Coefficient

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For other senses of this word, see coefficient (disambiguation).

In mathematics, a coefficient is a constant multiplicative factor of a certain object. For example, in the expression 9x², the coefficient of x² is 9.

The object can be such things as a variable, a vector, a function, etc. In some cases, the objects and the coefficients are indexed in the same way, leading to expressions such as:

$a_1 x_1 + a_2 x_2 + a_3 x_3 + \cdots$

where a_n is the coefficient of the variable x_n for each n = 1, 2, 3, …

In a polynomial P(x) of one variable x, the coefficient of x^k can be indexed by k, giving the convention that for example:

$P(x) = a_k x^k + \cdots + a_1 x^1 + a_0.$

For the largest k where a_k ≠ 0, a_k is called the leading coefficient of P because most often, polynomials are written starting from the left with the largest power of x. So for example the leading coefficient of the polynomial

$\, 4x^5 + x^3 + 2x^2$

is 4.

The coefficients of polynomial also may be in the other order:

$Q(x) = a_0 x^k + a_1 x^{k-1} + \cdots + a_{k-1} x^1 + a_k$

and must be a₀≠0 and a₀ is the leading coefficient of Q.

Important coefficients in mathematics include the binomial coefficients which are coefficients in the statement of the binomial theorem. These can be partially found with Pascal's triangle.

Linear algebra

In linear algebra, the leading coefficient of a row in a matrix is the first nonzero entry in that row. So, for example, given

$M = \begin{bmatrix}1 & 2 & 0 & 6 \\ 0 & 2 & 9 & 4 \\ 0 & 0 & 0 & 4 \\ 0 & 0 & 0 & 0 \end{bmatrix}.$

The leading coefficient of the first row is 1, 2 is the leading coefficient of the second row, 4 is the leading coefficient of the third row, and the last row does not have a leading coefficient.

Examples of Physical Coefficients

Coefficient of Thermal Expansion (thermodynamics) (dimensionless) - Relates the change in temperature to the change in a material's dimensions.
Partition Coefficient (K_D) (chemistry) - The ratio of concentrations of a compound in two phases of a mixture of two immiscible solvents at equilibrium.
Hall coefficient (electrical physics) - Relates a magnetic field applied to an element to the voltage created, the amount of current and the element thickness. It is a characteristic of the material from which the conductor is made.
Lift Coefficient (C_L or C_Z) (Aerodynamics) (dimensionless) - Relates the lift generated by an airfoil with the dynamic pressure of the fluid flow around the airfoil, and the planform area of the airfoil.
Ballistic coefficient (BC) (Aerodynamics) (units of kg/m²) - A measure of a body's ability to overcome air resistance in flight. BC is a function of mass, diameter, and drag coefficient.
Transmission Coefficient (quantum mechanics) (dimensionless) - Represents the probability flux of a transmitted wave relative to that of an incident wave. It is often used to describe the probability of a particle tunnelling through a barrier.
Damping Factor a.k.a. viscous damping coefficient (Physical Engineering) (units of Newton-seconds per meter) - relates a damping force with the velocity of the object whose motion is being dampened.

Coefficient

Linear algebra

Examples of Physical Coefficients

See also