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Copyrighted material reprinted with permission from The Soft Warehouse:
DERIVE Help File
.
Main Menu
Special key commands:
Backspace delete character left of cursor
Del delete character at the cursor
Enter enter line of text
Ctrl-Enter enter and simplify line of text
Esc abort edit and return to menu
Ins toggle text input mode (insert or overwrite)
F1 display help on line-editing, functions, etc.
F3 copy highlighted expression onto author line
F4 copy highlighted expression in parentheses
F6 toggle arrow key mode (line-edit or subexpression)
Back to main menu, function menu or Derive Basics
Cursor movement commands (arrow keys NOT available in subexpression mode):
Ctrl-S or <-- move cursor left a character
Ctrl-D or --> move cursor right a character
Ctrl-A or Ctrl<-- move cursor left a word
Ctrl-F or Ctrl--> move cursor right a word
Ctrl-], Ctrl-Q S or Home move cursor to left end of line
Ctrl-\, Ctrl-Q D or End move cursor to right end of line
Text deletion commands:
Ctrl-H or Backspace delete character left of cursor
Ctrl-G or Del delete character at cursor
Ctrl-T delete word beginning at cursor
Ctrl-Y delete all text on line
Ctrl-Q Y delete text at and to right of cursor
Ctrl-Q H delete text to the left of cursor
Miscellaneous commands:
Ctrl-M or Enter enter line of text
Ctrl-J or Ctrl-Enter enter and simplify line of text
Ctrl-[ or Esc abort edit and return to menu
Ctrl-V or Ins toggle insert/overwrite modes
Ctrl-U insert previous line of text
Ctrl-P insert control or special character
Back to main menu, function menu or Derive Basics
Expression highlighting commands (Section 3.3):
Ctrl-E or Up arrow move up one expression
Ctrl-X or Down arrow move down one expression
Ctrl-R or PgUp move up one half screen
Ctrl-C or PgDn move down one half screen
Ctrl-PgUp or Ctrl-Home move up to first expression
Ctrl-PgDn or Ctrl-End move down to last expression
Ctrl-F or Ctrl--> scroll expression left one half screen
Ctrl-A or Ctrl<-- scroll expression right one half screen
Sub-expression highlighting commands (Section 3.3):
--> move right an operand
<-- move left an operand
Home move to leftmost operand
End move to rightmost operand
Ctrl-E or Up arrow move up one level
Ctrl-X or Down arrow move down one level
.
Function Menu
Entering Greek letters, special constants
and operators (Section 4.1)
(Note: type English equivalent name or, on PC compatibles only,
use Alt key):
Alt-A alpha Alt-M mu
Alt-B beta Alt-P pi
Alt-G GAMMA Alt-S sigma
Alt-D delta Alt-T tau
Alt-N epsilon Alt-F phi
Alt-H theta Alt-O omega
Alt-E #e - base of natural logs
Alt-I #i - square root of -1
Alt-P pi - area of a unit circle
Alt-0 inf - plus infinity
Alt - "+-" - plus-or-minus operator
Alt-V SUB - subscript operator
Alt-Q SQRT - square root operator
Alt-G GAMMA - Gamma function
Back to main menu, function menu or Derive Basics
Functions, Constants, and Operators
DERIVE can approximate and/or simplify the following functions,
constants, and operators. Given numeric arguments in approximate
mode, they are numerically approximated to the current precision.
Otherwise they are algebraically simplified. Many transformations
are applied automatically; some are applied only when requested
by a Manage command:
#e - base of natural logs (=2.71828...) (Alt-E)
#i - square root of -1 (Alt-I)
pi - area of unit circle (=3.14159...) (Alt-P)
inf - plus infinity (Alt-0)
deg - radians per degree (=pi/180)
unit_circle - arbitrary point on unit circle
euler_gamma := Euler's constant (=0.577215...)
u + v u plus v
- u minus u
u - v u minus v
u * v u times v
u v u times v
u / v u divided by v
u ^ v u raised to v
u % u percent
u ! u factorial
Back to main menu, function menu or Derive Basics
Relational operators (Section 4.13):
u = v u equals v
u /= v u not equal v
u < v u less than v
u <= v u less than or equal to v
u > v u greater than v
u >= v u greater than or equal to v
Boolean operator and functions (Section 19.5):
NOT p true iff p is false
p AND q true iff both p and q are true
p OR q true iff either p or q is true
p XOR q true iff p or q is true but not both
p IMP q true iff p is false or q is true
If p and q are integers, bit-wise logical operations are performed
TRUTH_TABLE (p,q,...,u,v,...) p,q,... are variables and u,v,..
are Booleans
Solving equations and relations (Section 4.14):
SOLVE (u,x) - solve u=0 for x
SOLVE (u=v,x) - solve u=v for x
SOLVE (u<v,x) - solve u<v for x
SOLVE (u=v,x,a,b) - solve u=v for x in [a,b], if in approximate mode
SOLVE ([u1=v1,u2=v2,...], [x1,x2,...]) - solve system linear in
x1, x2, ...
