$s      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~                !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqr(c) Scott N. Walck 2012-2014BSD3 (see LICENSE)Scott N. Walck <walck@lvc.edu> experimentalSafe A type for vectors. x component y component z component3Form a vector by giving its x, y, and z components.Cross product.Unit vector in the x direction.Unit vector in the y direction.Unit vector in the z direction. x component y component z component   7(c) Scott N. Walck 2011-2018BSD3 (see LICENSE)Scott N. Walck <walck@lvc.edu> experimentalSafeFT    (c) Scott N. Walck 2012-2018BSD3 (see LICENSE)Scott N. Walck <walck@lvc.edu> experimentalSafe<"Composite Trapezoid RuleComposite Simpson's RuleFnumber of intervals (one less than the number of function evaluations) lower limit upper limitfunction to be integrateddefinite integralKnumber of half-intervals (one less than the number of function evaluations) lower limit upper limitfunction to be integrateddefinite integral(c) Scott N. Walck 2012-2018BSD3 (see LICENSE)Scott N. Walck <walck@lvc.edu> experimentalSafe;=]AA coordinate system is a function from three parameters to space.tSometimes we want to be able to talk about a field without saying whether it is a scalar field or a vector field. ?A vector field associates a vector with each position in space.!?A scalar field associates a number with each position in space."A displacement is a vector.#\A type for position. Position is not a vector because it makes no sense to add positions.$+Add two scalar fields or two vector fields.%:The Cartesian coordinate system. Coordinates are (x,y,z).&The cylindrical coordinate system. Coordinates are (s,phi,z), where s is the distance from the z axis and phi is the angle with the x axis.'The spherical coordinate system. Coordinates are (r,theta,phi), where r is the distance from the origin, theta is the angle with the z axis, and phi is the azimuthal angle.(vA helping function to take three numbers x, y, and z and form the appropriate position using Cartesian coordinates.)zA helping function to take three numbers s, phi, and z and form the appropriate position using cylindrical coordinates.*|A helping function to take three numbers r, theta, and phi and form the appropriate position using spherical coordinates.+DReturns the three Cartesian coordinates as a triple from a position.,FReturns the three cylindrical coordinates as a triple from a position.-DReturns the three spherical coordinates as a triple from a position..5Displacement from source position to target position./#Shift a position by a displacement.0An object is a map into #.1A field is a map from #.2 The vector field in which each point in space is associated with a unit vector in the direction of increasing spherical coordinate r, while spherical coordinates theta and phi are held constant. Defined everywhere except at the origin. The unit vector 2 points in different directions at different points in space. It is therefore better interpreted as a vector field, rather than a vector.3The vector field in which each point in space is associated with a unit vector in the direction of increasing spherical coordinate theta, while spherical coordinates r and phi are held constant. Defined everywhere except on the z axis.4-The vector field in which each point in space is associated with a unit vector in the direction of increasing (cylindrical or spherical) coordinate phi, while cylindrical coordinates s and z (or spherical coordinates r and theta) are held constant. Defined everywhere except on the z axis.5The vector field in which each point in space is associated with a unit vector in the direction of increasing cylindrical coordinate s, while cylindrical coordinates phi and z are held constant. Defined everywhere except on the z axis.6The vector field in which each point in space is associated with a unit vector in the direction of increasing Cartesian coordinate x, while Cartesian coordinates y and z are held constant. Defined everywhere.7The vector field in which each point in space is associated with a unit vector in the direction of increasing Cartesian coordinate y, while Cartesian coordinates x and z are held constant. Defined everywhere.8The vector field in which each point in space is associated with a unit vector in the direction of increasing Cartesian coordinate z, while Cartesian coordinates x and y are held constant. Defined everywhere.