Mathematics

MIT — Mathematics for Computer Science ↗ · Google this topic ↗

Foundations & discrete mathematics

Combinations and binomial coefficients

{n\choose k}=n!/[k!(n-k)!]

Definitions, derivation & worked example

Symbols. n distinct objects, k selected objects, order ignored.

Derivation. Count ordered selections n!/(n−k)! and divide by the k! internal permutations.

Assumptions. Integer 0≤k≤n, no repeated selection.

{5\choose2}=5(4)/(2(1))

Numerical result: 10 in the example’s stated units or dimensionless convention.

Used in physical modeling: Statistics · Reliability calculations

MIT — Mathematics for Computer Science ↗ · Google this topic ↗

Foundations & discrete mathematics

Modular arithmetic and orders

a^r\equiv1\pmod N

Definitions, derivation & worked example

Symbols. r least positive return exponent, gcd(a,N)=1.

Derivation. Repeated multiplication is a permutation of invertible residue classes; its orbit eventually returns to one.

Assumptions. Modular order is not usually N; validate the smallest positive exponent.

2^4\bmod15=16\bmod15=1\quad\Rightarrow r=4

Numerical result: 4 in the example’s stated units or dimensionless convention.

Used in physical modeling: Quantum computing · Particle unification

MIT — Mathematics for Computer Science ↗ · Google this topic ↗

Foundations & discrete mathematics

Graphs and network balance

B\mathbf f=\mathbf s

Definitions, derivation & worked example

Symbols. B oriented incidence matrix, f edge flows, s node net outflow.

Derivation. Sum signed incident edge flows at each node to express conservation; internal edges cancel globally.

Assumptions. Closed network has zero sum of node sources; constitutive edge laws are separate.

f_{in}=7,\ f_{out,1}=3:\quad f_{out,2}=7-3

Numerical result: 4 in the example’s stated units or dimensionless convention.

Used in physical modeling: Electrical engineering · Heat transfer

MIT — Single Variable Calculus ↗ · Google this topic ↗

Calculus & approximation

Derivatives and chain rule

(f\circ g)'(x)=f'(g(x))g'(x)

Definitions, derivation & worked example

Symbols. Prime denotes derivative with respect to its argument.

Derivation. Apply the first-order changes df≈f′dg and dg≈g′dx, then take the limit.

Assumptions. Both maps differentiable at the relevant points.

y=(3x+1)^2:\quad y'(1)=2(4)(3)

Numerical result: 24 in the example’s stated units or dimensionless convention.

Used in physical modeling: Continuum mechanics · Uncertainty quantification

MIT — Single Variable Calculus ↗ · Google this topic ↗

Calculus & approximation

Integration and the fundamental theorem

\int_a^b f(x)dx=F(b)-F(a),\quad F'=f

Definitions, derivation & worked example

Symbols. F antiderivative, a and b integration limits.

Derivation. Summed local changes telescope into an endpoint difference.

Assumptions. Continuous integrand suffices for the elementary theorem; singular integrals require extra care.

\int_0^2x^2dx=[x^3/3]_0^2=8/3

Numerical result: 2.6666667 in the example’s stated units or dimensionless convention.

Used in physical modeling: Heat transfer · Astrodynamics

MIT — Single Variable Calculus ↗ · Google this topic ↗

Calculus & approximation

Taylor expansion and local error

f(x+h)=f(x)+hf'(x)+h^2f''(\xi)/2

Definitions, derivation & worked example

Symbols. ξ lies between x and x+h for the second-order remainder.

Derivation. Integrate the derivative expansion to retain the linear term and bound the curvature remainder.

Assumptions. A suitable continuous second derivative is required.

e^{0.1}\simeq1+0.1=1.1

Numerical result: 1.1 in the example’s stated units or dimensionless convention.

Used in physical modeling: Uncertainty quantification · Control theory

MIT — Single Variable Calculus ↗ · Google this topic ↗

Calculus & approximation

Geometric series and convergence

\sum_{k=0}^\infty r^k=1/(1-r),\quad |r|<1

Definitions, derivation & worked example

Symbols. r common ratio, k nonnegative integer index.