Back to main menu, function menu or Derive Basics
Exponential functions (Section 6.1):
SQRT (z) - square root of z (or press Alt-Q)
EXP (z) - #e raised to the power z
Logarithmic functions (Section 6.2):
LN (z) - natural log of z
LOG (z) - natural log of z
LOG (z,w) - log of z to the base w
Trigonometric functions (Section 6.3):
SIN (z*deg) - sine of z degrees
SIN (z) - sine of z radians
COS (z) - cosine of z radians
TAN (z) - tangent of z radians
COT (z) - cotangent of z radians
SEC (z) - secant of z radians
CSC (z) - cosecant of z radians
Inverse trigonometric functions (radians) (Section 6.4):
ATAN (z) - angle whose tangent is z
ATAN (y,x) - angle of the point (x,y)
ACOT (z) - angle whose cotangent is z
ACOT (x,y) - angle of the point (x,y)
ASIN (z) - angle whose sine is z
ACOS (z) - angle whose cosine is z
ASEC (z) - angle whose secant is z
ACSC (z) - angle whose cosecant is z
Back to main menu, function menu or Derive Basics
Hyperbolic functions (Section 6.5):
SINH (z) - hyperbolic sine of z
COSH (z) - hyperbolic cosine of z
TANH (z) - hyperbolic tangent of z
COTH (z) - hyperbolic cotangent of z
SECH (z) - hyperbolic secant of z
CSCH (z) - hyperbolic cosecant of z
Inverse hyperbolic functions (Section 6.6):
ASINH (z) - inverse hyperbolic sine of z
ACOSH (z) - inverse hyperbolic cosine of z
ATANH (z) - inverse hyperbolic tangent of z
ACOTH (z) - inverse hyperbolic cotangent of z
ASECH (z) - inverse hyperbolic secant of z
ACSCH (z) - inverse hyperbolic cosecant of z
Piecewise continuous functions (Section 6.7):
ABS (x) - absolute value of x
SIGN (x) - sign of x
MAX (x, y, ...) - maximum of arguments
MIN (x, y, ...) - minimum of arguments
STEP (x) - returns 1 if x>0, 0 if x<0
CHI (a, x, b) - returns 1 if a<x<b, 0 if x<a or x>b
FLOOR (m, n) - greatest integer <= m/n
FLOOR (m) - integer part of m
MOD (m, n) - m modulo n (nonnegative remainder of m/n)
MOD (m) - fractional part of m
MODS (m, n) - symmetric m modulo n
Back to main menu, function menu or Derive Basics
Complex variable functions (Section 6.8):
ABS (z) - magnitude of z
SIGN (z) - radial projection of z on unit circle
RE (z) - real part of z
IM (z) - imaginary part of z
CONJ (z) - complex conjugate of z
PHASE (z) - phase angle of z
Probability functions (Section 6.9):
z! - z factorial
GAMMA (z) - gamma function of z
PERM (z,w) - permutations of z things taken w at a time
COMB (z,w) - combinations of z things taken w at a time
RANDOM (n) - if n>1, a random INTEGER in the interval [0,n)
RANDOM (n) - if n=1, a random NUMBER in the interval [0,1)
RANDOM (n) - if n<1, initialize random number seed to n
RANDOM (n) - if n=0, initialize seed based on current time
Statistical functions (Section 6.10):
AVERAGE (z1, ..., zn) - arithmetic mean (average)
RMS (z1, ..., zn) - root mean square
VAR (z1, ..., zn) - variance
STDEV (z1, ..., zn) - standard deviation
FIT (v, A) - least squares fit of label vector v to data matrix A
Back to main menu, function menu or Derive Basics
Error functions (Section 6.11):
ERF (z) - error function
ERF (z,w) - generalized error function
ERFC (z) - complementary error function
NORMAL (z,m,s) - normal distribution with mean m and standard deviation s
NORMAL (z) - cumulative distribution of z
ZETA (s) - the Riemann zeta function of s
Financial functions (Section 6.12):
PVAL (i, nper, pmt, fval, time) - present value of contract
FVAL (i, nper, pmt, pval, time) - future value of contract
PMT (i, nper, pval, fval, time) - periodic payment
NPER (i, pmt, pval, fval, time) - number of payment periods
RATE (nper, pmt, pval, fval, time, min, max) - periodic interest rate
Number theory functions (Section 6.13)
GCD (m, n, ...) - greatest common divisior of m, n, ...
LCM (m, n, ...) - least common multiple of m, n, ...
PRIME (n) - true if n is probably prime, false if not
PRIME (n,k) - do k iterations of probabilistic primality test before assuming n is prime
(default k=5)
NEXT_PRIME (n) - next prime larger than n
NEXT_PRIME (n,k) - do k iterations of probabilistic primality test
before assuming a number is prime (default k=5)
Expression decomposition functions (Section 6.14)
NUMERATOR (u) - syntactic numerator of u
DENOMINATOR (u) - syntactic denominator of u
QUOTIENT (u,v) - polynomial quotient of u divided by v
REMAINDER (u,v) - polynomial remainder of u divided by v
POLY_GCD (u,v) - polynomial gcd of u and v
TERMS (u) - vector of syntactic terms of u
FACTORS (u) - vector of syntactic factors of u
VARIABLES (u) - a vector of the free variables in u
LHS (r) - the left (hand) side of relation r
RHS (r) - the right (hand) side of relation r
Back to main menu, function menu or Derive Basics
Calculus functions
7.1 LIM (u,x,a) - limit of u(x) as x approaches a
7.1 LIM (u,x,a,1) - limit of u(x) as x approaches a from above
7.1 LIM (u,x,a,-1) - limit of u(x) as x approaches a from below
7.2 DIF (u,x) - derivative of u(x) wrt x
7.2 DIF (u,x,n) - nth order derivative of u(x) wrt x
7.3 TAYLOR (u,x,a,n) - nth order Taylor approximation of u(x)
about x=a
7.4 INT (u,x) - antiderivative of u(x) wrt x
7.4 DIF (u,x,-n) - nth-order antiderivative of u(x) wrt x
7.4 INT (u,x,a,b) - definite integral of u(x) from x=a to b
7.5 SUM (u,k) - antidifference of u(k) wrt k
7.5 SUM (u,k,m,n) - definite sum of u(k) from k=m to n
7.5 SUM (u,k,v) - sum of u(k) for k an element of vector v
7.5 SUM (v) - sum of the elements of vector v
7.6 PRODUCT (u,k) - antiquotient of u(k) wrt k
7.6 PRODUCT (u,k,m,n) - definite product of u(k) from k=m to n
7.6 PRODUCT (u,k,v) - product of u(k) for k an element of vector v
7.6 PRODUCT (v) - product of the elements of vector v
Back to main menu, function menu or Derive Basics
Generating vectors and matrices (Section 8.2)
[x1, x2, ..., xn] - n-element vector
[[a, b], [c, d]] - 2x2 matrix
VECTOR (u,k,n) - vector of u(k) as k goes from 1 thru n in steps of 1
VECTOR (u,k,m,n) - vector of u(k) as k goes from m thru n in steps of 1
VECTOR (u,k,m,n,s) - vector of u(k) as k goes from m thru n in steps of s
VECTOR (u,k,v) - vector of u(k) applied to elements of vector v
IDENTITY_MATRIX (n) - nxn identity matrix
DIMENSION (v) - number of elements of vector v
ABS (v) - magnitude (length) of vector v
Vector manipulation functions (Section 8.3)
v SUB n - nth element of vector v
ELEMENT(v,n) - nth element of vector v
A SUB j SUB k - element in jth row and kth column of matrix A
ELEMENT(A,j,k) - element in jth row and kth column of matrix A
APPEND (v,w) - vector of elements of v followed by elements of w
DELETE_ELEMENT (v,n) - delete element n from vector v
INSERT_ELEMENT (u,v,n) - insert u before the nth element of v
REPLACE_ELEMENT (u,v,n) - replace the nth element of v with u
REVERSE_VECTOR (v) - reverse elements of vector v
SELECT (u,k,m,n,s) - vector of k as k goes from m thru n in steps
of s for which u(k)is true
SELECT (u,k,v) - vector of those elements of v for which u(k) is true
Vector operations (Section 8.4)
v . w dot product of vectors v and w (period NOT required if v
and w declared Nonscalar)
CROSS (v,w) - cross product of vectors v and w
Matrix operations
8.4 A.B - dot product of matrices
8.5 A` - transpose of A (On PC-9801 use Yen char)
8.5 DET (A) - determinant of A
8.5 TRACE (A) - trace of A (sum of diagonal elements)
8.5 A^-1 - inverse of A
8.6 ROW_REDUCE (A) - row echelon form of A
8.6 ROW_REDUCE (A, B) - row echelon form of A augmented by B
8.7 CHARPOLY (A, mu) - characteristic polynomial of A, using mu
8.7 EIGENVALUES (A, mu) - eigenvalues of A
Back to main menu, function menu or Derive Basics
Differential vector calculus (Section 8.9):
GRAD (expn) - gradient of expn depending on x, y, and z
GRAD (expn,v) - gradient of expn depending on variables in vector v
GRAD (expn,A) - gradient of expn in coordinate system A
DIV (v,A) - divergence of vector v
LAPLACIAN (expn,A) - divergence of the gradient of expn
CURL (v,A) - curl of vector v
Integral vector calculus (Section 8.10):
