Calculus

Calculus studies local change through derivatives and accumulated change through integrals.

The floor $\lfloor x\rfloor$ is the greatest integer not exceeding the real number $x$.

The fractional part of a real number is $\{x\}=x-\lfloor x\rfloor \in [0,1)$.

An integral accumulates a function over a domain and may be defined as a limit of finite sums.

An integral representation expresses a function through an integral of specified data and an integral kernel. Green-function representations solve differential equations this way, while the Bromwich inversion formula reconstructs a function from its Laplace transform.

Lebesgue integration constructs an integral from the measure of level sets or from approximation by simple functions. It is linear on integrable functions and is stable under the principal convergence theorems of measure theory.

If $f\in L^1$, then for every $\epsilon >0$ there is $\delta >0$ such that ${\int }_E|f|<\epsilon$ whenever the measure of $E$ is below $\delta$. In particular, integrals over a sequence of shrinking measurable sets tend to zero.

A time integral accumulates an integrand over a time coordinate. In perturbation theory it commonly sums the effect of an interaction over all possible vertex times.

A double integral integrates a function over a two-dimensional region. A change of variables formula can replace the region and area element using a Jacobian determinant.

If measurable functions satisfy $f\le g$ almost everywhere, thenwhenever the two sides are defined.

An antiderivative of $g$ is a differentiable function $p$ satisfying $p^{\prime }=g$.

Any two antiderivatives of the same function on a connected interval differ by an additive constant.

The Gaussian integral isSquaring the integral and changing to polar coordinates proves the formula.

The Dawson function obeys $F_D^{\prime }+2z{F}_D=1$ by differentiation. Its large-real-argument asymptotic expansion begins $F_D(z)=1/(2z)+1/(4z^3)+O(z^{-5})$. This inverse-power behavior contrasts with the separate exponentially growing factors in its definition.

Define $E_0(z)={\int }_0^z{e}^{-t^2}{\int }_0^t{e}^{u^2}dudt$ and $E_1(z)={\int }_0^z{e}^{-t^2}{\int }_0^t{e}^{u^2}\mathrm{erf}ududt$. They satisfy $(D^2+2zD)E_0=1$ and $(D^2+2zD)E_1=\mathrm{erf}z$. The first is an even function, the second an odd function, and their real tails are respectively $\frac{1}{2}\log |z|+C_1+o(1)$ and $\mathrm{sgn}z[\frac{1}{2}\log |z|+C_2+o(1)]$. Their derivatives have inverse-linear tails, explaining the logarithmic matching constants in a singular perturbation.

The error function isIt is odd and tends to $\pm 1$ as $x\to \pm \infty$.

The complementary error function avoids subtracting a quantity close to one when describing a small Gaussian integral tail. For real $x>0$, $\mathrm{erfc}x=(2/\sqrt{\pi }){\int }_x^{\infty }{e}^{-t^2}dt$; analytic continuation defines it for complex arguments.

The Faddeeva function is an entire function built from the complementary error function. Its integral representation for $\mathrm{Im}z>0$ is $w(z)=(i\pi {)}^{-1}{\int }_{-\infty }^{\infty }{e}^{-t^2}/(t-z)dt$, as normalized in NIST's integral representation. Its reflection identity is $w(-z)=2e^{-z^2}-w(z)$, directly from the odd function property of the error function.

For a pole off the real axis, substitution $s=2t$ into the Faddeeva function integral gives $G(a)=i\pi w(a/2)$ when $\mathrm{Im}a>0$, and $G(a)=-i\pi w(-a/2)$ when $\mathrm{Im}a<0$. The difference of boundary values is the residue theorem jump $2\pi i{e}^{-a^2/4}$. In particular, for a lower-half-plane pole, $G(a)+2\pi i{e}^{-a^2/4}=i\pi w(a/2)$, which packages a crossed residue and a nearby saddle into one smooth expression.

The multivariate Taylor polynomial of total degree $p$ at $x_0$ is

The exponential function has the everywhere-convergent power series

The complex exponential is the entire functionIt maps $\mathbb{C}$ onto ${\mathbb{C}}^{\ast }$ and has kernel $2\pi i\mathbb{Z}$.

The complex sine is the entire functionFor real $x,y$, $\sin (x+iy)=\sin x\cosh y+i\cos x\sinh y$.

The kernel $2\pi i\mathbb{Z}$ of the complex exponential function givesThe same period appears as the translation generated by the monodromy of the complex inverse sine.

The logarithm to base $b>0$, $b\ne 1$, is the inverse of the exponential function $x\mapsto b^x$.