Derivation. Multiply the finite sum by1−r and pass to the limit only when the remainder vanishes.

Assumptions. Diverges for real r≥1; convergence must be checked before interchanging sums and limits.

\sum_{k=0}^\infty(1/2)^k=1/(1-1/2)

Numerical result: 2 in the example’s stated units or dimensionless convention.

Used in physical modeling: Ray tracing & radiation · Quantum statistical physics

Linear algebra

Linear systems and elimination

A\mathbf x=\mathbf b

Definitions, derivation & worked example

Symbols. A coefficient matrix, x unknown vector, b right-hand side.

Derivation. Row operations preserve the solution set and expose pivots and free variables.

Assumptions. Existence and uniqueness depend on rank; numerical conditioning is a separate issue.

x+y=5,\ x-y=1\Rightarrow x=(5+1)/2

Numerical result: 3 in the example’s stated units or dimensionless convention.

Used in physical modeling: Ray tracing & radiation · Electrical engineering

Linear algebra

Eigenvalues and normal modes

A\mathbf v=\lambda\mathbf v,\quad\det(A-\lambda I)=0

Definitions, derivation & worked example

Symbols. v nonzero mode vector, λ eigenvalue.

Derivation. A nonzero null vector of A−λI requires its determinant to vanish.

Assumptions. Not every matrix has a full eigenbasis; symmetric/Hermitian matrices do.

A=\begin{pmatrix}2&1\\1&2\end{pmatrix}:\quad\lambda_{max}=2+1

Numerical result: 3 in the example’s stated units or dimensionless convention.

Used in physical modeling: Quantum mechanics · Solid mechanics

Linear algebra

Least squares and projections

A^TA\widehat x=A^Tb

Definitions, derivation & worked example

Symbols. Residual b−Ax is projected orthogonally to the column space.

Derivation. Differentiate the squared residual norm and set its gradient to zero.

Assumptions. Full column rank gives uniqueness; QR/SVD is often preferable to explicit normal equations.

\min_c[(1-c)^2+(3-c)^2]\Rightarrow c=(1+3)/2

Numerical result: 2 in the example’s stated units or dimensionless convention.

Used in physical modeling: Statistics · Sensors

Linear algebra

Singular values and conditioning

A=U\Sigma V^T,\quad\kappa_2=\sigma_{max}/\sigma_{min}

Definitions, derivation & worked example

Symbols. U,V orthogonal; diagonalΣ holds nonnegative singular values.

Derivation. Diagonalize AᵀA to find right singular directions and square-root its eigenvalues.

Assumptions. Stated condition number assumes full rank and the Euclidean norm.

A=\operatorname{diag}(4,1):\quad\kappa_2=4/1

Numerical result: 4 in the example’s stated units or dimensionless convention.

Used in physical modeling: Uncertainty quantification · Optimization

MIT — Multivariable Calculus ↗ · Google this topic ↗

Geometry & vector calculus

Dot products, projections and norms

a\cdot b=\lVert a\rVert\lVert b\rVert\cos\theta

Definitions, derivation & worked example

Symbols. a,b Euclidean vectors and θ their mutual angle.

Derivation. Resolve b into a component parallel to a and an orthogonal remainder.

Assumptions. Euclidean inner product; curved-space coordinates require a metric.

(1,2,3)\cdot(4,0,-1)=4+0-3

Numerical result: 1 in the example’s stated units or dimensionless convention.

Used in physical modeling: Astrodynamics · Robotics

MIT — Multivariable Calculus ↗ · Google this topic ↗

Geometry & vector calculus

Gradients and directional derivatives

D_uf=\nabla f\cdot u,\quad |u|=1

Definitions, derivation & worked example

Symbols. Gradient collects spatial partial derivatives; u is a unit direction.

Derivation. Taylor-expand f(x+hu) to first order and divide byh.

Assumptions. Smooth scalar field at the evaluation point.

f=x^2+y^2:\quad\partial_xf(3,4)=2(3)

Numerical result: 6 in the example’s stated units or dimensionless convention.

