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Mathematical Functions and Constants

FormulaGrader and NumericalGrader Default Functions

!!! note Below, expressions marked with a * may require our AsciiMath renderer definitions to display properly in edX.

By default, all of the following functions are made available to students in FormulaGrader problems.

  • sin(x) Sine
  • cos(x) Cosine
  • tan(x) Tangent
  • sec(x) Secant
  • csc(x) Cosecant
  • cot(x) Cotangent
  • sqrt(x) Square Root
  • log10(x) Log (base 10)*
  • log2(x) Log (base 2)*
  • ln(x) Natural logarithm
  • exp(x) Exponential
  • arccos(x) Inverse Cosine
  • arcsin(x) Inverse Sine
  • arctan(x) Inverse Tangent
  • arctan2(x, y) Four-quadrant Inverse Tangent*
  • arcsec(x) Inverse Secant*
  • arccsc(x) Inverse Cosecant*
  • arccot(x) Inverse Cotangent*
  • abs(x) Absolute value (real) or modulus (complex)
  • factorial(x) and fact(x) Factorial*
    • domain: all complex numbers except negative integers. Large outputs may raise OverflowErrors.
  • sinh(x) Hyperbolic Sine
  • cosh(x) Hyperbolic Cosine
  • tanh(x) Hyperbolic Tangent
  • sech(x) Hyperbolic Secant
  • csch(x) Hyperbolic Cosecant
  • coth(x) Hyperbolic Cotangent
  • arcsinh(x) Inverse Hyperbolic Sine*
  • arccosh(x) Inverse Hyperbolic Cosine*
  • arctanh(x) Inverse Hyperbolic Tangent*
  • arcsech(x) Inverse Hyperbolic Secant*
  • arccsch(x) Inverse Hyperbolic Cosecant*
  • arccoth(x) Inverse Hyperbolic Cotangent*
  • floor(x) Floor function (applies only to real numbers)
  • ceil(x) Ceiling function (applies only to real numbers)
  • min(x, y, z, ...) Minimum of the arguments (applies only to real numbers, 2 or more arguments)
  • max(x, y, z, ...) Maximum of the arguments (applies only to real numbers, 2 or more arguments)
  • re(x) Real part of a complex expression*
  • im(x) Imaginary part of a complex expression*
  • conj(x) Complex conjugate of a complex expression*
  • kronecker(x, y) Kronecker delta* (Note that we highly recommend integer sampling over a short range (eg, 1 to 4) when Kronecker deltas appear in an answer, and using many samples (eg, 30) so that most permutations appear in the sampling.)

MatrixGrader Default Functions

In MatrixGrader problems, all FormulaGrader functions are available by default, as are the following extra functions:

  • abs(x): absolute value of a scalar or magnitude of a vector
  • adj(x): Hermitian adjoint, same as ctrans(x)*
  • cross(x, y): cross product, inputs must be 3-component vectors*
  • ctrans(x): conjugate transpose, same as adj(x)*
  • det(x): determinant, input must be square matrix
  • norm(x): Frobenius norm, works for scalars, vectors, and matrices
  • trans(x): transpose*
  • trace(x): trace

Default Constants

Available in FormulaGrader, NumericalGrader, and MatrixGrader by default:

  • i: imaginary unit (same as j)
  • j: imaginary unit (same as i)
  • e: approximately 2.718281828
  • pi: approximately 3.141592654

Optional Constant Collections

We provide a few collections of constants that can be imported for convenience and reuse. For example, pauli is a dictionary with keys sigma_x, sigma_y, and sigma_z that are MathArray representations of the 2x2 Pauli matrices.

The collections of available mathematical constants are:

  • pauli: MathArray representations of the 2x2 Pauli matrices, sigma_x, sigma_y, and sigma_z
  • cartesian_xyz: MathArray representations of the three-dimensional Cartesian unit vectors, named hatx, haty, hatz
  • cartesian_ijk: MathArray representations of the three-dimensional Cartesian unit vectors, named hati, hatj, hatk

Each collection is a dictionary that can be provided as a value of user_constants:

>>> from mitxgraders import *
>>> grader = MatrixGrader(
...   answers='sigma_x + sigma_z',
...   user_constants=pauli
... )