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Constructing algebraic objects

How do I...

enter a piecewise expression or function

 • Example 1.1: Use the piecewise template from the Expression palette
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enter an algebraic equation

 • See Example 1.2.

create a sequence

 • Context Panel: Sequence
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construct a loop

 • See Example 1.3.

write the exponential function ${ⅇ}^{x}$

 • Example 1.4: Use the exponential template from the Expression palette or use Command Completion in Math mode

enter ${\mathrm{log}}_{a}\left(x\right)$

 • In Math mode, the expression can be entered normally with the $a$ entered as a subscript.  To enter the subscript level, hold down [Ctrl] and press underscore [_]; after typing the subscript, press the right arrow key to leave the subscript.  Finish the expression by entering $\left(x\right)$
 • Example 1.5: Use the log template from the Expression palette
 • In Maple Input mode, log[a](x) can be used to represent ${\mathrm{log}}_{a}x$

convert an expression to a function

 • Context Panel: See Example 1.6
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write a procedure

Algebraic manipulations

How do I...

obtain the equation of a line

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obtain the coordinates of the midpoint of a line segment

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obtain the slope of a line segment

 • Task: Line Segment - Slope

compute the distance between two points

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complete the square

 • Context Panel: Complete Square

square both sides of an equation

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 • Context Panel: Manipulate Equation

substitute into an expression

 • Context Panel: Evaluate at a Point
 • Context Panel: Constructions → Evaluate At → [variable name] (yields unevaluated evaluation)
 • Use the template $\genfrac{}{}{0}{}{{f}\left({x}\right)}{\phantom{x=a}}|\genfrac{}{}{0}{}{\phantom{\mathrm{f\left(x\right)}}}{{x}={a}}$from the Expression palette; replace with the expression on which to perform the substitution, and overwrite ${x}={a}$ with either the variable name equated to a value or a list of such equations
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 • Use the command eval to substitute a variable or value into an expression

substitute into an equation

 • Use any of the devices for substitution into an expression, except that the Context Panel for an equation does not provide the Constructions option

obtain the solution to RootOf

 • Context Panel: All Values
 • Context Panel: Conversions → To Radical
 • Use the allvalues command.

force an equation to be an identity

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 • Use the command solve(identity(eqn, x), vars); the expression (or equation) eqn is considered an identity in terms of the variable x, and solve attempts to find a solution in terms of vars that satisfies eqn for any value of x.

determine the inverse of a function

 • Function Inverse Tutor:

interpolate data

 • Context Panel: Curve Fitting → any of B-Spline, Interactive Curve Fitting, Least Squares, Polynomial Interpolation, Rational, Spline, Thiele
 • Curve Fitting Assistant: This assistant also allows you to import data into Maple from an external file to produce plots of various interpolating functions (Example 2.1)
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obtain a partial fraction decomposition of a rational function

 • Context Panel: Conversions → Partial Fractions → [variable name]
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 • The command convert(f, parfrac, x) converts a function, f with main variable, x into partial fractions.

obtain real values of ${x}^{1/3}$

 • Use the surd command.

Algebraic solvers

How do I...

solve algebraic equation(s)

 • Context Panel: Solve → any of Isolate Expression for, Numerically Solve, Numerically Solve from point, Obtain Solutions for, Solve, Solve (explicit), Solve (general solution), Solve for Variable
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solve an inequality

 • Context Panel: Solve

eliminate parameter in parametric equations

 • Context Panel: Solve → Eliminate a Variable → [parameter name]

eliminate selected variables in a set or list of equations

 • Context Panel: Solve → Eliminate Variables
 • Use the eliminate command

Polynomial arithmetic

How do I...

multiply out the factored form of a polynomial

 • Context Panel: Expand

factor a polynomial

 • Context Panel: Factor

find the zeros of a polynomial

 • Context Panel: Solve

obtain the quotient and remainder when dividing polynomials

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Plotting

How do I...