( x coordinate y coordinate z coordinate) s coordinatephi coordinate z coordinate* r coordinatetheta coordinatephi coordinate.source positiontarget position !"#$%&'()*+,-./012345678#"! %&'()*+,-./01$2345678#s(c) Scott N. Walck 2012-2018BSD3 (see LICENSE)Scott N. Walck <walck@lvc.edu> experimentalSafe<FT::R is a parametrized function into three-space, an initial limit, and a final limit.<&function from one parameter into space=starting value of the parameter>ending value of the parameter?4A dotted line integral. Convenience function for K.@DCalculates integral vf x dl over curve. Convenience function for L.A<A dotted line integral, performed in an unsophisticated way.BACalculates integral vf x dl over curve in an unsophisticated way.CGCalculates integral f dl over curve, where dl is a scalar line element.D"Reparametrize a curve from 0 to 1.EConcatenate two curves.F=Concatenate a list of curves. Parametrizes curves equally.GReverse a curve.H0Evaluate the position of a curve at a parameter.I Shift a curve by a displacement.J5The straight-line curve from one position to another.K}Quadratic approximation to vector field. Quadratic approximation to curve. Composite strategy. Dotted line integral.L~Quadratic approximation to vector field. Quadratic approximation to curve. Composite strategy. Crossed line integral. ?Nnumber of half-intervals (one less than the number of function evaluations) vector fieldcurve to integrate over scalar result@Nnumber of half-intervals (one less than the number of function evaluations) vector fieldcurve to integrate over vector resultAnumber of intervals vector fieldcurve to integrate over scalar resultBnumber of intervals vector fieldcurve to integrate over vector resultCnumber of intervalsscalar or vector fieldcurve to integrate overscalar or vector resultEgo first along this curvethen along this curveto produce this new curveH the curve the parameter4position of the point on the curve at that parameterIamount to shiftoriginal curve shifted curveJstarting positionending positionstraight-line curvetvector field lowvector field midvector field high dl low to middl mid to highquadratic approximationKNnumber of half-intervals (one less than the number of function evaluations) vector fieldcurve to integrate over scalar resultuvector field lowvector field midvector field high dl low to middl mid to highquadratic approximationLNnumber of half-intervals (one less than the number of function evaluations) vector fieldcurve to integrate over vector result:;<=>?@ABCDEFGHIJKL:;<=>DEFGHIJC?@ABKL:;<=>(c) Scott N. Walck 2012-2018BSD3 (see LICENSE)Scott N. Walck <walck@lvc.edu> experimentalSafeM}Specification of a coordinate system requires a map from coordinates into space, and a map from space into coordinates.O!a map from coordinates into spaceP!a map from space into coordinatesQ(The standard Cartesian coordinate systemR*The standard cylindrical coordinate systemS(The standard spherical coordinate systemTDefine a new coordinate system in terms of an existing one. First parameter is a map from old coordinates to new coordinates. Second parameter is the inverse map from new coordinates to old coordinates.T(x',y',z') = f(x,y,z)(x,y,z) = g(x',y',z')old coordinate systemMNOPQRSTMNOPQRSTMNOP(c) Scott N. Walck 2012-2018BSD3 (see LICENSE)Scott N. Walck <walck@lvc.edu> experimentalSafePU)The x Cartesian coordinate of a position.V)The y Cartesian coordinate of a position.W:The z Cartesian (or cylindrical) coordinate of a position.XdThe s cylindrical coordinate of a position. This is the distance of the position from the z axis.YThe phi cylindrical (or spherical) coordinate of a position. This is the angle from the positive x axis to the projection of the position onto the xy plane.ZbThe r spherical coordinate of a position. This is the distance of the position from the origin.[lThe theta spherical coordinate of a position. This is the angle from the positive z axis to the position.UVWXYZ[UVWXYZ[(c) Scott N. Walck 2016-2018BSD3 (see LICENSE)Scott N. Walck <walck@lvc.edu> experimental TrustworthyO)^The state resulting from a measurement of spin angular momentum in the x direction on a spin-1/2 particle when the result of the measurement is hbar/2._The state resulting from a measurement of spin angular momentum in the x direction on a spin-1/2 particle when the result of the measurement is -hbar/2.