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 ) f(x) f ( x )
Function composition: f ( g ( x ) ) f(g(x)) f ( g ( x ))
Limit: lim x → x 0 f ( x ) \displaystyle \lim_{x \to x_0} f(x) x → x 0 lim f ( x )
Vector quantity: E \mathbf{E} E E ⃗ \vec E E
Derivative: d f d x \dfrac{df}{dx} d x df f ′ ( x ) f'(x) f ′ ( x ) x ˙ = d f d t , x ¨ = d 2 f d t 2 \dot x = \dfrac{df}{dt}, \ddot x = \dfrac{d^2 f}{dt^2} x ˙ = d t df , x ¨ = d t 2 d 2 f
Nth-derivative: d n f d x n \dfrac{d^n f}{dx^n} d x n d n f f ( n ) ( x ) f^{(n)}(x) f ( n ) ( x )
Derivative operator: d d x \dfrac{d}{dx} d x d
Partial derivative: ∂ f ∂ x \dfrac{\partial f}{\partial x} ∂ x ∂ f
Partial derivative operator: ∂ ∂ x \dfrac{\partial}{\partial x} ∂ x ∂
Gradient: ∇ f \nabla f ∇ f
Divergence: ∇ ⋅ F \nabla \cdot \mathbf{F} ∇ ⋅ F
Curl: ∇ × F \nabla \times \mathbf{F} ∇ × F
Laplacian: ∇ 2 f \nabla^2 f ∇ 2 f
Integrals:
Integral type Symbol Alternative notation Indefinite integral ∫ 0 x f ( k ) d k \displaystyle \int_0^x f(k)~dk ∫ 0 x f ( k ) d k Integral without bounds (but not recommended) Definite integral ∫ a b f ( x ) d x \displaystyle \int_a^b f(x) dx ∫ a b f ( x ) d x Limits can be placed directly on integral sign Line integral (scalar) ∫ C f ( x , y , z ) d ℓ \displaystyle \int_C f(x, y, z) d\ell ∫ C f ( x , y , z ) d ℓ Yes, use d r dr d r d s ds d s Closed line integral (scalar) ∮ C f ( x , y , z ) d ℓ \displaystyle \oint_C f(x, y, z) d\ell ∮ C f ( x , y , z ) d ℓ Yes, use d r dr d r d s ds d s Line integral (vector) ∫ C F ⋅ d ℓ \displaystyle \int_C \mathbf{F} \cdot \mathbf{d\ell} ∫ C F ⋅ dℓ Yes, use d r \mathbf{dr} dr d s \mathbf{ds} ds Closed line integral (vector) ∮ C F ⋅ d ℓ \displaystyle \oint_C \mathbf{F} \cdot \mathbf{d\ell} ∮ C F ⋅ dℓ Yes, use d r \mathbf{dr} dr d s \mathbf{ds} ds Surface integral (scalar) ∬ Σ f ( x , y , z ) d S \displaystyle \iint_\Sigma f(x, y, z)~dS ∬ Σ f ( x , y , z ) d S Yes, ∫ Σ f ( x , y , z ) d S \displaystyle \int_\Sigma f(x, y, z)~dS ∫ Σ f ( x , y , z ) d S Closed surface integral (scalar) ∯ Σ f ( x , y , z ) d S \displaystyle \oiint_\Sigma f(x, y, z)~dS ∬ Σ f ( x , y , z ) d S Yes, ∮ Σ f ( x , y , z ) d S \displaystyle \oint_\Sigma f(x, y, z)~dS ∮ Σ f ( x , y , z ) d S Surface integral (vector) ∬ Σ F ⋅ d S \displaystyle \iint_\Sigma \mathbf{F} \cdot \mathbf{dS} ∬ Σ F ⋅ dS Yes, ∫ Σ F ⋅ d S \displaystyle \int_\Sigma \mathbf{F} \cdot \mathbf{dS} ∫ Σ F ⋅ dS Closed surface integral (vector) ∯ Σ F ⋅ d S \displaystyle \oiint_\Sigma \mathbf{F} \cdot \mathbf{dS} ∬ Σ F ⋅ dS Yes, ∮ Σ F ⋅ d S \displaystyle \oint_\Sigma \mathbf{F} \cdot \mathbf{dS} ∮ Σ F ⋅ dS Double integral ∬ R f ( x , y ) d A \displaystyle \iint_R f(x, y)~dA ∬ R f ( x , y ) d A Not recommended Area integral ∬ R d A \displaystyle \iint_R dA ∬ R d A Not recommended Triple integral ∭ Ω f ( x , y , z ) d V \displaystyle \iiint_\Omega f(x, y, z)~dV ∭ Ω f ( x , y , z ) d V Not recommended Volume integral ∭ Ω d V \displaystyle \iiint_\Omega dV ∭ Ω d V Not recommended Spacetime integral ∫ M − g d 4 x \displaystyle \int_{M} \sqrt{-g}~d^4 x ∫ M − g d 4 x None
For all multivariable integrals, the precise subscript of the integral, whether C C C Ω \Omega Ω M M M Σ \Sigma Σ