Maple Notes

Written by Luka Kerr on April 26, 2019

Before entering commands, load the relevant packages to ensure all needed functions are available:

// use LinearAlgebra package
with(LinearAlgebra);

// use VectorCalculus package
with(VectorCalculus);

// use plots package
with(plots);

// use geom3d package
with(geom3d);

Significant figures

To 45 significant figures:

x := 100 * sin(28);
evalf[45](x);

dy/dx

Given an equation eqn, to find $\frac{dy}{dx}$:

implicitdiff(eqn, y, x);

Constants

$e$ = exp(1);

$e^x$ = exp(x);

$\pi$ = Pi;

$\infty$ = infinity;

$i$ = I;

Trigonometric Functions

Inverse:

$sin^{-1}(x)$ = arcsin(x);

$cos^{-1}(x)$ = arccos(x);

$tan^{-1}(x)$ = arctan(x);

Unique Turning Point

For $5 sin(\frac{1}{4}x^4) - sin(\frac{5}{4}x)^4$, to find a unique turning point over the interval $[1, 2]$:

f := 5 * sin(1/4 * x^4) - sin(5/4 *x)^4;

// solve the derivate of f over [1, 2]
fsolve(diff(f, x), x=1..2);

To then find the value of the 2nd derivative at this unique turning point:

// where % is the unique turning point from the last solution above
evalf(subs(x = %, diff(f, x$2)));

Integration

To integrate a given equation eqn, with bottom limit a, and top limit b:

int(eqn, x=a..b);

To evaluate $\int_{-\infty}^{\infty} \frac{cos(\frac{x}{4})}{x^4 + 6 cot^{-1}x}$ to 10 significant figures:

evalf[10](int((cos(x/4))/(x^4 + 6*arccot(x)), x=-infinity..infinity));

Complex Numbers

Principal arguments for $z^5 - 4z^2 - 2z - 3$:

// R1 is an array of complex roots
R1 := [fsolve(z^5 -4*z^2 -2*z -3, complex)];

// get arguments from array
argument~(R1)

// get max from last
max(%);

Moduli of roots for $z^5 -4z^4 -4z^2 + 1$:

// R2 is an array of complex roots
R2 := [fsolve(z^5 -4*z^4 -4*z^2 +1, complex)];

// get absolute values from array
abs~(R2);

// get max from last
max(%);

Roots

Get complex roots of $z^5 + 4z^3 - z^2 - 4$:

fsolve(z^5 + 4*z^3 -z^2 -4, complex);

Limits

To find the limit of a function $f$ as $x \to a$:

limit(f, x=a)

To find $\lim_{n\to\infty} \bigg[ n^{11/2} \prod_{k = 1}^{n} \frac{2k}{2k + 11} \bigg]$

limit((n^(11/2) * product((2*k)/(2*k + 11), k=1..n)), n=infinity);

To find $\lim_{k\to\infty} \bigg[ \sum_{n = 1}^{k} \frac{1}{3n + \frac{k}{7}} \bigg]$

limit(sum(1/(3*n + k/7), n=1..k), k=infinity);

Vectors

Projection of $a = (70, 197, 236)$ onto $v = (19, 128, 56)$:

// vector a
a := <70, 197, 236>;

// vector b
b := <19, 128, 56>;

u := Normalize(b, Euclidean);
v := DotProduct(a, u) * u;

Dot product:

// vector a
a := <70, 197, 236>;

// vector b
b := <19, 128, 56>;

// dotproduct
a.b;

// or
DotProduct(a, b);

Cross product:

// vector a
a := <70, 197, 236>;

// vector b
b := <19, 128, 56>;

// calculate cross product
CrossProduct(a, b);

The area of the parallelogram spanned by $a$ and $b$:

// vector a
a := <70, 197, 236>;

// vector b
b := <19, 128, 56>;

cp := CrossProduct(a, b);

// magnitude of the cross product
Norm(cp);