Used in physical modeling: Heat transfer · Optimization

MIT — Multivariable Calculus ↗ · Google this topic ↗

Geometry & vector calculus

Divergence theorem and conservation

\int_V\nabla\cdot F\,dV=\oint_{\partial V}F\cdot n\,dA

Definitions, derivation & worked example

Symbols. F vector field, outward n on the closed boundary.

Derivation. Sum fluxes of small cells; shared internal faces cancel, leaving the outer boundary.

Assumptions. Sufficiently smooth field and piecewise smooth volume boundary.

F=(x,y,z),\ V=2:\quad\int_V3dV=6

Numerical result: 6 in the example’s stated units or dimensionless convention.

Used in physical modeling: Fluid mechanics · Continuum mechanics

MIT — Multivariable Calculus ↗ · Google this topic ↗

Geometry & vector calculus

Metrics and distances

ds^2=g_{ij}dx^idx^j

Definitions, derivation & worked example

Symbols. gij metric components; repeated indices sum.

Derivation. A quadratic form specifies local length; coordinates may rescale or mix components.

Assumptions. Positive-definite for spatial Riemannian distance; spacetime metrics have indefinite signature.

ds^2=dx^2+dy^2,\ dx=3,dy=4:\quad ds=\sqrt{9+16}

Numerical result: 5 in the example’s stated units or dimensionless convention.

Used in physical modeling: General relativity · Robotics

MIT — Differential Equations ↗ · Google this topic ↗

Differential equations & transforms

First-order ODEs and exponential response

\dot y=-ky\Rightarrow y(t)=y_0e^{-kt}

Definitions, derivation & worked example

Symbols. k positive rate, y0 initial value.

Derivation. Separate dy/y=−kdt, integrate, and determine the constant from the initial condition.

Assumptions. Constantk and a linear first-order law.

y_0=2,\ k=1,\ t=1:\quad y=2e^{-1}

Numerical result: 0.73575888 in the example’s stated units or dimensionless convention.

Used in physical modeling: Chemical kinetics · Heat transfer

MIT — Differential Equations ↗ · Google this topic ↗

Differential equations & transforms

Second-order ODEs and oscillations

m\ddot x+kx=0,\quad\omega_n=\sqrt{k/m}

Definitions, derivation & worked example

Symbols. m mass and k stiffness.

Derivation. Substitute exp(st), solve ms²+k=0 and combine conjugate roots into sine and cosine.

Assumptions. Undamped linear oscillator; nonlinear and forced systems differ.

m=2,\ k=8:\quad\omega_n=\sqrt{8/2}

Numerical result: 2 in the example’s stated units or dimensionless convention.

Used in physical modeling: Solid mechanics · Car design

MIT — Differential Equations ↗ · Google this topic ↗

Differential equations & transforms

Fourier analysis and orthogonality

a_n=\frac1\pi\int_{-\pi}^{\pi}f(x)\cos(nx)dx

Definitions, derivation & worked example

Symbols. aₙ is the cosine coefficient in a 2π-periodic series.

Derivation. Multiply by a basis cosine and integrate; orthogonality removes the other modes.

Assumptions. Appropriate square-integrability; discontinuities affect pointwise convergence.

f=3\cos x:\quad a_1=3\pi/\pi

Numerical result: 3 in the example’s stated units or dimensionless convention.

Used in physical modeling: Communications / signal processing · Liquid-state physics

MIT — Differential Equations ↗ · Google this topic ↗

Differential equations & transforms

Diffusion PDE and separated modes

\partial_tu=D\partial_{xx}u,\quad u=Ae^{-Dk^2t}\sin(kx)

Definitions, derivation & worked example

Symbols. D diffusivity,k compatible boundary mode.

Derivation. Insert a sine spatial eigenfunction and solve its exponential time-amplitude ODE.

Assumptions. ConstantD and boundary conditions compatible with the chosen mode.

D=1,\ k=\pi,\ t=1/\pi^2:\quad u/A=e^{-1}\text{ at an antinode}

Numerical result: 0.36787944 in the example’s stated units or dimensionless convention.