`The state resulting from a measurement of spin angular momentum in the y direction on a spin-1/2 particle when the result of the measurement is hbar/2.aThe state resulting from a measurement of spin angular momentum in the y direction on a spin-1/2 particle when the result of the measurement is -hbar/2.bThe state resulting from a measurement of spin angular momentum in the z direction on a spin-1/2 particle when the result of the measurement is hbar/2.cThe state resulting from a measurement of spin angular momentum in the z direction on a spin-1/2 particle when the result of the measurement is -hbar/2.dThe state resulting from a measurement of spin angular momentum in the direction specified by spherical angles theta (polar angle) and phi (azimuthal angle) on a spin-1/2 particle when the result of the measurement is hbar/2.eThe state resulting from a measurement of spin angular momentum in the direction specified by spherical angles theta (polar angle) and phi (azimuthal angle) on a spin-1/2 particle when the result of the measurement is -hbar/2.fDimension of a vector.g+Scale a complex vector by a complex number.h5Complex inner product. First vector gets conjugated.iLength of a complex vector.j4Return a normalized version of a given state vector.k:Return a vector of probabilities for a given state vector.l"Conjugate the entries of a vector.m2Construct a vector from a list of complex numbers.n0Produce a list of complex numbers from a vector.oThe Pauli X matrix.pThe Pauli Y matrix.qThe Pauli Z matrix.r+Scale a complex matrix by a complex number.sMatrix product.tMatrix-vector product.uVector-matrix productv Conjugate transpose of a matrix.w;Construct a matrix from a list of lists of complex numbers.x9Produce a list of lists of complex numbers from a matrix.ySize of a matrix.zApply a function to a matrix. Assumes the matrix is a normal matrix (a matrix with an orthonormal basis of eigenvectors).{Complex outer product|6Build a pure-state density matrix from a state vector.}Trace of a matrix.~4Normalize a density matrix so that it has trace one."The one-qubit totally mixed state.Given a time step and a Hamiltonian matrix, produce a unitary time evolution matrix. Unless you really need the time evolution matrix, it is better to use z, which gives the same numerical results without doing an explicit matrix inversion. The function assumes hbar = 1.Given a time step and a Hamiltonian matrix, advance the state vector using the Schrodinger equation. This method should be faster than using q since it solves a linear system rather than calculating an inverse matrix. The function assumes hbar = 1.Given a Hamiltonian matrix, return a function from time to evolution matrix. Uses spectral decomposition. Assumes hbar = 1.|The possible outcomes of a measurement of an observable. These are the eigenvalues of the matrix of the observable.fGiven an obervable, return a list of pairs of possible outcomes and projectors for each outcome.|Given an observable and a state vector, return a list of pairs of possible outcomes and probabilites for each outcome.cForm an orthonormal list of complex vectors from a linearly independent list of complex vectors.k state vectorvector of probabilities.\]^_`abcdefghijklmnopqrstuvwxyz{|}~.^_`abcdefghijklmnopqrstuvwxyz{|}~\]\] (c) Scott N. Walck 2016-2018BSD3 (see LICENSE)Scott N. Walck <walck@lvc.edu> experimentalSafe;=>?A An orthonormal basis of kets.GThe adjoint operation on complex numbers, kets, bras, and operators.Generic multiplication including inner product, outer product, operator product, and whatever else makes sense. No conjugation takes place in this operation.5A bra vector describes the state of a quantum system.`An operator describes an observable (a Hermitian operator) or an action (a unitary operator).5A ket vector describes the state of a quantum system.CMake an orthonormal basis from a list of linearly independent kets.fState of a spin-1/2 particle if measurement in the x-direction would give angular momentum +hbar/2.fState of a spin-1/2 particle if measurement in the x-direction would give angular momentum -hbar/2.fState of a spin-1/2 particle if measurement in the y-direction would give angular momentum +hbar/2.fState of a spin-1/2 particle if measurement in the y-direction would give angular momentum -hbar/2.fState of a spin-1/2 particle if measurement in the z-direction would give angular momentum +hbar/2.fState of a spin-1/2 particle if measurement in the z-direction would give angular momentum -hbar/2.State of a spin-1/2 particle if measurement in the n-direction, described by spherical polar angle theta and azimuthal angle phi, would give angular momentum +hbar/2.State of a spin-1/2 particle if measurement in the n-direction, described by spherical polar angle theta and azimuthal angle phi, would give angular momentum -hbar/2."The orthonormal basis composed of  and ."The orthonormal basis composed of  and ."The orthonormal basis composed of  and .