Time (in s).

A time step (in s).

Velocity of a particle (in m/s).

A simple one-particle state, to get started quickly with mechanics of one 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).

Time derivative of state for a single particle with a constant mass.

Single Runge-Kutta step

The state of a single particle is given by the position of the particle and the velocity of the particle.

The associated vector space for the state of a single particle.

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.

An acceleration function gives the particle's acceleration as a function of the particle's state.

Time derivative of state for a single particle with a constant mass.

Single Runge-Kutta step

List of system states

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.

An acceleration function gives a pair of accelerations (one for particle 1, one for particle 2) as a function of the system's state.

Time derivative of state for two particles with constant mass.

Single Runge-Kutta step for two-particle system

The 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 list of accelerations (one for each particle) as a function of the system's state.

Time derivative of state for many particles with constant mass.

Single Runge-Kutta step for many-particle system

Electric charge, in units of Coulombs (C)

A charge distribution is a point charge, a line charge, a surface charge, a volume charge, or a combination of these. The ScalarField describes a linear charge density, a surface charge density, or a volume charge density.

Total charge (in C) of a charge distribution.

Electric current, in units of Amperes (A)

A current distribution is a line current (current through a wire), a surface current, a volume current, or a combination of these. The VectorField describes a surface current density or a volume current density.

The 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).

The electric flux through a surface produced by a charge distribution.

Electric potential from electric field, given a position to be the zero of electric potential.

Electric potential produced by a charge distribution. The position where the electric potential is zero is taken to be infinity.

The 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).

The magnetic flux through a surface produced by a current distribution.

A type for vectors.

x component

y component

z component

Form a vector by giving its x, y, and z components.

Add vectors

Group subtraction

Scale a vector

Vector multiplied by scalar

Vector divided by scalar

Inner/dot product

Cross product.

Length of a vector. See also magnitudeSq.

The zero element: identity for '(^+^)'

Additive inverse

Sum over several vectors

Unit vector in the x direction.

Unit vector in the y direction.

Unit vector in the z direction.

A type for position. Position is not a vector because it makes no sense to add positions.

A displacement is a vector.

A scalar field associates a number with each position in space.

A vector field associates a vector with each position in space.

Sometimes we want to be able to talk about a field without saying whether it is a scalar field or a vector field.

A coordinate system is a function from three parameters to space.

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.

A helping function to take three numbers x, y, and z and form the appropriate position using Cartesian coordinates.

A 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.

Returns the three Cartesian coordinates as a triple from a position.

Returns the three cylindrical coordinates as a triple from a position.

Returns the three spherical coordinates as a triple from a position.

Displacement from source position to target position.

Shift a position by a displacement.

An object is a map into Position.

A field is a map from Position.

Add two scalar fields or two vector fields.

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 rHat points in different directions at different points in space. It is therefore better interpreted as a vector field, rather than a vector.

The 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.

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.

The 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.

The 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.

The 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.

The 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.

Curve is a parametrized function into three-space, an initial limit, and a final limit.

Reparametrize a curve from 0 to 1.

Concatenate two curves.

Concatenate a list of curves. Parametrizes curves equally.

Reverse a curve.

Evaluate the position of a curve at a parameter.

Shift a curve by a displacement.

The straight-line curve from one position to another.

Calculates integral f dl over curve, where dl is a scalar line element.

A dotted line integral. Convenience function for compositeSimpsonDottedLineIntegral.

Calculates integral vf x dl over curve. Convenience function for compositeSimpsonCrossedLineIntegral.

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).

A unit sphere, centered at the origin.

A sphere with given radius centered at the origin.

Sphere with given radius and center.

The upper half of a unit sphere, centered at the origin.

A disk with given radius, centered at the origin.

Shift a surface by a displacement.

A plane surface integral, in which area element is a scalar.

A dotted surface integral, in which area element is a vector.

Volume 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).

A unit ball, centered at the origin.

A unit ball, centered at the origin. Specified in Cartesian coordinates.

A ball with given radius, centered at the origin.

Ball with given radius and center.

Upper half ball, unit radius, centered at origin.

Cylinder 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.

Shift a volume by a displacement.

A volume integral

An instance of StateSpace is a data type that can serve as the state of some system. Alternatively, a StateSpace is a collection of dependent variables for a differential equation. A StateSpace has an associated vector space for the (time) derivatives of the state. The associated vector space is a linearized version of the StateSpace.

Point minus vector

The scalars of the associated vector space can be thought of as time intervals.

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.

An initial value problem is a differential equation along with an initial state.

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.

Given 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.

The Euler method is the simplest evolution method. It increments the state by the derivative times the time step.

Take a single 4th-order Runge-Kutta step

Solve a first-order system of differential equations with 4th-order Runge-Kutta

An Attribute with a given label at a given position.

An Attribute that requests postscript output.

An Attribute giving the postscript file name.

assumes radians coming in

theta=0 is positive x axis, output angle in radians

An arrow

A think arrow

Make a V3 object from a Vec.

Make a V3 object from a Position.

A VisObject arrow from a vector

Place a vector at a particular position.

Display a vector field.

A displayable VisObject for a curve.