Mapping Functions

Function that takes in a list of complex numbers and returns the largest modulus from that list:

x -> max(map(abs, x))

Function that takes in a list of complex numbers and returns the smallest cosine from that list:

x -> min(map(cos, x))

Mean Value Theorum

To find a real number $c$ that satisfies the conclusion of the MVT for $f(x) = x^4 -12x^3 +56x^2 -120x +81$:

// define f
f := x^4-12*x^3+56*x^2-120*x+81;

// solve for interval, and output the point
MeanValueTheorem(f, 3..4, output=points);

Plotting

To plot a function:

f := x^2;
plot(f);

// plot f in a region
plot(f, x=-10..10, y=-10..10);

To plot a polar function:

// define polar function
r := 3 * sin(theta) - cos(4 * theta);

// plot function
polarplot(r);

To plot two curves and find the number of intersections in $-10 \le x \le 10, -10 \le y \le 10$. After plotting, the number of intersections should be able to be seen.

// define both functions
f1 := -x * y^2 + 4*x = 5;
f2 := 1/3 * x^2 * y + tanh(y) = 1;

// plot both functions
implicitplot([f1, f2], x=-10..10, y=-10..10);

Piecewise Functions

To define a piece wise function

\[\begin{array}{cc} \Bigg\{ & \begin{array}{cc} x & x\lt 1 \\ x^3 & x\lt 3 \\ 3 - x & otherwise \end{array} \end{array}\]

piecewise(x < 1, x, x < 3, x^3, 3 - x);

Matrices

Given a matrix $A$, create a vector $b$ that is column 3 from $A$, and a matrix $C$ that is made from columns 1 to 2, and 4 to 11 of $A$.

Then solve $Cx = b$ and enter the 6th component of the unique vector solution for $x$.

// set a to the matrix given
A := <...>

// create b
b := Column(A, 3);

// create C
C := Submatrix(A, [1..10], [1..2, 4..11]);

// set v to be Cx = b
v := LinearSolve(C, b);

// get 6th component
v[6];

Functions

Using the arrow -> we can express the function $f(x, y) = 6x^9 y^4 \ sin(6x -3y)$ as

(x, y) -> 6*x^9 * y^4 * sin(6*x - 3*y);

Partial Derivatives

Given a function $f(x, y) = cos(7y^7 + 5x^2)$, we can find the partial derivative $\dfrac{\partial^3}{\partial y^2 \partial x} f(x, y)$

// Define f(x, y)
f := (x, y) -> cos(7*y^7 + 5*x^2);

// y = 2, x, = 1
D[2, 2, 1](f);

Partial Fraction Decomposition

Given a $p(x)$ and $q(x)$ find the partial fraction decomposition of $\dfrac{p(x)}{q(x)}$:

p := ...;
q := ...;
convert(p / q, parfrac, x);

Initial Value Problem

Find the solution to the initial value problem $y \dfrac{d^2 y}{d x^2} + (\dfrac{dy}{dx})^2 = 0$ with initial conditions $y(0) = 2$ and $y’(0) = 3$

ODE := y(x) * diff(y(x), x$2) + (diff(y(x), x))^2 = 0;
dsolve({ODE, y(0) = 2, D(y)(0) = 3}, y(x));

Taylor Series

Given a taylor series for $e^x \ sin(4x)$ about 0 $= a_0 + a_1 x + a_2 x^2 + \dots + a_8 x^8 + \dots$, we can find values for $a_n$:

// Enter the taylor series, from 0 to n + 1, e.g to find a_8
taylor(exp(x) * sin(4*x), x=0, 9);

Dot Product and Linear Combinations

Given a number of vertices $u_1, u_2 \dots u_n$, to find the dot product between $u_k$ and $u_j$:

uk := <...>;
uj := <...>;
uk . uj;