POTENTIAL (v) - scalar potential of vector v
POTENTIAL (v,w) - potential of vector v with starting coordinates w
POTENTIAL (v,w,A) - potential of vector v in coordinate system A
VECTOR_POTENTIAL(v,w,A) - vector potential of vector v
Programming functions and operators
3.10 APPROX (u) - approximate u using the current digits of precision
3.10 APPROX (u,n) - approximate u using n digits of precision
4.4 EXPAND (u,amount,x,y,...) - expand u(x,y) by amount wrt variables x,y,...
4.6 FACTOR (u,amount,x,y,...) - factor u(x,y) by amount wrt variables x,y,...
4.10 x :î domain - declare x an element of domain (Integer, Real, Complex or Nonscalar)
4.10 x :î domain interval - declare x an element of domain
(Integer or Real) and interval e.g. [-2,10)
4.11 x := u - assign variable x the value u
4.11 x := - make x an arbitrary variable
4.12 f(x,y,...) := u - define function f of variables x,y,... the definition u
4.12 f(x,y,...) := - make f an arbitrary function
10.1 ITERATES (u,x,x0) - returns the vector [x0,u(x0),u(u(x0)),...]
until an element is repeated
10.1 ITERATES (u,x,x0,n) - returns the first n+1 elements of the
vector [x0,u(x0),u(u(x0)),...]
10.2 ITERATE (u,x,x0) - returns the first repeated element of the
sequence x0, u(x0), u(u(x0)), ...
10.2 ITERATE (u,x,x0,n) - returns the n+1th element of the sequence
x0, u(x0), u(u(x0)), ...
10.2 ITERATE (u,[x1,x2,...],[x01,x02,...]) - first repeated element
of [x01,x02,...],u(x01,x02,...),...
10.3 IF (r) - if relation r is true, return 1; if false return 0
10.3 IF (r,t,f) - if relation r is true, return expression t; if
false, return expression f
10.3 IF (r,t,f,u) - if relation r is true, return expression t;
if false, return expression f; if truth unknown, return u
Back to main menu, function menu or Derive Basics
To issue a menu option command, press Space bar until desired
command is highlighted, then press Enter key. Alternatively,
type the letter that is capitalized in the desired command.
The following gives the section in the DERIVE User Manual that
describes each command menu option.
3.1 Author - enter new expression on author line
3.5 Build - combine expressions using operators and functions Calculus
7.2 Differentiate - differentiate an expression
7.4 Integrate - integrate an expression
7.1 Limit - find the limit of an expression
7.6 Product - find the closed-form product of an expression
7.5 Sum - find the closed-form sum of an expression
7.3 Taylor - find Taylor polynomal series approximation
8.2 Vector - generate a vector of values Declare
4.12 Function - define a function and assign it a definition Variable
4.11 Value - declare a variable and assign it a value
4.10 Integer - declare an integer-valued variable
4.10 Real - declare a real-valued variable
4.10 Complex - declare a complex-valued variable
4.10 Nonscalar - declare a variable nonscalar
8.1 Matrix - enter a matrix
8.1 vectoR - enter a vector
4.4 Expand - expand an expression wrt some or all variables
4.6 Factor - factor an expression wrt some or all variables
2.5 Help get on-line help on:
2.4 Editing - line editing commands
6 Functions - pre-defined functions and constants
2.2 Algebra - algebra window commands
5.1 2D-plot - 2D-plot window commands
5.2 3D-plot - 3D-plot window commands
9 Utility - utility file functions
2.14 State - current state of system control variables
2.5 Return - return to DERIVE
3.3 Jump - move highlight to selected expression
4.14 soLve - solve equation, relation, or linear system of equations Manage
3.12 Annotate - annotate an expression
4.5 Branch - select principal, real, or any branch simplification
6.1 Exponential - collect or expand exponentials
6.2 Logarithm - collect or expand logarithms
4.3 Ordering - specify variable ordering
3.7 Renumber - sequentially renumber expressions in algebra window
4.8 Substitute - substitute a value for variables or subexpressions
6.3 Trigonometry - collect or expand trig functions and select angle mode
Options
Color
2.12 Menu - set menu, message, and status line colors
2.12 Work - set foreground and background colors of expressions
2.12 Display - set graphics or text display mode
2.7 Execute - execute a DOS command
4.1 Input - set variable name, case sensitivity, and arrow key modes
2.3 Mute - turn on or off audible error beeps
3.9 Notation - set notation used to display numbers
3.13 Output - set display format and times operator
3.8 Precision - set precision used for arithmetic operations
3.11 Radix - set input and output radix bases
5.2 Plot - switch to a 2D-plot or 3D-plot window
2.6 Quit - quit DERIVE; return to DOS
3.7 Remove - delete a block of one or more expressions
3.2 Simplify - simplify an expression or subexpression
Back to main menu, function menu or Derive Basics
Transfer
Load
2.9 Derive - load expressions from MTH file
2.14 State - load control settings from INI file
2.9 daTa - load numeric data arrays from DAT file
2.9 Utility - load, but do not display, definitions from MTH file
Save
2.8 Derive - save expressions in MTH file
2.8 Basic - save expressions in BAS file
2.8 C - save expressions in a C file
2.8 Fortran - save expressions in FOR file
2.8 Pascal - save expressions in PAS file
2.8 Options - select save range, file linelength, and annotation saving
2.14 State - save current control settings in INI file
2.9 Merge - merge expressions from MTH file
2.9 Clear - clear all expressions from algebra window
2.9 Demo - run demonstration (DMO) file
Print
Expressions
2.10 Printer - print expressions in algebra window
2.10 File - send text image of expressions to an ASCII file
2.10 Options - select print range, annotation, and character set
Screen
2.11 Printer - print screen image (= Shift-F10 or Shift-F9)
2.11 File - save screen image in TIF file (= Ctrl-F10 or Ctrl-F9)
2.11 Options - select print region, positioning, TIF file name
2.10 Layout - set page size and margins
2.10 Options - select printer type, font size, orientation, paper size
3.7 Unremove - recover last removed expressions
3.6 moVe - move a block of one or more expressions
Window
2.13 Close - close a window
2.13 Designate - redesignate active window
2.13 Flip - flip overlaid windows (= F2 key)
2.13 Goto - go to given window
2.13 Next - go to next window (= F1 key)
2.13 Open - open new window
2.13 Previous - go to previous window (= Shift-F1)
Split
2.13 Horizontal - split active window horizontally
2.13 Vertical - split active window vertically
3.10 approX - approximate an expression or subexpression
Back to main menu, function menu or Derive Basics
Algebra window function key commands:
F1 - go to next window
F2 - flip overlaid windows
F5 - switch to previous display mode
Shift-F10 - print entire screen (not available on PC-9801)
Shift-F9 - print current window (not available on PC-9801)
Ctrl-F10 - send entire screen to TIF file
Ctrl-F9 - send current window to TIF file
To issue a menu option command, press Space bar until desired
command is highlighted, then press Enter key. Alternatively,
type the letter that is capitalized in desired command.