graph a curve or a surface

 • For an expression, Context Panel: Plots → Plot Builder
 • For a function, Context Panel: Plots → 2-D Plot or 3-D Plot
 • See the comprehensive Plotting Guide

create an animation

 • Context Panel: Plots → Plot Builder → Select Plot Type and Functions → Animation
 • Use the animate command in the plots package

animate the drawing of a plane curve

 • Use the animatecurve command in the plots package

trace coordinates along a plane curve

 • Context Panel (for graph): Probe Info → Nearest point on line

create a graph with one or more parameters controlled by sliders

 • Context Panel: Plots → Plot Builder → Select Plot Type and Functions → Interactive Plot with (n) parameter(s)

graph a rational function and its asymptotes

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 • Rational Function Tutor:

graph linear inequalities

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 • Linear Inequalities tutor
 • Use the inequal command in the plots package

graph conic sections

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 • Conic Sections tutor

graph the intersection of two surfaces

Use the intersectplot command from the plots package

Differential calculus in one variable

How do I...

construct a limit

 • Context Panel: Constructions → Limit → [variable name] and input the value
 • Limit Methods tutor: See Example 6.1

evaluate a limit

 • Context Panel: Limit
 • Limit Methods tutor
 • Task: Limit - Formal Rules

display an annotated stepwise evaluation of a limit

 • Load the Student[Calculus1] package (Tools → Load Package → Student Calculus 1) Calculus palette: enter and complete $\underset{{x}\to {a}}{lim}{f}$, the limit template Context Panel: 2-D Math → Convert To → Inert Form Context Panel: Solve → Show Solution Steps

differentiate

 • Context Panel: Differentiate → [variable name]
 • Differentiation Methods tutor  or See Example 6.2
 • Use the template $\frac{ⅆ}{ⅆ\phantom{\rule[-0.0ex]{0.2em}{0.0ex}}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}$ from the Calculus palette: See Example 6.3

exhibit annotated stepwise evaluation of a derivative

Load the Student[Calculus1] package (Tools → Load Package → Student Calculus 1)
Calculus palette: enter and complete $\frac{ⅆ}{ⅆ\phantom{\rule[-0.0ex]{0.2em}{0.0ex}}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}$, the differentiation template
Context Panel: 2-D Math → Convert To → Inert Form
Context Panel: Solve → Show Solution Steps

differentiate implicitly

 • Context Panel: Differentiate → Implicitly
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 • Use the implicitdiff command

graph a function and its derivative(s)

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 • Derivatives tutor  or See Example 6.4

obtain equations for tangent and normal lines along a curve

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analyze a plane curve

 • Curve Analysis tutor
 • Use the FunctionChart command from the Student Calculus 1 package
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obtain Taylor series and polynomials

 • Context Panel: Series
 • Taylor Approximation tutor  or See Example 6.5
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implement Newton's Method

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 • Use the NewtonsMethod command from the Student Calculus1 package

Integral calculus in one variable

How do I...

obtain a Riemann sum for $f\left(x\right)$

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 • Riemann Sum tutor

obtain the indefinite integral of $f\left(x\right)$

 • Context Panel: Integrate
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 • Integration Methods tutor  or See Example 7.1

obtain the definite integral of $f\left(x\right)$

 • Context Panel: Constructions → Definite Integral
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 • Integration Methods tutor  or See Example 7.1

display annotated stepwise evaluation of an integral

 • Load the Student[Calculus1] package (Tools → Load Package → Student Calculus 1) Calculus palette: enter and complete definite or indefinite integration templates: $\int {f}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}ⅆ{x}$ or ${\int }_{{a}}^{{b}}{f}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}ⅆ{x}$ Context Panel: 2-D Math → Convert To → Inert Form Context Panel: Solve → Show Solution Steps

enter and evaluate

 • Example 7.2: Use the template, $\int {f}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}ⅆ{x}$, from the Calculus palette
 • Example 7.3: Type $\mathrm{int}$ and use Command Completion

enter and evaluate

 • Example 7.2: Use the template, ${\int }_{{a}}^{{b}}{f}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}ⅆ{x}$, from the Calculus palette
 • Example 7.3: Type $\mathrm{int}$ and use Command Completion

approximate a definite integral numerically

 • Context Panel: Approximate
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 • Approximate Integration tutor  or See Example 7.4

integrate by parts

 • Integration Methods tutor  or See Example 7.1

integrate by trig substitution

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 • Integration Methods tutor  or See Example 7.1

compute the average value of a function

 • Function Average tutor    or See Example 7.5
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 • Use the FunctionAverage command from the Student Calculus1 package

calculate the length of a curve (arc length)

