Introductory mathematics

Up to this point, we’ve gone through the details of the Project and explained its work in broad terms. But to gain a deep understanding of how everything works, we do have to use a reasonable amount of mathematics (in particular, calculus) and physics, which, as you’ll soon see, are very closely-related fields. We will start everything at basic algebra, and build up to calculus-based physics. These can be difficult topics; take it at your own pace. We hope it will be an enjoyable read.

Mathematical notation

Unless otherwise indicated, mathematical symbols will be represented by the following notational conventions:

Function: \(f(x)\)

Function composition: \(f(g(x))\)

Limit: \(\displaystyle \lim_{x \to x_0} f(x)\)

Vector quantity: \(\mathbf{E}\) or \(\vec E\)

Derivative: \(\dfrac{df}{dx}\) (preferred), \(f'(x)\) (alternative); time derivative only: \(\dot x = \dfrac{df}{dt}, \ddot x = \dfrac{d^2 f}{dt^2}\).

Nth-derivative: \(\dfrac{d^n f}{dx^n}\) (preferred), \(f^{(n)}(x)\) (alternative)

Derivative operator: \(\dfrac{d}{dx}\)

Partial derivative: \(\dfrac{\partial f}{\partial x}\)

Partial derivative operator: \(\dfrac{\partial}{\partial x}\)

Gradient: \(\nabla f\)

Divergence: \(\nabla \cdot \mathbf{F}\)

Curl: \(\nabla \times \mathbf{F}\)

Laplacian: \(\nabla^2 f\)

Integrals:

Integral type	Symbol	Alternative notation
Indefinite integral	\(\displaystyle \int_0^x f(k)~dk\)	Integral without bounds (but not recommended)
Definite integral	\(\displaystyle \int_a^b f(x) dx\)	Limits can be placed directly on integral sign
Line integral (scalar)	\(\displaystyle \int_C f(x, y, z) d\ell\)	Yes, use \(dr\) or \(ds\) as differential
Closed line integral (scalar)	\(\displaystyle \oint_C f(x, y, z) d\ell\)	Yes, use \(dr\) or \(ds\) as differential
Line integral (vector)	\(\displaystyle \int_C \mathbf{F} \cdot \mathbf{d\ell}\)	Yes, use \(\mathbf{dr}\) or \(\mathbf{ds}\) as differential
Closed line integral (vector)	\(\displaystyle \oint_C \mathbf{F} \cdot \mathbf{d\ell}\)	Yes, use \(\mathbf{dr}\) or \(\mathbf{ds}\) as differential
Surface integral (scalar)	\(\displaystyle \iint_\Sigma f(x, y, z)~dS\)	Yes, \(\displaystyle \int_\Sigma f(x, y, z)~dS\)
Closed surface integral (scalar)	\(\displaystyle \oiint_\Sigma f(x, y, z)~dS\)	Yes, \(\displaystyle \oint_\Sigma f(x, y, z)~dS\)
Surface integral (vector)	\(\displaystyle \iint_\Sigma \mathbf{F} \cdot \mathbf{dS}\)	Yes, \(\displaystyle \int_\Sigma \mathbf{F} \cdot \mathbf{dS}\)
Closed surface integral (vector)	\(\displaystyle \oiint_\Sigma \mathbf{F} \cdot \mathbf{dS}\)	Yes, \(\displaystyle \oint_\Sigma \mathbf{F} \cdot \mathbf{dS}\)
Double integral	\(\displaystyle \iint_R f(x, y)~dA\)	Not recommended
Area integral	\(\displaystyle \iint_R dA\)	Not recommended
Triple integral	\(\displaystyle \iiint_\Omega f(x, y, z)~dV\)	Not recommended
Volume integral	\(\displaystyle \iiint_\Omega dV\)	Not recommended
Spacetime integral	\(\displaystyle \int_{M} \sqrt{-g}~d^4 x\)	None

Note

For all multivariable integrals, the precise subscript of the integral, whether \(C\) or \(\Omega\) or \(M\) or \(\Sigma\), isn’t really that important. A subscript should be placed but should also be elaborated on in the text describing the integral. It is the integrating differential that is most important.