Delving into the realm of Calculus and Analysis requires a solid understanding of the mathematical symbols that convey complex concepts. This comprehensive table serves as a valuable resource, offering a quick reference guide to symbols used in differentiation, integration, limits, series, and other fundamental aspects of mathematical analysis. Whether you're a student embarking on a calculus course or an enthusiast exploring advanced mathematical concepts, this guide will assist you in deciphering the symbolic language of calculus and analysis.
Symbol |
Symbol Name |
Meaning / definition |
Example |
\lim_{x\to x0}f(x) |
limit |
limit value of a function |
|
ε |
epsilon |
represents a very small number, near zero |
ε → 0 |
e |
e constant / Euler's number |
e = 2.718281828... |
e = lim (1+1/x)x , x→∞ |
y ' |
derivative |
derivative - Lagrange's notation |
(3x3)' = 9x2 |
y '' |
second derivative |
derivative of derivative |
(3x3)'' = 18x |
y(n) |
nth derivative |
n times derivation |
(3x3)(3) = 18 |
\frac{dy}{dx} |
derivative |
derivative - Leibniz's notation |
d(3x3)/dx = 9x2 |
\frac{d^2y}{dx^2} |
second derivative |
derivative of derivative |
d2(3x3)/dx2 = 18x |
\frac{d^ny}{dx^n} |
nth derivative |
n times derivation |
|
\dot{y} |
time derivative |
derivative by time - Newton's notation |
|
Dx y |
derivative |
derivative - Euler's notation |
|
Dx2y |
second derivative |
derivative of derivative |
|
\frac{\partial f(x,y)}{\partial x} |
partial derivative |
∂(x2+y2)/∂x = 2x |
|
∫ |
integral |
opposite to derivation |
|
∬ |
double integral |
integration of function of 2 variables |
|
∭ |
triple integral |
integration of function of 3 variables |
|
∮ |
closed contour / line integral |
||
∯ |
closed surface integral |
||
∰ |
closed volume integral |
||
[a,b] |
closed interval |
[a,b] = {x | a ≤ x ≤ b} |
|
(a,b) |
open interval |
(a,b) = {x | a < x < b} |
|
i |
imaginary unit |
i ≡ √-1 |
z = 3 + 2i |
z* |
complex conjugate |
z = a+bi → z*=a-bi |
z* = 3 + 2i |
z |
complex conjugate |
z = a+bi → z = a-bi |
z = 3 + 2i |
Re(z) |
real part of a complex number |
z = a+bi → Re(z)=a |
Re(3 - 2i) = 3 |
Im(z) |
imaginary part of a complex number |
z = a+bi → Im(z)=b |
Im(3 - 2i) = -2 |
| z | |
absolute value/magnitude of a complex number |
|z| = |a+bi| = √(a2+b2) |
|3 - 2i| = √13 |
arg(z) |
argument of a complex number |
The angle of the radius in the complex plane |
arg(3 + 2i) = 33.7° |
∇ |
nabla / del |
gradient / divergence operator |
∇f (x,y,z) |
vector |
|||
unit vector |
|||
x * y |
convolution |
y(t) = x(t) * h(t) |
|
Laplace transform |
F(s) = {f (t)} |
||
Fourier transform |
X(ω) = {f (t)} |
||
δ |
delta function |
||
∞ |
lemniscate |
infinity symbol |
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