CALCULUS

Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.

It has two major branches, differential calculus and integral calculus; the former concerns instantaneous rates of change, and the slopes of curves, while the latter concerns accumulation of quantities, and areas under or between curves. These two branches are related to each other by the fundamental theorem of calculus, and they make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit.

Differential Calculus
Integral Calculus

Calculus is usually developed by working with very small quantities. Historically, the first method of doing so was by infinitesimals. These are objects which can be treated like real numbers but which are, in some sense, “infinitely small”. For example, an infinitesimal number could be greater than 0, but less than any number in the sequence 1, 1/2, 1/3, … and thus less than any positive real number. From this point of view, calculus is a collection of techniques for manipulating infinitesimals. The symbols dx and dy are taken to be infinitesimals and the derivative dy/dx was their ratio.

Differential calculus is the study of the definition, properties, and applications of the derivative of a function. The process of finding the derivative is called differentiation.

HINDI	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	(3×3)' = 9×2
	$ y `$ \| second derivative \| derivative of derivative \| (3×3)` = 18x
	y(n)	nth derivative	n times derivation	(3×3)(3) = 18
	$ dy/dx $	derivative	derivative - Leibniz's notation	d(3×3)/dx = 9×2
	$d	2y/dx	2 $	second derivative	derivative of derivative	d2(3×3)/dx2 = 18x
	$ d	ny/dx	n $	nth derivative	n times derivation
	$ \dot{y} $	time derivative	derivative by time - Newton's notation
		time second derivative	derivative of derivative
	$ D_{\text{x}}y $	derivative	derivative - Euler's notation
	$ D_{\text{\x	2}}y $	second derivative	derivative of derivative
	$ deltaf(x,y)/deltax $	partial derivative		∂(x2+y2)/∂x = 2x
	∫	integral	opposite to derivation	∫ f(x)dx
	∫∫	double integral	integration of function of 2 variables	∫∫ f(x,y)dxdy
	∫∫∫	triple integral	integration of function of 3 variables	∫∫∫ f(x,y,z)dxdydz
	∮	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

CALCULUS

PRINCIPLES

Limits and infinitesimals

Differential Calculus

Integral Calculus

Leibniz Notation

SYMBOLS