The following gives the section in the DERIVE User Manual that
describes each
2D-plot window's command menu option.
5.1.4 Algebra - switch to an algebra window
5.1.3 Center - center window on the cross
Delete delete from plot list:
5.1.4 All - all expressions
5.1.4 Butlast - all but the last expression
5.1.4 First - first expression
5.1.4 Last - last expression
2.5 Help - (same as algebra window's Help command)
5.1.1 Move - move cross to the specified coordinates
Options
5.1.5 Accuracy - set plot accuracy
Color
5.1.2 Auto - turn off auto color change
5.1.2 Plot - set plot and axes colors
2.12 Menu - set menu, message, and status line colors
2.12 Work - set plot area background color
2.12 Display - set graphics or text display mode
2.7 Execute - execute a DOS command
2.3 Mute - turn on or off audible error beeps
3.9 Notation - set notation used to display numbers
3.8 Precision - set precision used for arithmetic operations
3.11 Radix - set input and output radix bases
5.1.6 State - select coordinates, follow mode, trace mode, and
point size
5.1.1 Plot - plot highlighted expression
2.6 Quit - quit DERIVE; return to DOS
5.1.3 Range - set plot range (field of view)
5.1.3 Scale - set plot scale
Transfer
Load
2.14 State - load control settings from INI file
Save
2.14 State - save current control settings in INI file
Print
Screen
2.11 Printer - print screen image (= Shift-F10 or Shift-F9)
2.11 File - save screen image in TIF file (= Ctrl-F10 or Ctrl-F9)
2.11 Options - select print region, positioning, and TIF file name
2.10 Layout - set page size and margins
2.10 Options - select printer, font size, orientation, paper size
2.13 Window - (same as Window command of algebra window)
5.1.2 aXes - set aspect ratio, and axes titles and labels
5.1.3 Zoom - zoom (adjust) plot scale in or out
Back to main menu, function menu or Derive Basics
F1 - go to next window
F2 - flip overlaid windows
F3 - toggle trace mode
F5 - switch to previous display mode
F7 - zoom y-axis in
F8 - zoom y-axis out
Shift-F7 - zoom x-axis in
Shift-F8 - zoom x-axis out
F9 - zoom both axes in
F10 - zoom both axes out
Shift-F10 - print entire screen (not available on PC-9801)
Shift-F9 - print current window (not available on PC-9801)
Ctrl-F10 - send entire screen to TIF file
Ctrl-F9 - send current window to TIF file
Back to main menu, function menu or Derive Basics
To issue a menu option command, press Space bar until desired
command is highlighted, then press Enter key. Alternatively,
type the letter that is capitalized in desired command.
The following gives the section in the DERIVE User Manual that
describes each 3D-plot window's command menu option.
5.2.4 Algebra - switch to algebra window
5.2.2 Center - set coordinates of box center
5.2.3 Eye - set coordinates of viewer's eye
5.2.3 Focal - set coordinates of focal point
5.2.1 Grids - set number of grid panels
5.2.1 Hide - control display of hidden lines
5.2.2 Length - set lengths of sides of transparent box
Options
Color
5.2.1 Plot - set top, bottom, and axes colors
2.12 Menu - set menu, message, and status line colors
2.12 Work - set background color of plot area
2.12 Display - set graphics or text display mode
2.7 Execute - execute a DOS command
2.3 Mute - turn on or off audible error beeps
3.9 Notation - set notation used to display numbers
3.8 Precision - set precision used for arithmetic operations
3.11 Radix - set input and output radix bases
5.2.1 Plot - plot highlighted expression
2.6 Quit - quit DERIVE; return to DOS
Transfer
Acrospin
5.2.4 Run - run AcroSpin with an existing ACD data file
5.2.4 Save - save 3D plot points in an ACD data file
Load
2.14 State - load control settings from INI file
Save
2.14 State - save current control settings in INI file
Print
Screen
2.11 Printer - print screen image (= Shift-F10 or Shift-F9)
2.11 File - save screen image in TIF file (= Ctrl-F10 or Ctrl-F9)
2.11 Options - select print region, positioning, and TIF file name
2.10 Layout - set page size and margins
2.10 Options - select printer, font size, orientation, paper size
2.13 Window - (same as Window command of algebra window)
5.2.1 aXes - control axes display
5.2.2 Zoom - zoom box side lengths in or out
Back to main menu, function menu or Derive Basics
3D-plot window function key commands:
F1 - go to next window
F2 - flip overlaid windows
F5 - switch to previous display mode
F7 - zoom y-axis in
F8 - zoom y-axis out
Shift-F7 - zoom x-axis in
Shift-F8 - zoom x-axis out
F9 - zoom both axes in
F10 - zoom both axes out
Shift-F10 - print entire screen (not available on PC-9801)
Shift-F9 - print current window (not available on PC-9801)
Ctrl-F10 - send entire screen to TIF file
Ctrl-F9 - send current window to TIF file
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9.1 SOLVE.MTH - Solving Nonlinear Systems: Simplify using approX
command. Use complex x0 for complex solutions.