The natural logarithm is the inverse of the real exponential function on the positive real numbers.

For $x>0$, concavity of the natural logarithm givesEquivalently, ${\log }_{b}x\le ({\log }_{b}e)(x-1)$ for $b>1$.

For positive-definite Hermitian operators,The same statement on supports follows by regularizing with a positive multiple of the identity and taking a limit.

A Gaussian function has the form $A\exp [-a(x-b{)}^2]$ with $a>0$. Its integral and moments reduce by translation and scaling to the Gaussian integral.

Hyperbolic functions are combinations of the exponential function analogous to trigonometric functions. The hyperbolic cosine and hyperbolic sine satisfy ${\cosh }^{2}x-{\sinh }^{2}x=1$ and parametrize a unit hyperbola.

The hyperbolic cosine is $\cosh x=(e^x+e^{-x})/2$.

For $x\ge 1$, the inverse hyperbolic cosine on its nonnegative real branch is $\mathrm{arcosh}x=\log (x+\sqrt{x^2-1})$.

The hyperbolic sine is $\sinh x=(e^x-e^{-x})/2$.

The hyperbolic cotangent is $\coth x=\cosh x/\sinh x$ where $\sinh x\ne 0$.

The hyperbolic secant is

If $g\in L^1(a,b)$, then $F(x)=F(a)+{\int }_a^{x}g(s)ds$ is absolutely continuous and satisfies $F^{\prime }=g$ almost everywhere. Conversely, every absolutely continuous function has this form.

If a continuous real function is differentiable between two distinct points at which it has the same value, its derivative vanishes at some point between them.

The Taylor theorem expresses a sufficiently differentiable function as a finite Taylor polynomial plus a remainder.

If $f$ has $k+1$ continuous derivatives, its Taylor remainder about $a$ is

A Taylor expansion about a point consists of successive derivative terms there, together with either a finite remainder or an infinite Taylor series when that series represents the function.

A parabolic approximation retains terms through second order in a local Taylor expansion. Near the closest point of a circle of radius $a$ and its tangent line, the separation is $x^2/(2a)+O(x^4/a^3)$.

For a polynomial $p$ of degree at most $m$, the Taylor formula is exact:

If $f$ is $n$ times differentiable between $a$ and $x$, then for some $\xi$ strictly between them,

The Taylor remainder is the difference between a function and a finite Taylor polynomial. Bounds on higher derivatives or their Hölder continuity control this error.

If $f^{(m-1)}$ is locally absolutely continuous, then its Taylor remainder can be written as an integral involving the weak derivative $f^{(m)}$. This form permits norm bounds when the highest derivative is only integrable.

The Taylor series of a sufficiently regular function about $a$ is

The limit records the value approached by a function as its argument approaches a point.

A one-sided limit restricts the argument to values above or below the limiting point. It is denoted ${\lim }_{x\downarrow a}f(x)$ from above and ${\lim }_{x\uparrow a}f(x)$ from below.

A function is continuous at a point when its limit there equals its value.

An extended-real functional $f$ is sequentially lower semicontinuous when $x_n\to x$ implies $f(x)\le {\mathrm{liminf}}_{n}f(x_n)$.

A sublevel set consists of points whose objective value is at most a chosen threshold. Coercivity makes finite sublevel sets bounded, while sequential lower semicontinuity makes them sequentially closed sets in the selected topology.

On a metric space, a functional is sequentially lower semicontinuous exactly when every sublevel set $\{x:f(x)\le \eta \}$ is closed.

A function is locally constant when every point has a neighbourhood on which the function is constant.

If $f:[a,b]\to \mathbb{R}$ is continuous and $y$ lies between $f(a)$ and $f(b)$, then some $c\in [a,b]$ satisfies $f(c)=y$.

If a continuous real-valued function on a real or complex vector space has opposite signs at two points, restricting it to the line segment between them and applying the intermediate value theorem gives a zero.

If a continuous $g:[0,na]\to \mathbb{R}$ satisfies $g(na)-g(0)=na$, then the incrementssatisfy ${\sum }_{k=0}^{n-1}h(ka)=0$. Continuity forces $h$ to vanish somewhere on $[0,(n-1)a]$.

Every continuous injective function from a real interval to the real line is strictly monotone. A continuous bijection $\mathbb{R}\to \mathbb{R}$ therefore has a continuous inverse.

If $f$ is continuous at $x$ and $g$ is continuous at $f(x)$, then $g\circ f$ is continuous at $x$.

