The product rule for exponents states that when multiplying two exponential expressions with the same base, you keep the base and add the exponents.
The quotient rule for exponents states that when dividing two exponential expressions with the same base, you keep the base and subtract the exponents.
The power of a power rule for exponents states that when raising an exponential expression to another power, you keep the base and multiply the exponents.
The power of a product rule for exponents states that when raising a product to a power, you raise each factor to that power.
Any non-zero number raised to the power of zero equals one.
A negative exponent indicates the reciprocal of the base raised to the positive exponent.
The difference of squares formula.
The formula for the square of a binomial sum.
The formula for the square of a binomial difference.
The formula for the sum of two cubes.
The formula for the difference of two cubes.
Slope of a line through two points.
Equation of a line in slope-intercept form.
Equation of a line in point-slope form.
Equation of a line in standard form.
Solve quadratic equations of the form ax² + bx + c = 0.
Equation of a quadratic function in vertex form.
Determines the nature of the roots of a quadratic equation.
Definition of the derivative of a function at a point.
The derivative of a constant is zero.
The derivative of a constant multiplied by a function is the constant times the derivative of the function.
The power rule for derivatives states that the derivative of x raised to the power of n is n times x raised to the power of n minus 1.
The product rule for derivatives states that the derivative of the product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function.
The quotient rule for derivatives states that the derivative of the quotient of two functions is the derivative of the numerator times the denominator minus the numerator times the derivative of the denominator, all divided by the square of the denominator.
The chain rule for derivatives states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.
The derivative of the sum or difference of two functions is the sum or difference of their derivatives.
The derivative of the exponential function is the exponential function itself.
The derivative of an exponential function with a variable exponent is the exponential function times the derivative of the exponent.
The derivative of the natural logarithm of a function is the derivative of the function divided by the function itself.
Basic derivative rule from calculus.
Indefinite integral of the exponential function.
A complex number z consists of a real part a and an imaginary part b. The conjugate of z is denoted by \bar{z} and is obtained by changing the sign of the imaginary part.
The modulus of a complex number z = a + bi is the distance from the origin to the point (a, b) in the complex plane.
A fundamental identity combining e, i, π, 1, and 0.
Relates complex exponentials to trigonometric functions.
Represents a complex number in polar form, where |z| is the modulus and θ is the argument.
A contour integral of a complex function f(z) along a closed curve C in the complex plane.
If f(z) is analytic (holomorphic) on and inside a closed curve C, then the contour integral of f(z) around C is zero.
A fundamental formula in complex analysis that relates the value of an analytic function at a point to its values on a closed curve surrounding that point.
A generalization of Cauchy’s integral formula that gives the nth derivative of an analytic function at a point in terms of a contour integral around that point.
A theorem in complex analysis that relates the residue of a function at a singularity to its behavior near that point.
The residue of a function f at a simple pole z₀ is the limit as z approaches z₀ of (z - z₀)f(z).
A series expansion of a complex function that represents the function as an infinite sum of terms calculated from the derivatives of the function at a single point.
A series expansion of a complex function that includes both positive and negative powers of (z - a), used to represent functions with singularities.
The ratio of a circle's circumference to its diameter. Approximately equal to 3.141592653589793
The base of the natural logarithm, approximately equal to 2.718281828459045
The golden ratio, approximately equal to 1.618033988749895
The imaginary unit, defined as the square root of -1.
The Euler-Mascheroni constant, approximately equal to 0.577215664915366
The speed of light in a vacuum, approximately equal to 299,792,458 meters per second.
The gravitational constant, approximately equal to 6.67430 × 10^-11 m^3 kg^-1 s^-2.
Planck’s constant, approximately equal to 6.62607015 × 10^-34 J·s.
The elementary charge, approximately equal to 1.602176634 × 10^-19 coulombs.
The Boltzmann constant, approximately equal to 1.380649 × 10^-23 J/K.
The force between two point charges.
The electric field at a point in space due to a point charge.
The electric flux through a closed surface is equal to the charge enclosed divided by the permittivity of free space.
The electric potential at a point in space due to a point charge.
The relationship between voltage, current, and resistance in an electrical circuit.
The resistance of an electrical component. The first formula relates resistance to voltage and current, while the second formula relates resistance to the material's resistivity, length, and cross-sectional area.
The power dissipated in an electrical component. The first formula relates power to voltage and current, while the second and third formulas relate power to current and resistance, and voltage and resistance, respectively.
The capacitance of a capacitor. The first formula relates capacitance to charge and voltage, while the second formula relates capacitance to the permittivity of free space, the area of the plates, and the distance between them.