`Given spherical polar angle theta and azimuthal angle phi, the orthonormal basis composed of  theta phi and  theta phi.The Pauli X operator.The Pauli Y operator.The Pauli Z operator.ZPauli operator for an arbitrary direction given by spherical coordinates theta and phi.GAlternative definition of Pauli operator for an arbitrary direction.Given a time step and a Hamiltonian operator, produce a unitary time evolution operator. Unless you really need the time evolution operator, it is better to use z, which gives the same numerical results without doing an explicit matrix inversion. The function assumes hbar = 1.Given a time step and a Hamiltonian operator, advance the state ket using the Schrodinger equation. This method should be faster than using q since it solves a linear system rather than calculating an inverse matrix. The function assumes hbar = 1.~The possible outcomes of a measurement of an observable. These are the eigenvalues of the operator of the observable.fGiven an obervable, return a list of pairs of possible outcomes and projectors for each outcome.yGiven an observable and a state ket, return a list of pairs of possible outcomes and probabilites for each outcome.vMForm an orthonormal list of kets from a list of linearly independent kets.,, wxyz7 (c) Scott N. Walck 2016-2018BSD3 (see LICENSE)Scott N. Walck <walck@lvc.edu> experimental Trustworthy*/A beam of randomly oriented spin-1/2 particles.+Return the intensities of a stack of beams.+Remove the most recent beam from the stack. Return the number of beams in a .3Interchange the two most recent beams on the stack.Given angles describing the orientation of the splitter, removes an incoming beam from the stack and replaces it with two beams, a spin-up and a spin-down beam. The spin-down beam is the most recent beam on the stack.sGiven angles describing the orientation of the recombiner, returns a single beam from an incoming pair of beams.Given angles describing the direction of a uniform magnetic field, and given an angle describing the product of the Larmor frequency and the time, return an output beam from an input beam.,A Stern-Gerlach splitter in the x direction.,A Stern-Gerlach splitter in the y direction.,A Stern-Gerlach splitter in the z direction.Given an angle in radians describing the product of the Larmor frequency and the time, apply a magnetic in the x direction to the most recent beam on the stack.Given an angle in radians describing the product of the Larmor frequency and the time, apply a magnetic in the y direction to the most recent beam on the stack.Given an angle in radians describing the product of the Larmor frequency and the time, apply a magnetic in the z direction to the most recent beam on the stack..A Stern-Gerlach recombiner in the x direction..A Stern-Gerlach recombiner in the y direction..A Stern-Gerlach recombiner in the z direction.0Filter for spin-up particles in the x direction.2Filter for spin-down particles in the x direction.0Filter for spin-up particles in the z direction.2Filter for spin-down particles in the z direction.{ (c) Scott N. Walck 2012-2017BSD3 (see LICENSE)Scott N. Walck <walck@lvc.edu> experimentalSafe={Given an initial bracketing of a root (an interval (a,b) for which f(a) f(b) <= 0), produce a bracket (c,d) for which |c-d| < desired accuracy.4Given a bracketed root, return a half-width bracket.mFind a single root in a bracketed region. The algorithm continues until it exhausts the precision of a |). This could cause the function to hang.Find a list of roots for a function over a given range. First parameter is the initial number of intervals to use to find the roots. If roots are closely spaced, this number of intervals may need to be large.yFind a list of roots for a function over a given range. There are no guarantees that all roots will be found. Uses  with 1000 intervals.desired accuracyfunctioninitial bracket final bracketfunctionoriginal bracket new bracketfunctioninitial bracketapproximate root"initial number of intervals to usefunctionrange over which to search list of rootsfunctionrange over which to search list of roots (c) Scott N. Walck 2012-2014BSD3 (see LICENSE)Scott N. Walck <walck@lvc.edu> experimentalSafeES The zero vector.!The additive inverse of a vector.Sum of a list of vectors.Vector addition.Vector subtraction.YScalar multiplication, where the scalar is on the left and the vector is on the right.YScalar multiplication, where the scalar is on the right and the vector is on the left.!Division of a vector by a scalar.Dot product of two vectors.Magnitude of a vector.  