A complete collection of common symbols in advanced mathematics and their meanings

Encyclopedia of commonly used symbols in advanced mathematics and the meaning of

symbols Symbol meaning The square root of
i
-1
f(x)
the value of the function f at the independent variable x
sin(x)
the sine function value at the independent variable x
exp(x)
at the independent variable x The value of the exponential function at , is often written as ex
a^x
a to the power of x; a rational number x is defined by the inverse function
ln x
exp x The inverse function
ax
is the same as a^x
logba
the logarithm of a base b; blogba = a
cos x
the value of the cosine function at the independent variable x
tan x
the value of which is equal to the value of the cosine function of sin x/cos x
cot x
or the value of the cosine function of cos x/sin x
sec x
the secant including the value of 1/cos x
csc x
The value of the cosecant function, whose value is equal to 1/sin x
asin x
y, the value of the inverse function of the sine function at x, that is, x = sin y
acos x
y, the value of the inverse function of the cosine function at x, that is, x = cos y
atan x
y, the value of the inverse function of the tangent function at x, that is, x = tan y
acot x
y, the value of the inverse function of the cotangent function at x, that is, x = cot y
asec x
y, the inverse function of the secant function at x. Value, that is x = sec y
acsc x
y, the value of the inverse function of the cosecant function at x, that is, a standard symbol of x = csc y
​​θ
angle, if not specified, all refer to radians, especially for atan x/y, When x, y, and z are used to represent points in space,
i, j, and k represent unit vectors (a, b, c) in
the x, y, and z directions, respectively. A vector with a, b, and c as elements (a , b) Vectors with a and b as elements (a, b) Dot product of a and b vectors a•b Dot product of a and b vectors (a•b) Dot product of a and b vectors |v| vector v The absolute value Σ of the number x modulo |x| represents the summation, usually an exponent of some term. The lower boundary value is written below it, and the upper boundary value is written above it. For example, the sum of j from 1 to 100 can be expressed as: a common mathematical symbol. This means that 1 + 2 + … + n M means a matrix or sequence or other |v>


















A column vector, i.e. a vector whose elements are written as columns or can be seen as a matrix of order k × 1
<v| is written as a row or can be seen as an infinitesimal variation of the variable x from a
vector
dx of a matrix of order 1 × k, dy, dz
, dr, etc. Small changes in the length of
ds ρ variable (x2 + y2 + z2)1/2 or the distance to the origin in spherical coordinates r variable (x2 + y2)1/2 or three-dimensional space or polar coordinates to the z-axis The distance |M| of the determinant of matrix M whose value is the area or volume of the parallel region determined by the rows and columns of the matrix ||M|| The value of the determinant of matrix M, which is an area, volume or hypervolume det M The determinant of M M-1 The inverse matrix of M The inverse matrix v×w The vector product of vectors v and w or the cross product θvw The angle between the vectors v and w A B×C scalar triple product, with A, B, C is the unit vector of the determinant uw of the column matrix in the direction of the vector w, i.e. the small variation of the w/|w| df function f, small enough to be suitable for a linear approximation of all correlation functions df/dx The derivative of f with respect to x, Also the linear approximation slope f ' of f
























The derivative of the function f with respect to the corresponding independent variable. The independent variable is usually the partial derivative of f with respect to x when x
∂f/∂x y and z are fixed.
Usually the partial derivative of f with respect to a variable q is the ratio of df to dq when several other variables are fixed. Any place that may lead to variable confusion should be clearly stated
(∂f/∂x)|r, z When r and z are kept constant, the partial derivatives of f with respect to x grad f The elements are
the partial derivatives of f with respect to x, y, and z, respectively Derivative [(∂f/∂x), (∂f/∂y), (∂f/∂z)] or (∂f/∂x)i + (∂f/∂y)j + (∂f/∂ z)k; , called the gradient of f ∇ vector operator (∂/∂x)i + (∂/∂x)j + (∂/∂x)k, pronounced "del" ∇f f Gradient; its dot product with uw is the directional derivative of f in the w direction ∇•w the divergence of the vector field w, which is the dot product of the vector operator ∇ with the vector w, or (∂wx /∂x) + (∂ wy /∂y) + (∂wz /∂z) curl w vector operator ∇ the cross product of the same vector w ∇×w the curl of w, whose elements are [(∂fz /∂y) - (∂fy /∂ z), (∂fx /∂z) - (∂fz /∂x), (∂fy /∂x) - (∂fx /∂y)] ∇•∇ Laplace differential operator: (∂2/ ∂x2) + (∂/∂y2) + (∂/∂z2) f "(x) the second derivative of f with respect to x, the derivative of f '(x)
















d2f/dx2
The second derivative of f with respect to x
f(2)(x)
is also the second derivative of f with respect to x
f(k)(x)
The k-th derivative of f with respect to x, f(k-1) (x ) of the derivative
T The unit vector in the tangent direction of the curve, if the curve can be described as r(t), then T = (dr/dt)/|dr/dt|
ds
The derivative of the distance along the curve direction
κ
The curvature of the curve, unit tangent The value of the derivative of the vector relative to the distance of the curve: |dT/ds|
N dT/ds The
unit vector of the projection direction, the unit normal vector of the plane T and N perpendicular to the TB , that is, the torsion rate of the plane τ curve of curvature: |dB/ds | g gravitational constant F standard notation of force in mechanics k spring constant of spring pi momentum of i-th object H Hamiltonian function of physical system i.e. Poisson of energy {Q, H} Q, H expressed by position and momentum Parentheses The integral of f(x) expressed as a function of x by a common mathematical notation

















Definite integral of mathematically commonly used symbolic function f from a to b.
When f is positive and a < b, it means that the area L(d) of the graph enclosed by the x-axis and the straight line y = a, y = b and the function curve between these straight lines is
equal to the sub-interval size d, and each sub-interval is equal to d. The Riemann and
R(d) equal subintervals with the value of the left endpoint of the interval are d, and the Riemann and M(d) equal subintervals
with the value of the right endpoint of each subinterval are d, and the size of each subinterval is d. The Riemann sum m(d) with the maximum value of f