# Calculus

### Functions

FunctionOperation
Integrate

["Integrate", expression]

["Integrate", expression, symbol]

Indefinite integral

["Integrate", ["Sin", x], "x"]

$$\int \sin x ,\mathrm{d}x$$

["Integrate", expression, predicate]

Definite integral.

["Integrate", ["Sin", "x"], ["Element", x, ["Interval", 0, +Infinity]]]

$$\int_0^\infty \sin x ,\mathrm{d}x$$

["Integrate", expression, predicate1, predicate2…]

Multiple integral.

["Integrate", ['"Multiply", x ["Sin", "y"]], ["Element", x, ["Interval", 0, 2]], ["Element", y, ["Interval", 0, 1]]]

$$\int_0^1 \int_0^2 x\sin y ,\mathrm{d}x,\mathrm{d}y$$

D

["D", f, x]

Evaluate to the partial derivative ( \frac{\partialD f}{\partialD x} ) or ( f^{\prime}(x) )

["D", f, ["Tuple", x, n]]

Evaluate to the multiple derivative ( \frac{\partialD^n f}{\partialD x^n} ) (Leibniz notation) or ( f^{(n)}(x) ) (Lagrange notation)

### Lagrange Notation

LaTeX MathJSON
f'(x) ["D", f, x]
f''(x)
f\prime(x)
f\prime\prime(x)
f\doubleprime(x)
f^{\prime}(x)
f^{\prime\prime}(x)
f^{\doubleprime}(x)

@todo: f^{(4)}

## Leibniz Notation

LaTeX MathJSON
\frac{\partial f}{\partial x}
\frac{\partial^2 f}{\partial x\partial y}

## Euler Modified Notation

This notation is used by Mathematica. The Euler notation uses D instead of \partial

LaTeX MathJSON
\partial_{x} f
\partial_{x,y} f

## Newton Notation (@todo)

\dot{v} -> first derivative relative to time t \ddot{v} -> second derivative relative to time t

## Indefinite Integral

\int f dx -> ["Integrate", f, x,] \int\int f dxdy -> ["Integrate", f, x, y]

Note: ["Integrate", ["Integrate", f , x], y] is equivalent to ["Integrate", f , x, y]

## Definite Integral

\int_{a}^{b} f dx -> ["Integrate", f, [x, a, b]] \int_{c}^{d} \int_{a}^{b} f dxdy -> ["Integrate", f, [x, a, b], [y, c, d]]

\int_{a}^{b}\frac{dx}{f} -> ["Integrate", ["Power", f, -1], [x, a, b]]

\int_{a}^{b}dx f -> ["Integrate", f, [x, a, b]]

If [a, b] are numeric, numeric methods are used to approximate the integral.

## Domain Integral

\int_{x\in D} -> ["Integrate", f, ["In", x, D]]

### Contour Integral

\oint f dx -> ["ContourIntegral", f, x,]

\varointclockwise f dx -> ["ClockwiseContourIntegral", f, x]

\ointctrclockwise f dx -> ["CounterclockwiseContourIntegral", f, x,]

\oiint f ds -> ["DoubleCountourIntegral", f, s] : integral over closed surfaces

\oiiint f dv -> ["TripleCountourIntegral", f, v] : integral over closed volumes

\intclockwise

\intctrclockwise

\iint

\iiint