 • Arc Length tutor  or See Example 7.6
 • Use the ArcLength command from the Student Calculus1 package

calculate the volume of a solid of revolution

 • Volume of Revolution tutor  or See Example 7.7
 • Use the VolumeOfRevolution command from the Student Calculus1 package

calculate the surface area of a surface of revolution

 • Surface of Revolution tutor  or See Example 7.8
 • Use the SurfaceOfRevolution command from the Student Calculus1 package

obtain the radius of convergence of a power series

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apply the Ratio test for convergence of a series

Multivariate calculus

How do I...

obtain partial derivatives of a multivariate expression

 • Context Panel: Differentiate
 • Use the partial-differentiation template in the Calculus palette
 • Use the diff command

obtain partial derivatives of a multivariate function

 • Use the D operator

find and test critical points of a multivariate function or expression

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 • Use the SecondDerivativeTest command in the Student Multivariate Calculus package

obtain the gradient vector for a multivariate function

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 • Gradients tutor  or See Example 8.1
 • Use the Gradient command from the Student MultivariateCalculus package

obtain the directional derivative of a multivariate scalar field

 • Use the DirectionalDerivative command from the Student MultivariateCalculus package
 • Directional Derivatives tutor  or See Example 8.2
 • Use the DirectionalDiff command from the VectorCalculus package
 • Use the DirectionalDiff command from the Physics[Vectors] package

implement the Lagrange Multiplier method

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 • Use the LagrangeMultipliers command from the Student MultivariateCalculus package

obtain a Taylor expansion of a multivariate expression

 • Context Panel: Series → Multivariate Taylor Polynomial
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 • Taylor Approximation tutor  or See Example 8.3
 • Use the TaylorApproximation command from the Student MultivariateCalculus package
 • Use the mtaylor command

obtain the Jacobian matrix and the Jacobian of a multivariate expression

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 • Use the Jacobian command from the Student MultivariateCalculus package
 • Use the Jacobian command from the VectorCalculus package

obtain the Hessian of a multivariate expression

 • Use the Hessian command from the VectorCalculus package

implement iterated integration

 • Example 8.4: Iterate an integral icon from Calculus palette
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evaluate iterated double integrals over pre-defined regions

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evaluate iterated triple integrals over pre-defined regions

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evaluate an iterated integral numerically

 • Context Panel: Approximate
 • Multivariate Approximate Integration tutor  or See Example 8.5
 • Use the ApproximateInt command from the Student MultivariateCalculus package

visualize the region of integration for an iterated integral

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compute the average value of a multivariate expression

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determine the center of mass of a plane or spatial region

calculate the surface area for a surface that is not a surface of revolution

 • Task: Surface is a Box
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 • Task: Surface is a Sphere
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Vector calculus

How do I...

designate a coordinate system

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 • Use the SetCoordinates command from the VectorCalculus package

enter a free vector (the equivalent of a point)

 • Use the Matrix palette
 • Type $⟨{x}_{1},\dots ,{x}_{n}⟩$, where inequality signs are used for angle brackets
 • Use the Vector command from the VectorCalculus package

attach a coordinate system to a free vector

 • Example 9.1: Use the Vector command from the VectorCalculus package

construct a vector field

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 • Use the VectorField command from the VectorCalculus package

evaluate a vector field at a point

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graph a vector field

 • Vector Fields tutor
 • Use the PlotVector command from the VectorCalculus package

obtain the dot product of two vectors

 • Use the period, or the dot ($·$) from the Common Symbols palette
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 • Use the DotProduct command from the VectorCalculus package