667777 (c) Scott N. Walck 2014BSD3 (see LICENSE)Scott N. Walck <walck@lvc.edu> experimental Trustworthy;<=FTb An evolution method is a way of approximating the state after advancing a finite interval in the independent variable (time) from a given state.A (numerical) solution method is a way of converting an initial value problem into a list of states (a solution). The list of states need not be equally spaced in time.PAn initial value problem is a differential equation along with an initial state.;A differential equation expresses how the dependent variables (state) change with the independent variable (time). A differential equation is specified by giving the (time) derivative of the state as a function of the state. The (time) derivative of a state is an element of the associated vector space.OThe scalars of the associated vector space can be thought of as time intervals.An instance of Q is a data type that can serve as the state of some system. Alternatively, a M is a collection of dependent variables for a differential equation. A  has an associated vector space for the (time) derivatives of the state. The associated vector space is a linearized version of the .Associated vector spaceSubtract pointsPoint plus vectorPoint minus vectorGiven an evolution method and a time step, return the solution method which applies the evolution method repeatedly with with given time step. The solution method returned will produce an infinite list of states.tThe Euler method is the simplest evolution method. It increments the state by the derivative times the time step.PPosition is not a vector, but displacement (difference in position) is a vector.differential equation time interval initial state evolved state  666(c) Scott N. Walck 2012-2014BSD3 (see LICENSE)Scott N. Walck <walck@lvc.edu> experimental Trustworthy<gH (Take a single 4th-order Runge-Kutta stepOSolve a first-order system of differential equations with 4th-order Runge-Kutta  (c) Scott N. Walck 2014BSD3 (see LICENSE)Scott N. Walck <walck@lvc.edu> experimental TrustworthyFTyAn acceleration function gives a list of accelerations (one for each particle) as a function of the system's state.jThe state of a system of many particles is given by the current time and a list of one-particle states.An acceleration function gives a pair of accelerations (one for particle 1, one for particle 2) as a function of the system's state.The state of a system of two particles is given by the current time, the position and velocity of particle 1, and the position and velocity of particle 2.dAn acceleration function gives the particle's acceleration as a function of the particle's state.The state of a system of one particle is given by the current time, the position of the particle, and the velocity of the particle. Including time in the state like this allows us to have time-dependent forces.BThe associated vector space for the state of a single particle.lThe state of a single particle is given by the position of the particle and the velocity of the particle.|An acceleration function gives the particle's acceleration as a function of the particle's state. The specification of this function is what makes one single-particle mechanics problem different from another. In order to write this function, add all of the forces that act on the particle, and divide this net force by the particle's mass. (Newton's second law).VA simple one-particle state, to get started quickly with mechanics of one particle. Velocity of a particle (in m/s).A time step (in s). Time (in s). GTime derivative of state for a single particle with a constant mass.!Single Runge-Kutta step"GTime derivative of state for a single particle with a constant mass.#Single Runge-Kutta step$List of system states%ATime derivative of state for two particles with constant mass.