The equivalent capacitance of capacitors connected in series or parallel. The first formula relates the equivalent capacitance to the individual capacitances in series, while the second formula relates the equivalent capacitance to the individual capacitances in parallel.
The equivalent resistance of resistors connected in series or parallel. The first formula relates the equivalent resistance to the individual resistances in series, while the second formula relates the equivalent resistance to the individual resistances in parallel.
The magnetic force on a moving charged particle. The force is perpendicular to both the velocity of the particle and the magnetic field.
The magnetic force on a current-carrying wire. The force is perpendicular to both the current and the magnetic field.
The line integral of the magnetic field around a closed loop is equal to the permeability of free space times the enclosed current.
The magnetic field produced at a point in space by a small segment of current-carrying wire.
The magnetic flux through a surface. It is the integral of the magnetic field over the area.
The induced electromotive force in a closed loop is equal to the negative rate of change of the magnetic flux through the loop.
The electric flux through a closed surface is equal to the charge enclosed divided by the permittivity of free space.
The magnetic flux through a closed surface is always zero.
The induced electromotive force in a closed loop is equal to the negative rate of change of the magnetic flux through the loop.
The line integral of the magnetic field around a closed loop is equal to the permeability of free space times the enclosed current plus the displacement current.
Distance between two points in the plane.
Coordinates of the midpoint between two points.
Properties of a square.
Properties of a rectangle.
Properties of a triangle.
Properties of a circle.
Properties of a trapezoid.
Properties of a parallelogram.
Properties of a cube.
Properties of a rectangular prism or box.
Properties of a sphere with radius r.
Properties of a cylinder with radius r and height h.
Properties of a cone with radius r and height h.
Definitions of scalar, vector, matrix, and tensor.
A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns.
The sum of two matrices of the same dimensions is obtained by adding corresponding elements.
The norm of a matrix is a measure of its size, calculated as the square root of the sum of the squares of its elements.
The transpose of a matrix is obtained by swapping its rows and columns.
The transpose of the sum of two matrices is equal to the sum of their transposes.
The inverse of the product of two matrices is equal to the product of their inverses in reverse order.
The determinant of the product of two matrices is equal to the product of their determinants.
The determinant of a 2x2 matrix is calculated as the product of the diagonal elements minus the product of the off-diagonal elements.
The determinant of a 3x3 matrix is calculated using the rule of Sarrus or cofactor expansion.
The determinant of a 3x3 matrix is calculated using the rule of Sarrus or cofactor expansion.
The inverse of a matrix is a matrix that, when multiplied by the original matrix, gives the identity matrix.
The product of two matrices is obtained by multiplying corresponding elements and summing them.
The product of two matrices is obtained by multiplying corresponding elements and summing them.
The dot product of two matrices is the sum of the products of their corresponding components.
The cross product of two matrices is a vector that is perpendicular to both matrices.
The projection of one matrix onto another is a measure of how much of the first matrix points in the direction of the second.
Eigenvalues and eigenvectors are special values and vectors associated with a matrix.
The average velocity of an object over a given time interval.
The average acceleration of an object over a given time interval.
A formula for calculating displacement when initial velocity and acceleration are known.
A formula for calculating the final velocity of an object given its initial velocity, acceleration, and time.
A formula for calculating the time taken for an object to reach a certain velocity given its initial velocity, acceleration, and final velocity.
A formula for calculating the acceleration of an object given its initial and final velocities and the time taken.
An object at rest stays at rest and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force.
The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.
The force exerted on an object due to gravity.
The force of kinetic friction acting on an object.
The maximum force of static friction acting on an object.
The work done by a constant force acting on an object.
The energy possessed by an object due to its motion.
The energy possessed by an object due to its position in a gravitational field.
The work done by a net force acting on an object equals the change in its kinetic energy.
The rate at which work is done or energy is transferred.
The product of an object's mass and velocity.
The change in momentum of an object due to a force acting over a time interval.
The total linear momentum of a closed system remains constant over time.
The rate of change of angular displacement with respect to time.
The rate of change of angular velocity with respect to time.
The relationship between linear and rotational quantities.
The acceleration of an object moving in a circular path.
The relationship between angular displacement, initial angular velocity, angular acceleration, and time.
The probability of an event A occurring.
The probability of the complement of an event A occurring.
The probability of the union of two events A and B occurring.
The probability of the union of two mutually exclusive events A and B occurring.
The probability of the intersection of two events A and B occurring.
The probability of the intersection of two independent events A and B occurring.
The probability of event A occurring given that event B has occurred.
The factorial of a positive integer n is the product of all positive integers less than or equal to n.