If $A = (u_1 | u_2 | u_3 | u_4)$ and $v = (17, 59, 75, -74)$ , then $A v$ is a linear combination of the form $\lambda_1 u_1 + \lambda_2 u_2 + \lambda_3 u_3 + \lambda_4 u_4$, where the lambda values are $\lambda_1 = 17$, $\lambda_2 = 59$, $\lambda_3 = 75$, $\lambda_4 = -74$

Suppose that $Av = (b_1 \ b_2 \ \dots \ b_5)^T$, to find the value of $b_2$

// Find Av first
Av := A . v;

// Calculate transpose
Transpose(Av);

Kernel and Nullity

Given an $m \times n$ matrix $A$ we can find:

A := <<...> | <...> | <...> | <...>>;

// nullity(A)
nops(NullSpace(A));

// rank(A)
Rank(A);

To determine if a vector $v$ is in the kernel of $A$, $Av$ must produce the zero vector:

A := <<...> | <...> | <...> | <...>>;
v := <....>;
Av := A . v;

If $Av = (0, \dots, 0)$, then $v$ is in $ker(A)$

To determine if a vector $v$ is in the image of $A$, there must be a linear system for $Av$:

A := <<...> | <...> | <...> | <...>>;
v := <....>;

// If this produces an 'inconsistent system', then v is not in im(A)
LinearSolve(A, v);

Sets

Given a set $S = { u_1, u_2, u_3, u_4, u_5, u_6} \subset \mathbb{R^5}$

$S$ is linearly dependent since the number of vectors in $S \gt dim(\mathbb{R^5})$

To find lambda values for $\lambda_1 u_1 + \lambda_2 u_2 + \lambda_3 u_3 + \lambda_4 u_4 = 0$:

A := <<...> | <...> | <...> | <...>>;
B := <0, 0, 0, 0>;
LinearSolve(A, B);

Eigenvalues

Given an $m \times n$ matrix $A$ we can find its eigenvalues:

A := <<...> | <...> | <...> | <...>>;
Eigenvalues(A);

Geom3D

Plotting:

// Plot point
point(A, [1, 2, 3]);
point(B, [4, 5, 6]);
point(C, [7, 8, 9]);

// Plot line through A and parallel to [-1, -2, -3]
line(L1, [A, [-1, -2, -3]]);

// Plot plane through B with normal [-4, -5, -6]
plane(P, [B, [-4, -5, -6]]);

// Plot intersection of L1 and P
intersection(E, L1, P);

// Plot sphere through A B C E
sphere(S, [A, B, C, E]);

// Plot sphere with center A and radius 8
sphere(S, [A, 8]);

// Plot center of S
center(F, S);

// Plot line through C and F
line(L2, [C, F]);

Evaluating:

// Angle between line and point to 10 dp
evalf(FindAngle(L, P), 10);

// Coordinates of F
coordinates(F);

// Distance between A and L2
distance(A, L2);

For Loops

Write a for loop for $\sum^{21}_{n = 16} sin(\dfrac{k}{n})$ for k from 2 to 60

for
  k
from
  2
to
  60
do
  evalf(add(sin(k/n), n=16..21)
end do;

Another example:

f := proc(m)
  local a, i;
  a[0] := 0;

  for i to m do
    a[i] := evalf(sin((1+(1/4) * a[i-1])^2));
  end do;

  if abs(a[m] - a[m-1]) < 10^(-16) then
    a[m]
  else
    -1
  end if
end proc;

f(80);

Given a recurrence relation $a_{n+1} = a_n - 4a_{n-1} + a_{n-2}$ for $n = 3, 4, 5$, write a for loop to find the value of $a_{70}$ given that $a_1 = 5$, $a_2 = 2$ and $a_3 = -1$

a := proc(n)
  local a, i;
  a[1] := 5;
  a[2] := 2;
  a[3] := -1;

  for i from 3 to n - 1 do
    a[i + 1] := a[i] - 4 * a[i - 1] + a[i - 1]
  end do;

  return a[n]
end proc;

a(70);