&/Single Runge-Kutta step for two-particle system'BTime derivative of state for many particles with constant mass.(0Single Runge-Kutta step for many-particle system  &acceleration function for the particledifferential equation!&acceleration function for the particle time step initial statestate after one time step"&acceleration function for the particledifferential equation#&acceleration function for the particle time step initial statestate after one time step$&acceleration function for the particle time step initial statestate after one time step%'acceleration function for two particlesdifferential equation&acceleration function time step initial statestate after one time step'(acceleration function for many particlesdifferential equation(acceleration function time step initial statestate after one time step !"#$%&'( !"#$%&'((c) Scott N. Walck 2012-2014BSD3 (see LICENSE)Scott N. Walck <walck@lvc.edu> experimental TrustworthyFT.Surface is a parametrized function from two parameters to space, lower and upper limits on the first parameter, and lower and upper limits for the second parameter (expressed as functions of the first parameter).0-function from two parameters (s,t) into space1s_l2s_u3t_l(s)4t_u(s)5&A unit sphere, centered at the origin.62A sphere with given radius centered at the origin.7$Sphere with given radius and center.88The upper half of a unit sphere, centered at the origin.91A disk with given radius, centered at the origin.:<A plane surface integral, in which area element is a scalar.;=A dotted surface integral, in which area element is a vector.<"Shift a surface by a displacement.:*number of intervals for first parameter, s+number of intervals for second parameter, t'the scalar or vector field to integrate#the surface over which to integratethe resulting scalar or vector;*number of intervals for first parameter, s+number of intervals for second parameter, tthe vector field to integrate#the surface over which to integratethe resulting scalar./0123456789:;<./0123456789<:;./01234None=assumes radians coming in>6theta=0 is positive x axis, output angle in radians?An arrow}9Rotate takes its angle in degrees, and rotates clockwise.@ A think arrow?location of base of arrowdisplacement vector}displacement vector@arrow thicknesslocation of base of arrowdisplacement vector=>?@=>?@(c) Scott N. Walck 2011-2014BSD3 (see LICENSE)Scott N. Walck <walck@lvc.edu> experimentalSafeAAn ~( with a given label at a given position.BAn ~! that requests postscript output.CAn ~! giving the postscript file name.DAn example of the use of A. See the source code.EAn example of the use of B and C. See the source code.F*Plot a Curve in the xy plane using GnuplotABCDEFABCDEF(c) Scott N. Walck 2012-2014BSD3 (see LICENSE)Scott N. Walck <walck@lvc.edu> experimental TrustworthyFTGPVolume is a parametrized function from three parameters to space, lower and upper limits on the first parameter, lower and upper limits for the second parameter (expressed as functions of the first parameter), and lower and upper limits for the third parameter (expressed as functions of the first and second parameters).I#function from 3 parameters to spaceJs_aKs_bLt_a(s)Mt_b(s)Nu_a(s,t)Ou_b(s,t)P$A unit ball, centered at the origin.QIA unit ball, centered at the origin. Specified in Cartesian coordinates.R1A ball with given radius, centered at the origin.S"Ball with given radius and center.T1Upper half ball, unit radius, centered at origin.UCylinder with given radius and height. Circular base of the cylinder is centered at the origin. Circular top of the cylinder lies in plane z = h.VA volume integral n+1 pointsW!Shift a volume by a displacement.Sradiuscenter!ball with given radius and centerV-number of intervals for first parameter (s)-number of intervals for second parameter (t)-number of intervals for third parameter (u)scalar or vector field the volumescalar or vector resultGHIJKLMNOPQRSTUVWGHIJKLMNOPQRSTUWVGHIJKLMNO(c) Scott N. Walck 2012-2014BSD3 (see LICENSE)Scott N. Walck <walck@lvc.edu> experimental Trustworthy XA current distribution is a line current (current through a wire), a surface current, a volume current, or a combination of these. The  D describes a surface current density or a volume current density.Ycurrent through a wireZ ! is surface current density (A/m)[ " is volume current density (A/m^2)\$combination of current distributions])Electric current, in units of Amperes (A)^TMagnetic field produced by a line current (current through a wire). The function aR calls this function to evaluate the magnetic field produced by a line current._>Magnetic field produced by a surface current. The function a calls this function to evaluate the magnetic field produced by a surface current. This function assumes that surface current density will be specified parallel to the surface, and does not check if that is true.