NEWTONS(u,x,x0,n) := n iterations of Newton's method for vector
u(x)=0
FIXED_POINT(g,x,x0,n) := n iterations of vector x=g(x), starting
at x=x0
TAYLOR_SOLVE(u,x,y,x0,y0,n) := nth order series solution y(x)
of u(x,y)=0
TAYLOR_INVERSE(u,x,y,x0,n) := nth order series expansion of
inverse of y=u(x)
9.2 VECTOR.MTH - Additional Vector and Matrix Functions. If
v and w are vectors, and A and B are matrices:
i_ := [1, 0, 0]: unit vector for x-axis
j_ := [0, 1, 0]: unit vector for y-axis
k_ := [0, 0, 1]: unit vector for z-axis
OUTER (v,w) := outer product of v and w
KRONECKER (i,j) := Kronecker delta function
ADJOIN_ELEMENT (e,v) := adjoin e to front of v
APPEND_COLUMNS (A,B) := append columns of A and B
MINOR (A,i,j) := delete row i and column j from A
SWAP_ELEMENTS (v,i,j) := interchange elements i and j of v
SCALE_ELEMENT (v,i,s) := multiply element i of v by s
SUBTRACT_ELEMENTS (v,i,j,s) := subtract element j*s from element i of v
FORCE0 (A,i,j,p) := force element i,j of A to 0 using pivot row p
PIVOT (A,i,j) := force column j below row i to 0 by pivoting
MATPROD (A,B,i,j) := element i,j of the dot product of A and B
COFACTOR (A,i,j) := numerator of element i,j of A inverse
ADJOINT (A) := adjoint of square matrix A
RANK (A) := rank of matrix A
EXACT_EIGENVECTOR (A,mu) := eigenvector of A < 4x4 for eigenvalue mu
APPROX_EIGENVECTOR (A,mu) := approximate eigenvector of A: Use
approX command
JACOBIAN (u,v) := Jacobian for vector x=u(c), curvilinear coord c and Cart. x
COVARIANT_METRIC_TENSOR (A) := covariant metric tensor of Jacobian
matrix A
GEOMETRY_MATRIX (c,G) := geometry matrix of curv coord c and metric
tensor G
cylindrical := geometry matrix for cylindrical coordinates r,theta, z
spherical := geometry matrix for spherical coordinates r, theta, phi
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9.3 NUMERIC.MTH - Numeric Differentiation and Integration: Use approX command.
DIF_NUM (y,x,x0,h) := 1st derivative of y wrt x at x0 using step size h
DIF2_NUM (y,x,x0,h) := 2nd derivative of y wrt x
SMOOTH_VECTOR (v) := smoothed copy of vector v
SMOOTH_COLUMN (A,j) := matrix A with column j smoothed
DIF_DATA (A) := 1st derivative of 2-column numeric data matrix A
DIF2_DATA (A) := 2nd derivative of A
INT_DATA (A) := antiderivative of A
9.4 DIF_APPS.MTH - Applications of Differentiation
CURVATURE (y,x) := curvature of y(x)
CENTER_OF_CURVATURE (y,x) := center of curvature of y(x)
TANGENT (y,x,x0) := line tangent to y(x) at x=x0
PERPENDICULAR (y,x,x0) := line perpendicular to y(x) at x=x0
OSCULATING_CIRCLE (y,x,theta) := circle osculating y(x), in terms
of theta
PARA_DIF (v,t,n) := nth derivative of v wrt t, where v=[x(t),y(t)]
PARA_CURVATURE (v,t) := curvature of v
PARA_CENTER_OF_CURVATURE (v,t) := center of curvature of v
PARA_TANGENT (v,t,t0,x) := line tangent to v at t=t0, in terms of x
PARA_PERPENDICULAR (v,t,t0,x) := line perpendicular to v at t=t0
PARA_OSCULATING_CIRCLE (v,t,t0,phi) := circle osculating v, in terms of phi
POLAR_DIF (r,theta,n) := nth derivative of r(theta), in terms of theta
POLAR_CURVATURE (r,theta) := curvature of r
POLAR_CENTER_OF_CURVATURE (r,theta) := center of curvature of r
POLAR_TANGENT(r,theta,theta0,x):=line tangent to r at theta=theta0 in terms of x
POLAR_PERPENDICULAR(r,theta,theta0,x):=line perpendicular to r at theta=theta0
POLAR_OSCULATING_CIRCLE(r,theta,theta0,phi):=osculating circle in
terms of phi
IMP_DIF (u,x,y,n) := implicit derivative DIF(y,x,n) for u(x,y)=0
IMP_CURVATURE (u,x,y) := curvature of implicit curve u=0
IMP_CENTER_OF_CURVATURE (u,x,y) := center of curvature of u=0
IMP_TANGENT (u,x,y,x0,y0) := line tangent to u=0 at x=x0 and y=y0
IMP_PERPENDICULAR (u,x,y,x0,y0) := line perpendicular to u=0 at x0
and y0
IMP_OSCULATING_CIRCLE (u,x,y,x0,y0,phi) := osculating circle in terms
of phi
TANGENT_PLANE (u,v,v0) := plane tangent to u(x,y,z)=0 at [x,y,z]=v=v0
NORMAL_LINE (u,v,v0,t) := line normal to u=0 at v=v0, using parameter t
9.5 INT_APPS.MTH - Applications of Integration
FOURIER (y,t,t1,t2,n) := nth-harmonic Fourier series of y(t) from
t=t1 to t2
LAPLACE (y,t,s) := Laplace transform of y(t) for transform domain
variable s.
s must be declared sufficiently large for the integral to converge.
To find the inverse Laplace transform, use the Expand command
to find the partial fraction expansion, then lookup the inverse
in a table.
ARC_LENGTH (y,x,x1,x2) := arc length of y(x) from x=x1 to x2
ARC_LENGTH (y,x,x1,x2,mu) := integral of mu(x) along arc y(x) from
x=x1 to x2
POLAR_ARC_LENGTH (r,th,th1,th2) := arc length of polar r(th) from
th1 to th2
POLAR_ARC_LENGTH (r,th,th1,th2,mu) := integral of mu(th) along arc r
PARA_ARC_LENGTH (v,t,t1,t2) := arc length of vector v(t) from t=t1
to t2
PARA_ARC_LENGTH (v,t,t1,t2,mu) := integral of mu(t) along v
AREA (x,x1,x2,y,y1,y2) := area of region x=x1 to x2 and y=y1(x) to
y2(x)
AREA (x,x1,x2,y,y1,y2,mu) := integral of mu(x,y) over region
AREA_CENTROID (x,x1,x2,y,y1,y2) := areal centroid of region
AREA_CENTROID (x,x1,x2,y,y1,y2,mu) := centroid of density mu(x,y)
over region
AREA_INERTIA (x,x1,x2,y,y1,y2) := areal inertia tensor of region
AREA_INERTIA (x,x1,x2,y,y1,y2,mu) := inertia tensor of density mu(x,y)
POLAR_AREA (r,r1,r2,th,th1,th2) := area of th=th1 to th2 and r=r1(th)
to r2(th)
POLAR_AREA (r,r1,r2,th,th1,th2,mu) := integral of mu(th) over region
SURFACE_AREA (z,x,x1,x2,y,y1,y2) := area of surface z(x,y)
SURFACE_AREA (z,x,x1,x2,y,y1,y2,mu) := integral of mu(x,y) over
surface z
VOLUME(x,x1,x2,y,y1,y2,z,z1,z2):=volume y=y1(x) to y2(x), z=z1(x,y)
to z2(x,y)
VOLUME (x,x1,x2,y,y1,y2,z,z1,z2,mu) := integral of mu(x,y,z)
over region
VOLUME_CENTROID (x,x1,x2,y,y1,y2,z,z1,z2) := volumetric centroid
of region
VOLUME_CENTROID (x,x1,x2,y,y1,y2,z,z1,z2,mu) := centroid of density mu(x,y,z)
VOLUME_INERTIA (x,x1,x2,y,y1,y2,z,z1,z2) := volumetric inertia tensor
VOLUME_INERTIA (x,x1,x2,y,y1,y2,z,z1,z2,mu) := inertia tensor of
mu(x,y,z)
SPHERICAL_VOLUME(r,r1,r2,th,th1,th2,phi,phi1,phi2):=volume in
spherical coord.