Used in physical modeling: Heat transfer · Materials science

Probability & statistics

Expectation and variance

E[X]=\sum_xxp_x,\quad V[X]=E[X^2]-E[X]^2

Definitions, derivation & worked example

Symbols. px probability mass, X a discrete random variable.

Derivation. Compute weighted moments and expand the squared deviation from the mean.

Assumptions. Probabilities sum to one; finite second moment for variance.

X\in\{0,1\},\ p=0.3:\quad V=.3(.7)

Numerical result: 0.21 in the example’s stated units or dimensionless convention.

Used in physical modeling: Statistics · Statistical physics

Probability & statistics

Bayes’ theorem

P(A|B)=P(B|A)P(A)/P(B)

Definitions, derivation & worked example

Symbols. A event or hypothesis,B observed evidence.

Derivation. Write the joint probability in two conditional factorizations and equate them.

Assumptions. P(B)>0; priors and likelihoods must describe the problem.

P(A)=.2,\ P(B|A)=.8,\ P(B|\bar A)=.1:\quad P(A|B)=.16/(.16+.08)

Numerical result: 0.66666667 in the example’s stated units or dimensionless convention.

Used in physical modeling: Statistics · Uncertainty quantification

Probability & statistics

Gaussian distributions and standardization

f(x)=e^{-(x-\mu)^2/(2\sigma^2)}/(\sigma\sqrt{2\pi})

Definitions, derivation & worked example

Symbols. μ mean,σ positive standard deviation.

Derivation. Rescale the normalized standard Gaussian by x=μ+σz, including the Jacobian1/σ.

Assumptions. A model assumption, not a universal data distribution.

x=14,\mu=10,\sigma=2:\quad z=(14-10)/2

Numerical result: 2 in the example’s stated units or dimensionless convention.

Used in physical modeling: Statistics · Sensors

Probability & statistics

Monte Carlo estimation

\widehat\mu=\frac1N\sum_kf(X_k),\quad SE\simeq s_f/\sqrt N

Definitions, derivation & worked example

Symbols. Independent samplesXk from the target distribution.

Derivation. Linearity gives an unbiased sample mean; independence makes its varianceVar(f)/N.

Assumptions. Finite variance; rare events and correlated samples need additional analysis.

s_f=2,\ N=10000:\quad SE=2/100

Numerical result: 0.02 in the example’s stated units or dimensionless convention.

Used in physical modeling: Uncertainty quantification · Ray tracing & radiation

Boyd and Vandenberghe — Convex Optimization ↗ · Google this topic ↗

Optimization & numerical analysis

Convexity and optimality

f(\theta x+(1-\theta)y)\le\theta f(x)+(1-\theta)f(y)

Definitions, derivation & worked example

Symbols. θ between0and1.

Derivation. Convexity puts the function below every chord; differentiable stationary points are global minima on unconstrained convex domains.

Assumptions. Strict convexity ensures at most one minimizer; constraints alter optimality conditions.

f=(x-3)^2+2:\quad f'(x)=2(x-3)=0\Rightarrow x=3

Numerical result: 3 in the example’s stated units or dimensionless convention.

Used in physical modeling: Optimization · Control theory

Boyd and Vandenberghe — Convex Optimization ↗ · Google this topic ↗

Optimization & numerical analysis

Lagrange multipliers

\nabla f+\lambda\nabla g=0,\quad g=0

Definitions, derivation & worked example

Symbols. g equality constraint,λ multiplier.

Derivation. Feasible first-order changes are tangent to the constraint, so the objective gradient is normal at a regular optimum.

Assumptions. Constraint regularity required; stationary points must still be classified.

\min(x^2+y^2),\ x+y=2:\quad x=y=1,\ f=2

Numerical result: 2 in the example’s stated units or dimensionless convention.

Used in physical modeling: Optimization · Statistical physics

Boyd and Vandenberghe — Convex Optimization ↗ · Google this topic ↗

Optimization & numerical analysis

Newton iteration and local convergence

x_{n+1}=x_n-f(x_n)/f'(x_n)

Definitions, derivation & worked example

Symbols. xn current root estimate.

