# Calculus¶

In this chapter, some facilities for doing calculus are described. These include functions implementing differentiation, integration, calculating limits etc.

D(variable[, n=1]) expression

derivative

Param variable

variable

Param expression

expression to take derivatives of

Param n

order

Returns

n-th derivative of expression with respect to variable

D(variable) expression

derivative

Param variable

variable

Param list

a list of variables

Param expression

expression to take derivatives of

Param n

order of derivative

Returns

derivative of expression with respect to variable

This function calculates the derivative of the expression {expr} with respect to the variable {var} and returns it. If the third calling format is used, the {n}-th derivative is determined. Yacas knows how to differentiate standard functions such as {Ln} and {Sin}. The {D} operator is threaded in both {var} and {expr}. This means that if either of them is a list, the function is applied to each entry in the list. The results are collected in another list which is returned. If both {var} and {expr} are a list, their lengths should be equal. In this case, the first entry in the list {expr} is differentiated with respect to the first entry in the list {var}, the second entry in {expr} is differentiated with respect to the second entry in {var}, and so on. The {D} operator returns the original function if $$n=0$$, a common mathematical idiom that simplifies many formulae.

Example

In> D(x)Sin(x*y)
Out> y*Cos(x*y);
In> D({x,y,z})Sin(x*y)
Out> {y*Cos(x*y),x*Cos(x*y),0};
In> D(x,2)Sin(x*y)
Out> -Sin(x*y)*y^2;
In> D(x){Sin(x),Cos(x)}
Out> {Cos(x),-Sin(x)};

Curl(vector, basis)

curl of a vector field

Param vector

vector field to take the curl of

Param basis

list of variables forming the basis

This function takes the curl of the vector field vector with respect to the variables basis. The curl is defined in the usual way, Curl(f,x) = {D(x) f - D(x) f, D(x) f - D(x)f, D(x) f - D(x) f}. Both vector and basis should be lists of length 3.

Diverge(vector, basis)

divergence of a vector field

Param vector

vector field to calculate the divergence of

Param basis

list of variables forming the basis

This function calculates the divergence of the vector field vector with respect to the variables basis. The divergence is defined as Diverge(f,x) = D(x) f + ... + D(x[n]) f[n], where n is the length of the lists vector and basis. These lists should have equal length.

HessianMatrix(function, var)

create the Hessian matrix

Param function

a function in $$n$$ variables

Param var

an $$n$$-dimensional vector of variables

The function HessianMatrix() calculates the Hessian matrix of a vector. If $$f(x)$$ is a function of an $$n$$-dimensional vector $$x$$, then the $$(i,j)$$-th element of the Hessian matrix of the function $$f(x)$$ is defined as :math: Deriv(x[i]) Deriv(x[j]) f(x). If the second order mixed partials are continuous, then the Hessian matrix is symmetric (a standard theorem of calculus). The Hessian matrix is used in the second derivative test to discern if a critical point is a local maximum, a local minimum or a saddle point.

Example

In> HessianMatrix(3*x^2-2*x*y+y^2-8*y, {x,y} )
Out> {{6,-2},{-2,2}};
In> PrettyForm(%)
/                \
| ( 6 )  ( -2 )  |
|                |
| ( -2 ) ( 2 )   |
\                /

JacobianMatrix(functions, variables)

calculate the Jacobian matrix of $$n$$ functions in $$n$$ variables

Param functions

an $$n$$-dimensional vector of functions

Param variables

an $$n$$-dimensional vector of variables

The function {JacobianMatrix} calculates the Jacobian matrix of n functions in n variables. The $$(i,j)$$-th element of the Jacobian matrix is defined as the derivative of $$i$$-th function with respect to the $$j$$-th variable.

Example

In> JacobianMatrix( {Sin(x),Cos(y)}, {x,y} );
Out> {{Cos(x),0},{0,-Sin(y)}};
In> PrettyForm(%)
/                                 \
| ( Cos( x ) ) ( 0 )              |
|                                 |
| ( 0 )        ( -( Sin( y ) ) )  |
\                                 /

Integrate(var) expr
Integrate(var, x1, x2) expr

integral

Param expr

expression to integrate

Param var

atom, variable to integrate over

Param x1

first point of definite integration

Param x2

second point of definite integration

This function integrates the expression expr with respect to the variable var. In the case of definite integral, the integration is carried out from $$var=x1$$ to $$var=x2$$”. Some simple integration rules have currently been implemented. Polynomials, some quotients of polynomials, trigonometric functions and their inverses, hyperbolic functions and their inverses, {Exp}, and {Ln}, and products of these functions with polynomials can be integrated.