r=r1 to r2, th=th1(r) to th2(r), phi=phi1(r,th) to phi2(r,th)
SPHERICAL_VOLUME (r,r1,r2,th,th1,th2,phi,phi1,phi2,mu) := integral
of mu
CYLINDRICAL_VOLUME(r,r1,r2,th,th1,th2,z,z1,z2) := volume in
ylindrical coord.
z=z1 to z2, th=th1(z) to th2(z), r=r1(th,z) to r2(th,z)
CYLINDRICAL_VOLUME (r,r1,r2,th,th1,th2,z,z1,z2,mu) := integral of mu
VOLUME_OF_REVOLUTION (y,x,x1,x2) := volume of y(x) revolved about
x-axis
AREA_OF_REVOLUTION (y,x,x1,x2) := area of y(x) revolved about x-axis
VOLUMEY_OF_REVOLUTION (y,x,x1,x2) := volume of y(x) revolved about
y-axis
AREAY_OF_REVOLUTION (y,x,x1,x2) := area of y(x) revolved about y-axis
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9.6 ODE1.MTH - First Order Ordinary Differential Equations (Elementary Methods)
Specific solution for the initial condition y=y0 at x=x0:
DSOLVE1 (p,q,x,y,x0,y0) := solves p(x,y)+q(x,y)y'=0
SEPARABLE (p,q,x,y,x0,y0) := solves y'=p(x)q(y)
LINEAR1 (p,q,x,y,x0,y0) := solves y'+p(x)y=q(x)
HOMOGENEOUS (r,x,y,x0,y0) := solves y'=r(x,y) if r is homogeneous
EXACT (p,q,x,y,x0,y0) := solves p(x,y)+q(x,y)y'=0 if it is exact
INTEGRATING_FACTOR (p,q,x,y,x0,y0) := solves p(x,y)+q(x,y)y'=0 if integrating
factor exists
General solution in terms of the constant c:
DSOLVE1_GEN (p,q,x,y,c) := solves p(x,y)+q(x,y)y'=0
SEPARABLE_GEN (p,q,x,y,c) := solves y'=p(x)q(y)
LINEAR1_GEN (p,q,x,y,c) := solves y'+p(x)y=q(x)
HOMOGENEOUS_GEN (r,x,y,c) := solves y'=r(x,y) if r is homogeneous
EXACT_GEN (p,q,x,y,c) := solves p(x,y)+q(x,y)y'=0 if it is exact
INTEGRATING_FACTOR_GEN (p,q,x,y,c) := solves p(x,y)+q(x,y)y'=0 if
integrating factor exists
9.6 ODE1.MTH - First Order Ordinary Differential Equations (Advanced Methods)
MONOMIAL_TEST (p,q,x,y) := integrating factor of p(x,y)+q(x,y)y'=0
if of the form x^m*y^n
Specific solution for the initial condition y=y0 at x=x0:
BERNOULLI (p,q,k,x,y,x0,y0) := solves y'+p(x)y=q(x)y^k
GEN_HOM (r,x,y,x0,y0) := solves y'=r(x,y) if r is generalized
homogeneous
FUN_LIN_CCF (r,p,q,k,x,y,x0,y0) := solves y'=r(p*x+q*y+k) if p,q,k constant
LIN_FRAC (r,a,b,c,p,q,k,x,y,x0,y0) := solves y'=r((ax+by+c)/(px+qy+k))
ALMOST_LIN (r,b,p,q,x,y,x0,y0) := solves r(x,y)y'+p(x)b(y)=q(x) if
almost linear
General solution in terms of the constant c:
BERNOULLI_GEN (p,q,k,x,y,c) := solves y'+p(x)y=q(x)y^k
GEN_HOM_GEN (r,x,y,c) := solves y'=r(x,y) if r is generalized
omogeneous
FUN_LIN_CCF_GEN (r,p,q,k,x,y,c) := solves y'=r(p*x+q*y+k) if p,q,k constant
LIN_FRAC_GEN (r,a,b,c,p,q,k,x,y,c) := solves y'=r((ax+by+c)/(px+qy+k))
ALMOST_LIN_GEN (r,b,p,q,x,y,c) := solves r(x,y)y'+p(x)b(y)=q(x) if
almost linear
CLAIRAUT (p,q,x,y,v,c) := solves p(x*v-y)=q(v) where v represents y'
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9.7 ODE2.MTH - Second Order Ordinary Differential Equations
DSOLVE2 (p,q,r,x,c1,c2) := solves y"+p(x)y'+q(x)y=r(x)
DSOLVE2_BV (p,q,r,x,x0,y0,x2,y2) := solves y"+p(x)y'+q(x)y=r(x)
DSOLVE2_IV (p,q,r,x,x0,y0,v0) := solves y"+p(x)y'+q(x)y=r(x)
AUTONOMOUS_CONSERVATIVE(q,x,y,x0,y0,v0):=solves y"=q(y),
y=y0 and y'=v0 at x=x0
LIOUVILLE (p,q,x,y,c1,c2) := solves y"+p(x)y'+q(y)(y')^2=0
AUTONOMOUS (r,v) := dv/dy, given y"=r(y,v), reducing to sequence
of 2 1st ord
EXACT2 (p,q,x,y,v,c) reduces order of p(x,y,v)y"+q(x,y,v)=0
with v=y' if exact
9.8 ODE_APPR.MTH - Approximate Series Solutions of ODEs
Given the equation y'=r(x,y) with y=y0 at x=x0:
TAYLOR_ODE1(r,x,y,x0,y0,n) := nth order Taylor series solution
PICARD(r,p,x,y,x0,y0) := improved series approximation, given the
series p(x)
Given m equations yi'=ri(x,y1,...,ym) with yi=yi0 at x=x0:
TAYLOR_ODES (r,x,y,x0,y0,n) := vector of nth order Taylor series
solutions
PICARD(r,p,x,y,x0,y0) := improved series approximation, given the
series p(x)
Given the equation y''=r(x,y,y') with y=y0 and y'=v=v0 at x=x0:
TAYLOR_ODE2(r,x,y,v,x0,y0,v0,n) := nth order Taylor series solution
9.8 ODE_APPR.MTH - Approximate Numerical Solutions of ODEs
Given the equation y'=r(x,y) with y=y0 at x=x0:
EULER(r,x,y,x0,y0,h,n) := [[x0,y0],...,[xn,yn]] with step size h
Given m equations yi'=ri(x,y1,...,ym) with yi=yi0 at x=x0:
RK (r,v,v0,h,n) := [[x0, y0(x0), ... ym(x0)], ... [xn, y0(xn), ... ym(xn)]] classic
Runge-Kutta with step h.