The number of ways to arrange r objects from a set of n distinct objects.
The number of ways to choose r objects from a set of n distinct objects.
The mean (average) of a set of n numbers is the sum of the numbers divided by n.
The variance of a set of n numbers is the average of the squared differences from the mean.
The standard deviation of a set of n numbers is the square root of the variance.
The Z-score of a data point is the number of standard deviations it is from the mean.
The probability density function of a normal distribution with mean μ and standard deviation σ.
The probability of getting exactly k successes in n independent Bernoulli trials, each with success probability p.
The mean and standard deviation of a binomial distribution with n trials and success probability p.
The probability of observing k events in a fixed interval of time or space, given the average number of events (λt) in that interval.
The basic mathematical operators for arithmetic, comparison, and other operations.
The symbol Δ is used in mathematics to represent change or difference. It is often used to denote the difference between two values, such as Δx (the change in x) or Δy (the change in y).
The differential is a fundamental concept in calculus used to represent an infinitesimally small change in a variable.
The integral is a fundamental concept in calculus used to calculate the area under a curve or the accumulation of a quantity.
The summation is a fundamental concept in mathematics used to calculate the sum of a sequence of numbers.
The product is a fundamental concept in mathematics used to calculate the product of a sequence of numbers.
A collection of distinct objects, considered as an object in its own right.
The intersection of two sets A and B is the set of elements that are in both A and B.
The union of two sets A and B is the set of elements that are in either A or B (or both).
Set A is a proper subset of set B if every element of A is in B, but A is not equal to B.
Set A is a subset of set B if every element of A is in B.
Set A is a proper superset of set B if every element of B is in A, but A is not equal to B.
Set A is a superset of set B if every element of B is in A.
Set A is not a subset of set B.
Set A is not a superset of set B.
The complement of set A is the set of all elements in the universal set that are not in A.
The relative complement of set B in set A is the set of elements that are in A but not in B.
The symmetric difference of two sets A and B is the set of elements that are in either A or B, but not in both.
Element x is a member of set A.
Element x is not a member of set A.
An ordered pair is a collection of two elements where the order matters.
The Cartesian product of two sets A and B is the set of all ordered pairs (a, b) where a is in A and b is in B.
The cardinality of a set A is the number of elements in A.
The empty set is the set that contains no elements.
The universal set is the set that contains all elements under consideration.
The set of all natural numbers.
The set of all rational numbers.
The set of all real numbers.
The set of all complex numbers.
The set of all integer numbers.
The relationship between pressure, volume, temperature, and the amount of an ideal gas.
The change in internal energy of a system is equal to the heat added to the system minus the work done by the system.
Relationships between heat, work, and internal energy in thermodynamics.
The entropy of an isolated system always increases over time.
The enthalpy and Gibbs free energy of a system.
Relates the sides of a right triangle.
Properties of a right triangle with angle θ.
For this definition θ is any angle. The hypotenuse is always 1, the opposite is the y-coordinate, and the adjacent is the x-coordinate.
Tangent and cotangent in terms of sine and cosine.
Reciprocal identities for trigonometric functions.
Pythagorean identities for trigonometric functions.
Properties of even and odd trigonometric functions.
Properties of periodic trigonometric functions.
Converting between degrees and radians. If x is an angle in degrees and t is an angle in radians.
Formulas for the sine, cosine, and tangent of half an angle.
Formulas for the sine, cosine, and tangent of double an angle.
Formulas for the sine and cosine of the sum or difference of two angles.
Formulas for converting products of trigonometric functions to sums.
Formulas for converting sums of trigonometric functions to products.
Formulas relating sine and cosine of complementary angles.
The speed of a wave is equal to the product of its frequency and wavelength.
The period of a wave is equal to the reciprocal of its frequency.
The angular frequency of a wave is equal to 2π times its frequency.
The wave number is equal to the angular frequency divided by the wave speed.
The general equation for a wave traveling in the x-direction with speed v.
The index of refraction is equal to the speed of light in a vacuum divided by the speed of light in the material.
The relationship between the angles of incidence and refraction when light passes through the interface of two different media.
The critical angle is the angle of incidence at which light is completely reflected at the interface of two different media.
The relationship between the focal length of a spherical mirror or thin lens and the object and image distances.
The magnification of an image formed by a spherical mirror or thin lens.
The condition for constructive interference in a double-slit experiment.
The condition for destructive interference in a double-slit experiment.
The separation between adjacent bright fringes in a double-slit experiment.
The condition for minima in a single-slit diffraction pattern.
The condition for constructive interference in X-ray diffraction.
The intensity of polarized light after passing through a polarizer.