frame );
SymmetricWaveFunction := cos(Pi*x)*sin(2*Pi*y) + sin(2*Pi*x)*cos(Pi*y);
plot3d( SymmetricWaveFunction, x=-0.5..0.5, y=-0.5..0.5, axes=frame ); NiM+STpBbnRpU3ltbWV0cmljV2F2ZUZ1bmN0aW9uRzYiLCYqJi1JJGNvc0c2JEkqcHJvdGVjdGVkR0YrSShfc3lzbGliR0YlNiMqJkkjUGlHRisiIiJJInhHRiVGMEYwLUkkc2luR0YqNiMsJComRi9GMEkieUdGJUYwIiIjRjBGMComLUYzNiMsJEYuRjhGMC1GKTYjRjZGMCEiIg== 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# Solve n linear equations in n variables
# Example uses:
# x + y + z = 10
# y + 2z = 12
# 2x + z = 9
with(linalg);
gaussjord(matrix([[1,1,1,10],[0,1,2,12],[2,0,1,9]])); 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 NiMtSSdtYXRyaXhHNiRJKnByb3RlY3RlZEdGJkkoX3N5c2xpYkc2IjYjNyU3JiIiIiIiIUYtIyIiKCIiJDcmRi1GLEYtIyIjNUYwNyZGLUYtRiwjIiM4RjA=