Example

In> Integrate(x,a,b) Cos(x)
Out> Sin(b)-Sin(a);
In> Integrate(x) Cos(x)
Out> Sin(x);


Limit(var, val) expr

limit of an expression

Param var

variable

Param val

number or Infinity

Param dir

direction (Left or Right)

Param expr

an expression

This command tries to determine the value that the expression “expr” converges to when the variable “var” approaches “val”. One may use {Infinity} or {-Infinity} for “val”. The result of {Limit} may be one of the symbols {Undefined} (meaning that the limit does not exist), {Infinity}, or {-Infinity}. The second calling sequence is used for unidirectional limits. If one gives “dir” the value {Left}, the limit is taken as “var” approaches “val” from the positive infinity; and {Right} will take the limit from the negative infinity.

Example

In> Limit(x,0) Sin(x)/x
Out> 1;
In> Limit(x,0) (Sin(x)-Tan(x))/(x^3)
Out> -1/2;
In> Limit(x,0) 1/x
Out> Undefined;
In> Limit(x,0,Left) 1/x
Out> -Infinity;
In> Limit(x,0,Right) 1/x
Out> Infinity;

Add(val1, val2, ...)
Add(list)

find sum of a list of values

Param val1 val2

expressions

Param list

This function adds all its arguments and returns their sum. It accepts any number of arguments. The arguments can be also passed as a list.

Example

In> Add(1,4,9);
Out> 14;
Out> 55;

Multiply(val1, val2, ...)
Multiply(list)

product of a list of values

Param val1 val2

expressions

Param list

Multiply all arguments and returns their product. It accepts any number of arguments. The arguments can be also passed as a list.

Example

In> Multiply(2,3,4);
Out> 24
In> Multiply(1 .. 10)
Out> 3628800

Sum(var, from, to, body)

find sum of a sequence

Param var

variable to iterate over

Param from

integer value to iterate from

Param to

integer value to iterate up to

Param body

expression to evaluate for each iteration

The command finds the sum of the sequence generated by an iterative formula. The expression “body” is evaluated while the variable “var” ranges over all integers from “from” up to “to”, and the sum of all the results is returned. Obviously, “to” should be greater than or equal to “from”. Warning: {Sum} does not evaluate its arguments {var} and {body} until the actual loop is run.

Example

In> Sum(i, 1, 3, i^2);
Out> 14;

Factorize(list)

product of a list of values

Param list

list of values to multiply

Param var

variable to iterate over

Param from

integer value to iterate from

Param to

integer value to iterate up to

Param body

expression to evaluate for each iteration

The first form of the {Factorize} command simply multiplies all the entries in “list” and returns their product. If the second calling sequence is used, the expression “body” is evaluated while the variable “var” ranges over all integers from “from” up to “to”, and the product of all the results is returned. Obviously, “to” should be greater than or equal to “from”.

Example

In> Factorize({1,2,3,4});
Out> 24;
In> Factorize(i, 1, 4, i);
Out> 24;


Taylor(var, at, order) expr

univariate Taylor series expansion

Param var

variable

Param at

point to get Taylor series around

Param order

order of approximation

Param expr

expression to get Taylor series for

This function returns the Taylor series expansion of the expression “expr” with respect to the variable “var” around “at” up to order “order”. This is a polynomial which agrees with “expr” at the point “var = at”, and furthermore the first “order” derivatives of the polynomial at this point agree with “expr”. Taylor expansions around removable singularities are correctly handled by taking the limit as “var” approaches “at”.

Example

In> PrettyForm(Taylor(x,0,9) Sin(x))
3    5      7       9
x    x      x       x
x - -- + --- - ---- + ------
6    120   5040   362880
Out> True;


D(), InverseTaylor(), ReversePoly(), BigOh()

InverseTaylor(var, at, order) expr

Taylor expansion of inverse

Param var

variable

Param at

point to get inverse Taylor series around

Param order

order of approximation

Param expr

expression to get inverse Taylor series for

This function builds the Taylor series expansion of the inverse of the expression “expr” with respect to the variable “var” around “at” up to order “order”. It uses the function {ReversePoly} to perform the task.