`=Magnetic field produced by a volume current. The function aT calls this function to evaluate the magnetic field produced by a volume current.aThe magnetic field produced by a current distribution. This is the simplest way to find the magnetic field, because it works for any current distribution (line, surface, volume, or combination).bGThe magnetic flux through a surface produced by a current distribution.^current (in Amps)geometry of the line currentmagnetic field (in Tesla)_surface current densitygeometry of the surface currentmagnetic field (in T)`volume current densitygeometry of the volume currentmagnetic field (in T) XYZ[\]^_`ab ]XYZ[\a^_`bXYZ[\(c) Scott N. Walck 2011-2014BSD3 (see LICENSE)Scott N. Walck <walck@lvc.edu> experimental TrustworthycA charge distribution is a point charge, a line charge, a surface charge, a volume charge, or a combination of these. The !\ describes a linear charge density, a surface charge density, or a volume charge density.d point chargee! is linear charge density (C/m)f!" is surface charge density (C/m^2)g!! is volume charge density (C/m^3)h#combination of charge distributionsi)Electric charge, in units of Coulombs (C)j-Total charge (in C) of a charge distribution.k;Electric field produced by a point charge. The function oR calls this function to evaluate the electric field produced by a point charge.l:Electric field produced by a line charge. The function oQ calls this function to evaluate the electric field produced by a line charge.m=Electric field produced by a surface charge. The function oT calls this function to evaluate the electric field produced by a surface charge.n<Electric field produced by a volume charge. The function oS calls this function to evaluate the electric field produced by a volume charge.oThe electric field produced by a charge distribution. This is the simplest way to find the electric field, because it works for any charge distribution (point, line, surface, volume, or combination).pFThe electric flux through a surface produced by a charge distribution.qaElectric potential from electric field, given a position to be the zero of electric potential.rElectric potential produced by a charge distribution. The position where the electric potential is zero is taken to be infinity. kcharge (in Coulombs)of point chargeelectric field (in V/m)llinear charge density lambdageometry of the line chargeelectric field (in V/m)msurface charge density sigmageometry of the surface chargeelectric field (in V/m)nvolume charge density rhogeometry of the volume chargeelectric field (in V/m)q)position where electric potential is zeroelectric fieldelectric potentialcharge (in Coulombs)of point chargeelectric potentiallinear charge density lambdageometry of the line chargeelectric potentialsurface charge density sigmageometry of the surface chargeelectric potentialvolume charge density rhogeometry of the volume chargeelectric potentialcdefghijklmnopqricdefghjoklmnpqrcdefgh(c) Scott N. Walck 2014-2018BSD3 (see LICENSE)Scott N. Walck <walck@lvc.edu> experimental Trustworthy o  !"#$%&'()*+,-./012345678:;<=>?@CDEFGHIJ  !"#$%&'(./0123456789:;<=>?@ABCGHIJKLMNOPQRSTUVWXYZ[\]abcdefghijopqr !"#$%&'(icdefghj]XYZ[\opqrab   #"! %&'()*+,-./01$2345678:;<=>DEFGHIJC?@./0123456789<:;GHIJKLMNOPQRSTUWV ABC=>?@ ! " # $ %&'&(&)&*&+,,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXXYZ[\]^_`abcdefghi<<jklmnopqrstuvwxyz{|}~  x         ! y z { | } ~   + ) ' * ( $ " # % !                     !"#$%&''()*+,-./0123456789:;<=>??@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijkl m    nopqrstuvwxyz*learn-physics-0.6.2-4tFdKVFpA3K6W7slR7IoXoPhysics.Learn.QuantumMatPhysics.Learn.CarrotVecPhysics.Learn.CommonVec!Physics.Learn.CompositeQuadraturePhysics.Learn.PositionPhysics.Learn.CurvePhysics.Learn.CoordinateSystemPhysics.Learn.CoordinateFieldsPhysics.Learn.KetPhysics.Learn.BeamStackPhysics.Learn.RootFindingPhysics.Learn.SimpleVecPhysics.Learn.StateSpacePhysics.Learn.RungeKuttaPhysics.Learn.MechanicsPhysics.Learn.SurfacePhysics.Learn.Visual.GlossToolsPhysics.Learn.Visual.PlotToolsPhysics.Learn.VolumePhysics.Learn.CurrentPhysics.Learn.Charge Physics.Learn&hmatrix-0.19.0.0-4fS2XrDxhQP73ElsI1QKZInternal.MatrixMatrixInternal.VectorC&vector-0.12.0.1-LflPw1fguMb6as60UrZpxNData.Vector.StorableVector(vector-space-0.13-GXOaDRWEbYx2S8Sp58P20BData.VectorSpace magnitude^*^/*^<.