calculate the magnitude of a vector

 • Context Panel: Norm
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 • Use the Norm command from the VectorCalculus package

obtain the cross product of two vectors

 • In Math mode, use from the Common Symbols palette
 • In text mode, use &x as the cross-product operator
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 • Use the CrossProduct command from the VectorCalculus package

visualize the cross-product vector

obtain the gradient of a scalar field

 • Example 9.2: Gradient via the Nabla $\left(\nabla \right)$ or Del operator
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 • Use the Gradient command from the VectorCalculus package

obtain the divergence of a vector field

 • Example 9.3: Divergence via the Nabla $\left(\nabla \right)$ or Del operator
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 • Use the Divergence command from the VectorCalculus package

obtain the curl of a vector field

 • Example 9.4: Curl via the Nabla $\left(\nabla \right)$ or Del operator
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 • Use the Curl command from the VectorCalculus package

obtain the Laplacian of a scalar field

 • Example 9.5: Laplacian via the Nabla $\left(\nabla \right)$ or Del operator
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 • Use the Laplacian command from the VectorCalculus package

obtain the Laplacian of a vector field

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evaluate iterated double integrals over pre-defined regions

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evaluate iterated triple integrals over pre-defined regions

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evaluate an iterated integral using the int command as modified by the VectorCalculus packages

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compute a line integral along a plane curve

 • Task: Along a Line Segment
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compute a line integral along a space curve

 • Task: Along a Line Segment
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calculate a surface integral

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compute the flux of a vector field through a plane curve

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compute the flux of a vector field through a surface

 • Task: Flux Through a Box
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visualize the TNB (tangent-normal-binormal) frame for a space curve

 • Space Curve tutor

interactively implement the Frenet-Serret formalism for a space-curve

 • Example 9.6: Interactive Frenet-Serret formalism

programmatically implement the Frenet-Serret formalism for a space-curve

 • Use the TNBFrame command from the VectorCalculus package
 • Use the TangentVector command from the VectorCalculus package
 • Use the PrincipalNormal command from the VectorCalculus package
 • Use the Binormal command from the VectorCalculus package
 • Use the Curvature command from the VectorCalculus package
 • Use the Torsion command from the VectorCalculus package
 • Use the RadiusOfCurvature command from the VectorCalculus package

obtain the coordinates of a point in a different coordinate system

 • Use the MapToBasis command in the VectorCalculus package: See Example 9.7.

change coordinates in a vector field

 • Use the MapToBasis command in the VectorCalculus package: See Example 9.8.

Complex arithmetic

How do I...

enter a complex number

 • Example 10.1: Enter a complex number using $i$, $ȷ$, or $I$ from the Common Symbols palette

obtain the real and imaginary parts of a complex number

obtain the magnitude and argument of a complex number

express a complex number in polar form

convert a complex number to rectangular form

 • Use the evalc command

ODEs

How do I...

enter an ordinary differential equation

 • Example 11.1: Enter a differential equation using dot notation, prime notation, or the command diff

obtain a direction field for $y\prime =f\left(x,y\right)$

obtain the Picard iterates for $y\prime =f\left(x,y\right),y\left({x}_{0}\right)={y}_{0}$

solve an ordinary differential equation

 • Context Panel: Solve DE
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 • Use the dsolve command

solve an initial or boundary value problem

 • Context Panel: Solve DE Interactively
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classify the type of an ODE

 • Context Panel: Classify the ODE
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solve an ODE numerically

 • Context Panel: Solve DE Interactively, then choose Solve Numerically
 • Example 11.2:  Use the dsolve command with the numeric option

obtain the Wronskian for a fundamental set of solutions

Use the Wronskian command from the VectorCalculus package

generate a phase portrait for an autonomous system of ODEs

 • Task: Phase Portrait of ODEs (Interactively, for planar system)
 • Use the DEplot command from the DEtools package

explore phase portraits for autonomous systems of ODEs

 • DE Plots tutor

Linear algebra

How do I...

for a given vector, find its coordinates with respect to a specific basis

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obtain the dot product of two vectors