Example

In> PrettyPrinter'Set("PrettyForm")
True
In> exp1 := Taylor(x,0,7) Sin(x)
3    5      7
x    x      x
x - -- + --- - ----
6    120   5040
In> exp2 := InverseTaylor(x,0,7) ArcSin(x)
5      7     3
x      x     x
--- - ---- - -- + x
120   5040   6
In> Simplify(exp1-exp2)
0


ReversePoly(), Taylor(), BigOh()

ReversePoly(f, g, var, newvar, degree)

solve $$h(f(x)) = g(x) + O(x^n)$$ for $$h$$

Param f

function of var

Param g

function of var

Param var

a variable

Param newvar

a new variable to express the result in

Param degree

the degree of the required solution

This function returns a polynomial in “newvar”, say “h(newvar)”, with the property that “h(f(var))” equals “g(var)” up to order “degree”. The degree of the result will be at most “degree-1”. The only requirement is that the first derivative of “f” should not be zero. This function is used to determine the Taylor series expansion of the inverse of a function “f”: if we take “g(var)=var”, then “h(f(var))=var” (up to order “degree”), so “h” will be the inverse of “f”.

Example

In> f(x):=Eval(Expand((1+x)^4))
Out> True;
In> g(x) := x^2
Out> True;
In> h(y):=Eval(ReversePoly(f(x),g(x),x,y,8))
Out> True;
In> BigOh(h(f(x)),x,8)
Out> x^2;
In> h(x)
Out> (-2695*(x-1)^7)/131072+(791*(x-1)^6)/32768 +(-119*(x-1)^5)/4096+(37*(x-1)^4)/1024+(-3*(x-1)^3)/64+(x-1)^2/16;


InverseTaylor(), Taylor(), BigOh()

BigOh(poly, var, degree)

drop all terms of a certain order in a polynomial

Param poly

a univariate polynomial

Param var

a free variable

Param degree

positive integer

This function drops all terms of order “degree” or higher in “poly”, which is a polynomial in the variable “var”.

Example

In> BigOh(1+x+x^2+x^3,x,2)
Out> x+1;


Taylor(), InverseTaylor()

LagrangeInterpolant(xlist, ylist, var)

polynomial interpolation

Param xlist

list of argument values

Param ylist

list of function values

Param var

free variable for resulting polynomial

This function returns a polynomial in the variable “var” which interpolates the points “(xlist, ylist)”. Specifically, the value of the resulting polynomial at “xlist” is “ylist”, the value at “xlist” is “ylist”, etc. The degree of the polynomial is not greater than the length of “xlist”. The lists “xlist” and “ylist” should be of equal length. Furthermore, the entries of “xlist” should be all distinct to ensure that there is one and only one solution. This routine uses the Lagrange interpolant formula to build up the polynomial.

Example

In> f := LagrangeInterpolant({0,1,2}, \
{0,1,1}, x);
Out> (x*(x-1))/2-x*(x-2);
In> Eval(Subst(x,0) f);
Out> 0;
In> Eval(Subst(x,1) f);
Out> 1;
In> Eval(Subst(x,2) f);
Out> 1;
In> PrettyPrinter'Set("PrettyForm");
True
In> LagrangeInterpolant({x1,x2,x3}, {y1,y2,y3}, x)
y1 * ( x - x2 ) * ( x - x3 )
----------------------------
( x1 - x2 ) * ( x1 - x3 )
y2 * ( x - x1 ) * ( x - x3 )
+ ----------------------------
( x2 - x1 ) * ( x2 - x3 )
y3 * ( x - x1 ) * ( x - x2 )
+ ----------------------------
( x3 - x1 ) * ( x3 - x2 )

n!