EXTRACT_2_COLUMNS (A,j,k) := columns j and k from A for plots of RK results
DIRECTION_FIELD(r,x,x0,xm,m,y,y0,yn,n): grid (x0,y0) to (xm,yn)
9.9 RECUREQN.MTH - Recurrence Equations
LIN1_DIFFERENCE (p,q,x,x0,y0) := solves y(x+1)=p(x)y(x)+q(x), y(x0)=y0
RECURRENCE1 (r,x,y,x0,y0,n) := n steps of y(x+1)=r(x,y(x)), y(x0)=y0
GEOMETRIC1 (k,p,q,x,x0,y0) := solves y(k*x)=p(x)y(x)+q(x), y(x0)=y0
CLAIRAUT_DIF (p,q,d,x,y,c) := solves p(y-xd)=q(d), d represents
y(x+1)-y(x)
LIN2_CCF (p,q,r,x,c1,c2) := solves y(x+2)+p*y(x+1)+q*y(x)=r(x)
LIN2_CCF_BV (p,q,r,x,x0,y0,x2,y2) := solves y(x+2)+p*y(x+1)+q*y(x)=r(x)
9.10 APPROX.MTH - Pade Rational Approximation
PADE(y,x,x0,n,d):=approx y(x) near x=x0, n=numr deg, d=denr deg, n=d
or d-1
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9.11 EXP_INT.MTH - Exponential, Log, Sine, and Cosine Integrals:
use approX
EI(x,m):=m terms of series for exponential integral
INT(#e^-t/t,t,-x,inf), x>0
LI(x,m):=m terms of series for logarithmic integral
INT(1/LN t,t,0,x), x>1
EN (n,z) := nth exponential integral INT(exp(-zt)/t^n,t,1,inf), RE(z)>0, n>=0
EN_ASYMP (n,z,m) := m+1 terms of asymptotic series for EN(n,z),
|z| large
E1(z,m) := m terms of series for exponential integral E1(z)=EN(1,z)
SI(z) := sine integral INT(SIN(t)/t,t,0,z)
CI(z) := cosine integral INT(COS(t)/t,t,0,z), |phase z| < pi
9.12 PROBABIL.MTH - Additional Probability Functions: use approX
POCHHAMMER (a,x) := Pochhammer symbol (a)x = GAMMA(a+x)/GAMMA(a)
PSI (z) := DIF(GAMMA(z),z)/GAMMA(z), |phase z| < pi/2
POLYGAMMA (n,z,m) := m+1 terms of series for DIF(PSI(z),z,n),
z/=0,-1,-2,...
INCOMPLETE_GAMMA (z,w) := P(z,w)=INT(#e^-t*t^(z-1),t,0,w)/GAMMA(z),
RE(z)>0
INCOMPLETE_GAMMA_SERIES (z,w,m) := m+1 terms of series for above
BETA (z,w) := beta function B(z,w)=GAMMA(z)*GAMMA(w)/GAMMA(z+w)
INCOMPLETE_BETA (x,z,w) := Bx(z,w) = INT(t^(z-1)*(1-t)^(w-1),t,0,x)
POISSON_DENSITY (k,t) := Poisson probability density #e^-t*t^k/k!
POISSON_DISTRIBUTION (k,t) := SUM (#e^-t*t^j/j!,j,0,k)
BINOMIAL_DENSITY (k,n,p) := COMB(n,k)*p^k*(1-p)^(n-k)
BINOMIAL_DISTRIBUTION (k,n,p) := SUM (COMB(n,j)*p^j*(1-p)^(n-j), j,
0, MIN(k,n))
HYPERGEOMETRIC_DENSITY (k,n,m,j) := COMB(m,k)*COMB(j-m,n-k)/COMB(j,n)
HYPERGEOMETRIC_DISTRIBUTION (k,n,m,j) := cumulative hypergeometric
distribution
STUDENT (t,v) := student's cumulative probability distribution A(t|v)
F_DISTRIBUTION (f,v1,v2) := the F cumulative distribution P(f|v1,v2)
CHI_SQUARE (u,v) := Chi-square distribution P(u|v), u = Chi^2
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9.13 FRESNEL.MTH - Fresnel Integrals: use approX
FRESNEL_SIN (z) := Fresnel sine integral S(z)=INT(SIN(pi*t^2/2),t,0,z)
FRESNEL_SIN_SERIES (z,m) := m+1 terms of series for sine integral
FRESNEL_COS (z) := Fresnel cosine integral C(z)=INT(COS(pi*t^2/2),t,0,z)
FRESNEL_COS_SERIES (z,m) := m+1 terms of series for cosine integral
FRESNEL_SIN_ASYMP (z) := two-term asymptotic series for sine integral
FRESNEL_COS_ASYMP (z) := two-term asymptotic series for cosine integral
9.14 BESSEL.MTH - Bessel and Airy Functions: use approX
BESSEL_J (n,z) := Bessel function of first kind Jn(z), n numeric
BESSEL_J_SERIES (n,z,m) := m+1 terms of series for above
BESSEL_J_ASYMP (n,z) := 1 term of asymptotic series for above for large |z|
BESSEL_Y (n,z) := Bessel function of second kind Yn(z), n fractional
BESSEL_Y_SERIES (n,z,m) := m+1 terms of series for above
BESSEL_Y_ASYMP (n,z) := 1 term of asymptotic series for above for large |z|
BESSEL_I (n,z) := modified Bessel function of first kind In(z), n numeric
BESSEL_I_SERIES (n,z,m) := m+1 terms of series for above
BESSEL_I_ASYMP (n,z) := 1 term of asymptotic series for above for
large |z|
BESSEL_K (n,z) := modified Bessel function of second kind Kn(z), n fractional
BESSEL_K_ASYMP (n,z) := 1 term of asymptotic series for above for
large |z|
SPHERICAL_BESSEL_J(n,z) := closed-form spherical Bessel fcn of 1st kind,jn(z)
SPHERICAL_BESSEL_Y(n,z) := closed-form spherical Bessel fcn of 2nd kind,yn(z)
AI_SERIES (z,m) := m+1 terms of series for Airy function Ai(z)
BI_SERIES (z,m) := m+1 terms of series for Airy function Bi(z)
9.15 HYPERGEO.MTH - Hypergeometric Functions: use approX
KUMMER (a,b,z) := Kummer's confluent hypergeometric function M(a,b,z)
KUMMER_SERIES (a,b,z,m) := m+1 terms of series for M(a,b,z)
HYPERGEOMETRIC (a,b,c,z) := Gauss hypergeometric function F(a,b;c;z)
HYPERGEOMETRIC (a,b,c,z,m) := m+1 terms of series for F(a,b;c;z)
9.16 ELLIPTIC.MTH - Elliptic Integrals: use approX
ELLIPTIC_F (phi,m) := elliptic integral of 1st kind F(phi\m)
ELLIPTIC_E (phi,m) := elliptic integral of 2nd kind E(phi\m)
ELLIPTIC_PI (phi,m,n) := elliptic integral of 3rd kind II(n;phi\m)
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9.17 ORTH_POL.MTH - Orthogonal Polynomials
CHEBYCHEV_T (n,x) := nth Chebychev polynomial of 1st kind Tn(x)
CHEBYCHEV_U (n,x) := nth Chebychev polynomial of 2nd kind Un(x)
LEGENDRE_P (n,x) := nth Legendre polynomial Pn(x)
ASSOCIATED_LEGENDRE_P (n,m,x) := nth associated Legendre polynomial
Pn^m(x)
HERMITE_H (n,x) := nth Hermite polynomial Hn(x)
HERMITE_HE (n,x) := nth associated Hermite polynomial HEn(x)
WEBER_D (n,x) := Weber's nth parabolic cylinder function Dn(x)
GENERALIZED_LAGUERRE(n,a,x):=nth generalized Laguerre polynomial
n^a(x).