The continuity set of a function is the set of points where it is continuous. For a real function $f$, define its oscillation at $x$ byThe sets $\{x:{\omega }_f(x)<\epsilon \}$ are open, and $f$ is continuous at $x$ exactly when ${\omega }_f(x)=0$. Hence its continuity set is the $G_{\delta }$ Borel set

A function $f$ on the real line is right-continuous at $t$ when $f(s)\to f(t)$ as $s\downarrow t$.

A càdlàg function is right-continuous and has a finite left limit at every positive time. The name abbreviates the French phrase /continue à droite, limites à gauche/.

A function is uniformly continuous when one input tolerance works at every point of its domain.

If two functions with the same limit bound a third nearby, the bounded function has that limit too.

The derivative is the limit of the difference quotient and gives the best linear approximation to local change.

The derivative of the first derivative. For a twice differentiable function, its sign governs local convexity and its magnitude controls the second-order Taylor remainder.

A difference quotient divides a finite change in a function by the corresponding change in its argument. Its limit, when it exists, is the derivative.

A total derivative accounts for every dependence of a quantity on the variable of differentiation. Its integral contributes only boundary terms.

For an indeterminate quotient whose numerator and denominator both tend to zero or both diverge, suitable differentiability and denominator conditions allow its limit to be computed from the limit of the quotient of their derivatives.

Every derivative has the intermediate-value property, even when the derivative is discontinuous.

If $f$ and $g$ are differentiable at $a$, then

The higher-order product rule is

At zero, $f(x)=x$ and $g(x)=1+|x|$ give a differentiable productwith derivative one, although $g$ is not differentiable.

If $f$ and $g$ are differentiable and $g\ne 0$, then

If $f$ is differentiable at $a$ and $g$ is differentiable at $f(a)$, then

At zero, $f(x)=\sqrt[3]{x}$ is not differentiable and $g(y)=y^3$ is differentiable, while $g\circ f(x)=x$ has derivative one.

For twice differentiable functions,

A function is increasing when $x\le y$ implies $f(x)\le f(y)$.

For an unbounded nondecreasing function $f:[0,\infty )\to \mathbb{R}$, its strict generalized inverse is $G(a)=\inf \{t\ge 0:f(t)>a\}$. It is nondecreasing and right-continuous in $a$. If $a_n\downarrow a$, then for any $t>G(a)$ monotonicity gives $f(t)>a$, so eventually $f(t)>a_n$ and $G(a_n)\le t$; also $G(a_n)\ge G(a)$. Local finiteness gives finite left limits. Flat parts of $f$ create jumps of $G$. For the continuous running maximum of Brownian motion, this gives the càdlàg Brownian first-passage subordinator.

A function is strictly increasing when $x<y$ implies $f(x)<f(y)$. It is therefore injective and each value has at most one preimage.

Every real-valued monotone function on a real interval is differentiable with a finite derivative at almost every point.

Multivariable calculus extends differentiation and integration to functions of several variables.

If $f:U\to {\mathbb{R}}^m$ is continuously differentiable and the segment from $x$ to $y$ lies in $U$, then

A partial derivative differentiates a multivariable function with respect to one variable while holding the others fixed.

A time derivative is the partial derivative of an evolving field with respect to time while holding its spatial coordinates fixed. For a quantity depending only on time, it is an ordinary derivative.

A spatial derivative is a partial derivative with respect to a spatial coordinate.

The directional derivative of $f$ at $x$ in the direction $v$ isFor a differentiable scalar function on Euclidean space, it equals $v\cdot \nabla f(x)$.

The gradient of a scalar-valued differentiable function is the vector of its first partial derivatives.

A gradient field is a vector field of the form $\nabla f$ for a scalar potential $f$. Its curl vanishes wherever the required second derivatives commute.

For a differentiable scalar function of several variables, the total differential is

The Hessian is the symmetric matrix of second partial derivatives; definiteness classifies nondegenerate critical points.

A symmetric matrix $A$ is negative semidefinite when $v^{T}Av\le 0$ for every vector $v$. Equivalently, all its eigenvalues are nonpositive.

At an interior local maximum of a twice differentiable function, its Hessian matrix is negative semidefinite; at a local minimum it is positive semidefinite. For a scalar function on Euclidean space, the Laplacian is therefore nonpositive at a local maximum and nonnegative at a local minimum.

A critical point of a twice differentiable function is nondegenerate when its Hessian matrix is invertible there. A positive-definite Hessian gives a strict local minimum, a negative-definite Hessian gives a strict local maximum, and an indefinite Hessian gives a saddle point.

Vector calculus studies differentiation and integration of scalar and vector fields.

For differentiable vector fields $A$ and $B$,

A vector field assigns a vector to every point of its domain.