factorial

Param m

integer

Param n

integer, half-integer, or list

Param a}, {b

numbers

The factorial function {n!} calculates the factorial of integer or half-integer numbers. For nonnegative integers, $$n! := n*(n-1)*(n-2)*...*1$$. The factorial of half-integers is defined via Euler’s Gamma function, $$z! := Gamma(z+1)$$. If $$n=0$$ the function returns $$1$$. The “double factorial” function {n!!} calculates $$n*(n-2)*(n-4)*...$$. This product terminates either with $$1$$ or with $$2$$ depending on whether $$n$$ is odd or even. If $$n=0$$ the function returns $$1$$. The “partial factorial” function {a * b} calculates the product :math:a*(a+1)*… which is terminated at the least integer not greater than :math:b. The arguments :math:a and :math:b do not have to be integers; for integer arguments, {a * b} = $$b! / (a-1)!$$. This function is sometimes a lot faster than evaluating the two factorials, especially if $$a$$ and $$b$$ are close together. If $$a>b$$ the function returns $$1$$. The {Subfactorial} function can be interpreted as the number of permutations of {m} objects in which no object appears in its natural place, also called “derangements.” The factorial functions are threaded, meaning that if the argument {n} is a list, the function will be applied to each element of the list. Note: For reasons of Yacas syntax, the factorial sign {!} cannot precede other non-letter symbols such as {+} or {*}. Therefore, you should enter a space after {!} in expressions such as {x! +1}. The factorial functions terminate and print an error message if the arguments are too large (currently the limit is $$n < 65535$$) because exact factorials of such large numbers are computationally expensive and most probably not useful. One can call {Internal’LnGammaNum()} to evaluate logarithms of such factorials to desired precision.

Example

In> 5!
Out> 120;
In> 1 * 2 * 3 * 4 * 5
Out> 120;
In> (1/2)!
Out> Sqrt(Pi)/2;
In> 7!!;
Out> 105;
In> 1/3 *** 10;
Out> 17041024000/59049;
In> Subfactorial(10)
Out> 1334961;


Bin(), Factorize(), Gamma(), !(), ***(), Subfactorial()

n!!

double factorial

x *** y

whatever

Bin(n, m)

binomial coefficients

Param n}, {m

integers

This function calculates the binomial coefficient “n” above “m”, which equals $$n! / (m! * (n-m)!)$$ This is equal to the number of ways to choose “m” objects out of a total of “n” objects if order is not taken into account. The binomial coefficient is defined to be zero if “m” is negative or greater than “n”; {Bin(0,0)}=1.

Example

In> Bin(10, 4)
Out> 210;
In> 10! / (4! * 6!)
Out> 210;


(), Eulerian()

Eulerian(n, m)

Eulerian numbers

The Eulerian numbers can be viewed as a generalization of the binomial coefficients, and are given explicitly by $$Sum(j,0,k+1,(-1)^j*Bin(n+1,j)*(k-j+1)^n)$$.

Example

In> Eulerian(6,2)
Out> 302;
In> Eulerian(10,9)
Out> 1;

KroneckerDelta(i, j)
KroneckerDelta({i, j, ...})

Kronecker delta

Calculates the Kronecker delta, which gives $$1$$ if all arguments are equal and $$0$$ otherwise.

LeviCivita(list)

totally anti-symmetric Levi-Civita symbol

Param list

a list of integers $$1,\ldots,n$$ in some order

LeviCivita() implements the Levi-Civita symbol. list should be a list of integers, and this function returns 1 if the integers are in successive order, eg. LeviCivita({1,2,3,...}) would return 1. Swapping two elements of this list would return -1. So, LeviCivita({2,1,3}) would evaluate to -1.

Example

In> LeviCivita({1,2,3})
Out> 1;
In> LeviCivita({2,1,3})
Out> -1;
In> LeviCivita({2,2,3})
Out> 0;

Permutations(list)

get all permutations of a list

Param list

a list of elements

Permutations returns a list with all the permutations of the original list.

Example

In> Permutations({a,b,c})
Out> {{a,b,c},{a,c,b},{c,a,b},{b,a,c},
{b,c,a},{c,b,a}};

Fibonacci(n)

Fibonacci sequence

The function returns $$n$$-th Fibonacci number

Example

In> Fibonacci(4)
Out> 3
In> Fibonacci(8)
Out> 21
In> Table(Fibonacci(i), i, 1, 10, 1)
Out> {1,1,2,3,5,8,13,21,34,55}