 • Context Panel: Dot Product (apply to sequence of two vectors)
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 • Example 12.1: Obtain the dot product of two vectors using the Common Symbols palette or using a period

determine the angle between two vectors

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 • Use the VectorAngle command in the Student LinearAlgebra package

calculate a vector norm

 • Context Panel: Norm
 • Example 12.2: Calculate the norm of a vector using symbols or a command

project one vector onto another

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project a vector onto a subspace spanned by two other vectors or onto a plane through the origin

obtain $\mathbf{A}×\mathbf{B}$, the cross-product of two vectors

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 • Math mode: use $×$ from Common Symbols, or Operators palettes Text mode: use &x  See Example 12.3

extract a maximal linearly independent subset from a set of vectors

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 • Use the Basis command from the Student LinearAlgebra package

obtain the determinant of a matrix

 • Context Panel: Standard Operations → Determinant
 • Example 12.4: Use the absolute value template from the Layout palette
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 • Use the Determinant command from the Student LinearAlgebra package

multiply a matrix by a scalar

 • In math mode, use a space as the multiplication operator
 • In text mode, use * as the multiplication operator
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apply the function $f$ to each element of a vector or matrix $A$

 • Use the element-wise operator: $f~\left(A\right)$
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obtain the product of two matrices $A$ and $B$

 • Use the period for noncommutative multiplication: $A.B$
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raise a square matrix $A$ to a positive integer power such as 3

 • Use ordinary exponentiation: ${A}^{3}$
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obtain the rank of a matrix

 • Context Panel: Queries → Rank
 • Task: Rank of a Matrix
 • Use the Rank command from the Student LinearAlgebra package

obtain the nullity of a matrix

 • Task: Nullity of a Matrix

obtain bases for row, column, and null spaces of a matrix

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 • Task: Null Space of a Matrix (kernel)
 • See also the RowSpace, ColumnSpace, NullSpace commands in the Student LinearAlgebra package

obtain the transpose or Hermitian transpose of a matrix

 • Context Panel: Standard Operations → Transpose
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 • Example 12.5: In Math mode, for a matrix A, its transpose can be found by typing ${A}^{\mathrm{%T}}$

construct a projection matrix

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perform augmentation or stacking operations on a matrix

 • Example 12.6: Stacking $A$ on top of $B$, where $A$ and $B$ are vectors or matrices is done by typing $⟨A,B⟩$; Augmenting $A$ with $B$ is done by typing $⟨A|B⟩$

solve the linear system $A\mathbf{x}=\mathbf{b}$

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 • Augment by using $⟨A|b⟩$ and apply Context Panel: Solvers and Forms → Row-Echelon Form (see Example 12.6)
 • Use the LinearSolve command from the Student LinearAlgebra package

implement Gaussian elimination

 • Gaussian Elimination tutor: See Example 12.7
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 • Context Panel: Solvers and Forms → Row-Echelon Form
 • Use the GaussianElimination command from the Student LinearAlgebra package

obtain the inverse of a square matrix $A$

 • In math mode, simply execute ${A}^{-1}$
 • In text mode, execute A^(-1)
 • Task: Inverse of a Matrix
 • Context Panel: Standard Operations → Inverse
 • Matrix Inverse tutor: See Example 12.8
 • Use the MatrixInverse command from the Student LinearAlgebra package

obtain the pseudoinverse of a singular or nonsquare matrix

 • Context Panel: Standard Operations → Pseudoinverse
 • Use the Pseudoinverse command from the Student LinearAlgebra package

obtain eigenvalues and eigenvectors for a matrix

 • Context Panel: Eigenvalues, etc → Eigenvalues
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 • Use the Eigenvalues and Eigenvectors commands from the Student LinearAlgebra package

compute  for a constant matrix $A$

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 • Use the MatrixExponential command from Student LinearAlgebra package

apply the Gram-Schmidt process to the columns of a matrix, or a list or set vectors

apply the Gram-Schmidt process to a list or set of vectors

 • Use the GramSchmidt command from the Student LinearAlgebra package

visualize the effect of multiplying a planar vector by a square matrix

Equate corresponding components in two vectors or matrices

 • Context Panel: Equate (applied to the sequence of objects)
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 • Use the Equate command

convert linear equations to matrix form

 • Context Panel: Student Linear Algebra → Constructions → Generate Matrix (applied to sequence of equations)
 • Use the GenerateMatrix command from the Student LinearAlgebra package