>Data.AdditiveGroupsumV^-^negateV^+^zeroVVecxCompyCompzCompvec><iHatjHatkHat $fShowVec$fEqVec$fInnerSpaceVec$fVectorSpaceVec$fAdditiveGroupVeccompositeTrapezoidcompositeSimpsonCoordinateSystemField VectorField ScalarField DisplacementPosition addFields cartesian cylindrical sphericalcartcylsphcartesianCoordinatescylindricalCoordinatessphericalCoordinates displacement shiftPosition shiftObject shiftFieldrHatthetaHatphiHatsHatxHatyHatzHat$fShowPositionCurve curveFuncstartingCurveParamendingCurveParamdottedLineIntegralcrossedLineIntegral$compositeTrapezoidDottedLineIntegral%compositeTrapezoidCrossedLineIntegralsimpleLineIntegralnormalizeCurve concatCurvesconcatenateCurves reverseCurve evalCurve shiftCurve straightLine"compositeSimpsonDottedLineIntegral#compositeSimpsonCrossedLineIntegral toPosition fromPositionstandardCartesianstandardCylindricalstandardSphericalnewCoordinateSystemxyzsphirtheta KroneckerkronxpxmypymzpzmnpnmdimscaleVinnernorm normalize probVectorconjVfromListtoListsxsyszscaleM<>#><#conjugateTranspose fromListstoListssizematrixFunctioncouterdmtrace normalizeDM oneQubitMixed timeEvMattimeEv timeEvMatSpecpossibleOutcomesoutcomesProjectorsoutcomesProbabilities gramSchmidt$fKroneckerMatrix$fKroneckerVectorKron RepresentablerepOrthonormalBasisHasNormDaggerdaggerMultBraOperatorKetimakeOB listBasisxBasisyBasiszBasisnBasissnsn'timeEvOp$fNumKet $fNumOperator$fNumBra $fShowBra$fMultOperatorOperatorOperator$fMultKetBraOperator$fMultOperatorKetKet$fMultBraOperatorBra$fMultBraKetComplex$fMultOperatorComplexOperator$fMultBraComplexBra$fMultKetComplexKet$fMultComplexOperatorOperator$fMultComplexBraBra$fMultComplexKetKet$fMultComplexComplexComplex$fDaggerComplexComplex$fDaggerOperatorOperator$fDaggerBraKet$fDaggerKetBra $fHasNormBra $fHasNormKet$fRepresentableOperatorMatrix$fRepresentableBraVector$fRepresentableKetVector$fShowOperator $fShowKet$fKronOperator $fKronBra $fKronKet$fShowOrthonormalBasis BeamStack randomBeamdetectdropBeamnumBeams flipBeamssplit recombine applyBFieldsplitXsplitYsplitZ applyBFieldX applyBFieldY applyBFieldZ recombineX recombineY recombineZxpFilterxmFilterzpFilterzmFilter$fShowBeamStack bracketRootbracketRootStepfindRoot findRootsN findRootsEvolutionMethodSolutionMethodInitialValueProblemDifferentialEquationTime StateSpaceDiff.-..+^.-^ stepSolution eulerMethod$fVectorSpace[]$fAdditiveGroup[]$fStateSpace[]$fStateSpacePosition$fStateSpaceVec$fStateSpaceDouble$fStateSpace(,,)$fStateSpace(,) rungeKutta4integrateSystem ManyParticleAccelerationFunctionManyParticleSystemStateTwoParticleAccelerationFunctionTwoParticleSystemStateOneParticleAccelerationFunctionOneParticleSystemStateDStStpositionvelocitySimpleAccelerationFunction SimpleStateVelocityTimeStepTheTimesimpleStateDerivsimpleRungeKuttaSteponeParticleStateDerivoneParticleRungeKuttaSteponeParticleRungeKuttaSolutiontwoParticleStateDerivtwoParticleRungeKuttaStepmanyParticleStateDerivmanyParticleRungeKuttaStep$fStateSpaceSt$fVectorSpaceDSt$fAdditiveGroupDSt$fShowSt $fShowDStSurface surfaceFunc lowerLimit upperLimit lowerCurve upperCurve unitSpherecenteredSpherespherenorthernHemispheredisksurfaceIntegraldottedSurfaceIntegral shiftSurface polarToCart cartToPolararrow thickArrowlabel postscriptpsFile examplePlot1 examplePlot2 plotXYCurveVolume volumeFuncloLimitupLimitloCurveupCurveloSurfupSurfunitBallunitBallCartesian centeredBallballnorthernHalfBallcenteredCylindervolumeIntegral shiftVolumeCurrentDistribution LineCurrentSurfaceCurrent VolumeCurrentMultipleCurrentsCurrentbFieldFromLineCurrentbFieldFromSurfaceCurrentbFieldFromVolumeCurrentbField magneticFluxChargeDistribution PointCharge LineCharge SurfaceCharge VolumeChargeMultipleChargesCharge totalChargeeFieldFromPointChargeeFieldFromLineChargeeFieldFromSurfaceChargeeFieldFromVolumeChargeeField electricFluxelectricPotentialFromFieldelectricPotentialFromChargeCart dottedSimp crossedSimpOBghc-prim GHC.TypesDouble originArrow&gnuplot-0.5.5.2-HMksWHbYfy54srWcPrRkWSGraphics.Gnuplot.Simple Attribute linSpacedePotFromPointChargeePotFromLineChargeePotFromSurfaceChargeePotFromVolumeCharge