Use a=0 for ordinary Laguerre polynomials
JACOBI_P (n,a,b,x) := nth Jacobi polynomial Pn^(a,b)(x)
GEGENBAUER_C (n,a,x) := nth Gegenbauer ultraspherical polynomial
Cn^a(x)
9.18 ZETA.MTH - Generalized Riemann Zeta Functions
HURWITZ_ZETA (s,a,m) := m+1 terms of series for SUM((k+a)^-s,k,0,inf)
POLYLOG (n,z,m) := m terms of series for Jonquire's polylogarithm Lin(z)
LERCH_PHI (z,s,a,m) := m terms of series for Lerch transcendent Phi(z,s,a)
DILOG (x) := dilogarithm function INT(LN(t)/(t-1),t,1,x)
9.19 GRAPHICS.MTH - Plotting Space Curves, Parametric Surfaces and
Complex Values
Isometric projections for 2D Plot: Use Options State Rectangular
Connected
axes := 3D coordinate axes
ISOMETRIC (v) := matrix for space curve v = [x(t),y(t),z(t)]:
Plot
ROTATE_X (phi) := matrix A so that A . [x,y,z] rotates angle phi
about x axis
ROTATE_Y (phi) := matrix A so that A . [x,y,z] rotates angle phi about y axis
ROTATE_Z (phi) := matrix A so that A . [x,y,z] rotates angle phi
about y axis
ISOMETRICS (v,s,s0,sm,m,t,t0,tn,n):=matrix for surface [x(s,t),y(s,t),z(s,t)]
s=s0...sm, t=t0...tn: approX then Plot
COPROJECTION (A) := convert matrix from lines of constant t to constant s
CYLINDER (r,theta,z) := eg. of v for ISOMETRICS (v,theta,...,z,...)
SPHERE (r,theta,phi) := eg. of v for ISOMETRICS (v,theta,...,phi,...)
TORUS (r,theta,phi) := eg. of v for ISOMETRICS (v,theta,...,phi,...)
CONE (phi,theta,z) := eg. of v for ISOMETRICS (v,theta,...,z,...)
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2D Plots of complex-expression w(z) where z = x + #i y = r #e^(#i theta)
Use approX and Options State Rectangular Connected
RAYS (w,z,z00,zmn,m,n):= matrix of w vectors for z = x0 + #i y0 ...
xm + #i y
HORIZONTALS (w,z,z00,zmn,m,n) := w-plane map of rectangular z-plane grid
ARCS (w,z,r0,rm,m,th0,thm,n) := w-plane map of polar z-plane grid r=r0...rm,
theta=th0...thn
9.20 NUMBER.MTH - Number Theory Functions
CEILING (m,n) := smallest integer >= m/n (n defaults to 1)
ROUND (m,n) := nearest integer to m/n (n defaults to 1)
SQUARE_WAVE (n) := 1 if MOD(n,2)<1, else -1
POWER_MOD (n,d,m) := n^d mod m
NTH_PRIME (n) := nth prime number
BERNOULLI (n) := nth Bernoulli number
CATALAN (n) := nth Catalan number
EULER (n) := nth Euler number
FIBONACCI (n) := nth Fibonacci number
MERSENNE (n) := nth Mersenne prime
9.21 MISC.MTH - Miscellaneous Utility Functions
RATIO_TEST (t,n) := if > 1, SUM(t,n,a,inf) converges; < 1, sum diverges
LIM2 (u,x,y,x0,y0) := limit of u as [x,y] -> [x0,y0] along slope of @1
LEFT_RIEMANN (u,x,a,b,n) := left Riemann sum for INT(u,x,a,b), n rectangles
INT_PARTS (u,v,x) := antiderivative of u(x)*v(x) using integration by parts
INT_SUBST (y,x,u) := antiderivative of y(x) by substituting x for u(x)
DEF_INT_SUBST (y,x,u,a,b) := integral of y(x) from x=a to b by substitution
INVERSE (u,x) := inverse of u(x) with respect to x
PROVE_SUM (t,k,a,n,s) := [0,0] if SUM(t,k,a,n)=s
POLY_COEFF (u,x,n) := coefficient of x^n term in polynomial u(x)
POLY_DEGREE (u,x) := degree of polynomial u(x)
random_sign := random 1 or -1
RANDOM_POLY (x,d,s) := poly of degree d in x with random coeffs from
-s to s
RANDOM_VECTOR (n,s) := n element vector with random elements from
-s to s
RANDOM_MATRIX (m,n,s) := m by n matrix with random elements from
-s to s
GOODNESS_OF_FIT (u,x,A) := fit of u(x) to data matrix A
POLY_INTERPOLATE (A,x) := polynomial interpolation of data matrix A
POLY_INTERPOLATE_EXPRESSION (u,x,a) := polynomial interpolation of u
given supporting points vector a
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