Numerical analysis

How do I...

approximate the roots of an expression to a given accuracy using Newton's method

 • Use the Newton command from the Student NumericalAnalysis package: See Example 13.1

approximate the roots of an expression using a specific method

 • Use the Roots command from the Student NumericalAnalysis package: See Example 13.2 Available methods: Newton, Modified Newton, Bisection, Secant, Fixed-Point Iteration, False-Position, and Steffensen

find the interpolating polynomial

 • Use the PolynomialInterpolation command from the Student NumericalAnalysis package and return the Interpolant: See Example 13.3  Available methods: Hermite, Lagrange, Neville, and Newton

find the error term for a polynomial interpolation problem

 • Example 13.4: Find the Polynomial Interpolation (see Example 13.3).  Then use the command RemainderTerm to find the error term.

find the divided difference table

 • Example 13.5: Find the Polynomial Interpolation (see Example 13.3).  Then use the command DividedDifferenceTable.

find the quadrature using a specific method

 • Use the Quadrature command from the Student NumericalAnalysis package: See Example 13.6  Available methods:  Boole's rule, Simpson's rule, Simpson's 3/8 rule, trapezoid rule, Newton-Cotes rule, Gaussian rule, and Romberg integration.  Adaptive quadrature can be applied to the first five methods.

solve an ODE initial value problem using Euler's method

 • Euler tutor
 • Use the Euler command from the Student NumericalAnalysis package: See Example 13.7

solve an ODE initial value problem using a specific method, or compare the numerical solutions found using various methods

factor a square matrix using matrix decomposition

 • Matrix Decomposition tutor

use a numerical method to solve $A\mathbf{x}=\mathbf{b}$

 • Iterative Formula tutor
 • Use the LinearSolve command from the Student NumericalAnalysis package: See Example 13.8  Available methods: Jacobi, Gauss-Seidel, SOR, LU, LU[tridiagonal], PLU, and PLU[scaled]

Statistics

How do I...

define a random variable

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evaluate the probability density function of a random variable

evaluate the probability function of a discrete random variable

evaluate the cumulative probability density function of a random variable

define a probability distribution

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sample a random variable with a given probability distribution

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compute moments for a random variable

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compute maximum likelihood estimates

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fit a regression model to data

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import data from a file

 • Import Data Assistant: Tools → Assistants → Import Data
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create statistical process control charts

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Integer manipulations

How do I...

decompose an integer into the product of its prime factors

obtain the greatest common divisor (GCD) of integers

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obtain the lowest common multiple (LCM) of integers

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obtain the value of an integer modulo $n$

solve an equation for integer values of the variables

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solve an equation for integers modulo $n$

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determine whether a specified integer is prime

Units, errors, and tolerances

How do I...

apply a unit to a quantity

 • Example 16.1: Apply units to quantities by using either of the two Units palettes or by using the Context Panel: Units → Affix Unit.

convert units

 • Example 16.2: Convert a quantity with units to another unit using the Context Panel: Units → Convert → System → [desired system of units] and Units → Replace Units.

use tolerances

 • Example 16.3: Add tolerances to quantities by inserting $±$ from a palette or by using Command Completion, and then perform computations using tolerances.

change a default unit in a system of units

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compute with quantities having units attached

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compute with quantities having errors attached

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access and use values of scientific constants

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change to equivalent units in a quantity carrying units

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convert between Celsius and Fahrenheit

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change the units associated with a quantity

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evaluate an expression at values having units

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switch from one unit system to another

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compute with quantities carrying tolerance limits

Recurrence equations

How do I...

enter a recurrence equation

 • Example 17.1: Entering a recurrence equation in Maple

solve a recurrence equation

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