Derivation. Intersect the local tangent line with zero.

Assumptions. Nonzero derivative and suitable initial guess; global convergence is not guaranteed.

f=x^2-2,\ x_0=1:\quad x_1=1-(-1)/2

Numerical result: 1.5 in the example’s stated units or dimensionless convention.

Used in physical modeling: Optimization · Astrodynamics

Boyd and Vandenberghe — Convex Optimization ↗ · Google this topic ↗

Optimization & numerical analysis

Finite-volume conservation

U_i^{n+1}=U_i^n-\frac{\Delta t}{\Delta x}(F_{i+1/2}-F_{i-1/2})

Definitions, derivation & worked example

Symbols. U cell-average conserved quantity,F numerical interface flux.

Derivation. Integrate the PDE over one cell and update using shared boundary fluxes.

Assumptions. Stability, accuracy and entropy depend on flux and time discretization.

U=1,\Delta t/\Delta x=.1,F_R=3,F_L=2:\quad U'=1-.1(3-2)

Numerical result: 0.9 in the example’s stated units or dimensionless convention.

Used in physical modeling: Shock capturing: solids & fluids · Fluid mechanics

MIT — Quantum Physics III, mathematical methods ↗ · Google this topic ↗

Structures & mathematical physics

Complex exponentials and phasors

e^{i\theta}=\cos\theta+i\sin\theta

Definitions, derivation & worked example

Symbols. i²=−1; θ phase angle.

Derivation. Separate even and odd terms of the exponential power series into cosine and sine series.

Assumptions. Phasor circuit methods assume a linear sinusoidal steady-state system.

\operatorname{Re}(e^{i\pi})=\cos\pi

Numerical result: -1 in the example’s stated units or dimensionless convention.

Used in physical modeling: Electrical engineering · Quantum mechanics

MIT — Quantum Physics III, mathematical methods ↗ · Google this topic ↗

Structures & mathematical physics

Tensors and change of basis

\sigma'=R\sigma R^T

Definitions, derivation & worked example

Symbols. σ second-rank Cartesian tensor,R orthogonal rotation.

Derivation. Rotate both input normal and output traction to preserve the same physical linear map.

Assumptions. Passive/active rotation conventions must remain consistent; nonorthogonal coordinates need metric care.

\operatorname{tr}[\operatorname{diag}(1,2,3)]=1+2+3

Numerical result: 6 in the example’s stated units or dimensionless convention.

Used in physical modeling: Continuum mechanics · General relativity

MIT — Quantum Physics III, mathematical methods ↗ · Google this topic ↗

Structures & mathematical physics

Groups, symmetry and generators

R(\theta_1)R(\theta_2)=R(\theta_1+\theta_2)

Definitions, derivation & worked example

Symbols. R planar rotation; angles compose modulo2π.

Derivation. Matrix multiplication implements successive rotations; identity, inverse, closure and associativity define the group.

Assumptions. General3Drotations do not commute; a generator is an infinitesimal transformation.

30^\circ+45^\circ=75^\circ

Numerical result: 75 in the example’s stated units or dimensionless convention.

Used in physical modeling: Particle unification · Quantum field theory

MIT — Quantum Physics III, mathematical methods ↗ · Google this topic ↗

Structures & mathematical physics

Variational calculus and Euler–Lagrange

\frac d{dt}\frac{\partial L}{\partial\dot q}-\frac{\partial L}{\partial q}=0

Definitions, derivation & worked example

Symbols. L Lagrangian,q generalized coordinate.

Derivation. Vary the action with fixed endpoints, integrate the velocity variation by parts and require the interior coefficient to vanish.

Assumptions. Differentiable action and appropriate boundary terms; constrained systems need extra structure.

L=m\dot q^2/2-kq^2/2,\ m=2,k=8,q=.5:\quad\ddot q=-kq/m

Numerical result: -2 in the example’s stated units or dimensionless convention.

Used in physical modeling: Astrodynamics · Continuum mechanics · Quantum field theory