Mathematical Functions and Constants #

FormulaGrader and NumericalGrader Default Functions #

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
... )