Calculus Symbols

Last Updated : 11 Mar, 2024

Calculus symbols are symbols used in calculus in mathematics. There are different calculus symbols including limits symbols, derivatives, integrals, definite integrals, vector calculus etc. In this article we will explore all the calculus symbols that are useful in representing different types of calculus. Let’s start our learning on the topic “Calculus Symbols.”

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Table of Content

What are Symbols in Math?
Basic Calculus Symbols
Limits Symbols
Derivatives Symbols
Integrals Symbols
Definite Integrals Symbols
Some Advanced Calculus Symbols

What are Symbols in Math?

The symbols are special characters or figures that are used to represent different mathematical relations. There are different symbols in Math which are used in different conditions according to the requirement. The different types of symbols in Math include arithmetic symbols, logical symbols, calculus symbols and many more.

Some common symbols in mathematics are +, −, ×, ÷, =, <, >, ≤, ≥, √, ∑, ∏, π, Σ, Π, ∈, ∞, ±, ∠, °, %, ≈, ≠, and ∀.

Read More about Math Symbols in detail.

Basic Calculus Symbols

The basic calculus symbols can be classified as:

Limits Symbols
Derivatives
Integrals
Definite Integrals

Let’s discuss these symbols in detail as follows:

Limits Symbols

Table for some of the most common limit symbols is given below:

Limit Symbol	Symbol Name	Example
x → a	x tends to a	x → 3
lim _x→a f(x)	Limit of f(x) function on x tends to a	lim_x→0 (x + 1)
lim _x→a+ f(x) or lim _x↓a f(x)	Right limit (limit x tends to a from right of a)	lim_x→1+ (x – 1) = 0
lim _x→a– f(x) or lim _x↑a f(x)	Left limit (limit x tends to a from left of a)	lim_x→1– (x – 1) = -2

Derivatives Symbols

Table for some of the most common derivative symbols is given below:

Derivative Symbol	Symbol Name	Description
f'(x)	First Derivative	f'(p) = lim h→0 [{f (c +h) – f(c)} / h]
f”(x)	Second Derivative
f⁽ⁿ⁾(x)	N^th Derivative
dy/dx	First Derivative	dy/dx = f'(x)
d²y/dx²	Second Derivative	d²y/dx² = f”(x)
dⁿy/dxⁿ	N^th Derivative	dⁿy/dxⁿ = f⁽ⁿ⁾(x)
y’	First Derivative	y’ = dy/dx
y”	Second Derivative	y” = d²y/dx²
y⁽ⁿ⁾	N^thDerivative	y⁽ⁿ⁾ = dⁿy/dxⁿ
D_xy	First Derivative	D_x y = dy/dx
D_x²y	Second Derivative	D_x²y = D_x(D_x y)
D_xⁿy	N^th Derivative	–
Δx	Increment of x	Δp ≈ f'(x) Δx
dx	Differential of x	df = [df/dx] dx
𝝏f(x, y)/𝝏x	Partial Derivative	–

Integrals Symbols

Some of the integral symbols are given below:

Integral Symbol	Symbol Name	Description
∫	Integral	–
∫f(x) dx	Integral of Function	Indefinite integral of a function f(x) with respect to the variable x.
∬	Double Integral	Double integral of a function f(x, y) over a region in the xy-plane.
∭	Triple Integral	Triple integral of a function f(x, y, z) over a region in three-dimensional space.
∮	Closed Line Integral	Line integral of a function f(x) along a curve, with respect to arc length s.
∯	Closed Surface Integral	Surface integral of a function f(x, y) over a surface S.
∰	Closed Volume Integral	Triple integral of a function f(x, y, z) over a solid region in three-dimensional space.

Definite Integrals Symbols

Some of the definite integral symbols are given below:

Definite Integral Symbol	Symbol Name	Description
∫_a^b f(x) dx	Definite Integral of f(x) with lower limit a and upper limit b	After integration of f(x) put the values a and b in resultant function.
[p(x)]^b_a	Put limits a and b in p(x)	p(b) – p(a)
p(x) \|^b_a	Put limits a and b in p(x)	p(b) – p(a)

Some Advanced Calculus Symbols

Some of the advanced calculus symbols includes vector calculus, various notations and other symbols. Below table represents some symbols of vector calculus, various notations and other symbols.

Vector Calculus Symbol	Symbol Name	Description
$\overrightarrow{\rm x}$	Vector	Vector x
$\hat x$	Unit Vector	Unit vector x i.e., $\frac{\vec x}{\|\vec x\|}$
⛛f or grad f	Gradient vector of f	⛛f = (𝝏f/𝝏x₁, … ,𝝏f/𝝏x_n)
⛛ · F or div F	Divergence of vector F	⛛· F = 𝝏F_x/𝝏x + 𝝏F_y/𝝏y + 𝝏F_z/𝝏z
⛛× F or Curl F	Curl of vector F	⛛× F = (𝝏/𝝏x, 𝝏/𝝏y, 𝝏/𝝏z) × (F_x , F_y, F_z)

Commonly Used Constants in Calculus

Some of the commonly used constants are given below:

Constant Symbol	Symbol Name
C	Constant of Integration
𝓛{f}	Laplace Transform of f
𝓕{f}	Fourier Transform of f

Some other constants are:

Constant	Symbol	Value	Description
Pi	π	approximately 3.14159	The ratio of the circumference of a circle to its diameter.
Euler’s Number	e	approximately 2.71828	The base of the natural logarithm, widely used in calculus.
Golden Ratio	φ	approximately 1.61803	A special irrational number related to geometry and aesthetics.
Imaginary Unit	i	√(-1)	A unit imaginary number used in complex analysis and calculus.
Euler-Mascheroni Constant	γ	approximately 0.57721	Appears in various mathematical contexts, including calculus.

Conclusion

In conclusion, calculus symbols serve as a precise and concise language for expressing mathematical ideas. From derivatives (∂) to integrals (∫), these symbols streamline complex concepts, aiding mathematicians in their analyses and problem-solving.