AcademyEquation Sheet

Academy

Equation Sheet

A curated Level 1 - Math I (Physics) formula reference with a print-ready PDF view.

Course Formula Reference

Curated equations grouped by section and linked to their source topics.

Curated from the Level 1 - Math I (Physics) topic content. Reusable definitions, identities, theorems, and computational formulas are included; derivation-only steps, narrow examples, and repeated rearrangements are intentionally omitted.

15 sections132 equationsDownload PDF

Real Numbers

6 equations

Absolute value

Real Numbers

\[|x|=\sqrt{x^2}\]

Triangle inequality

Real Numbers

\[|x+y|\le |x|+|y|\]

Difference of squares

Algebraic Manipulation

\[a^2-b^2=(a-b)(a+b)\]

Finite sum notation

Summation Notation

\[\sum_{k=m}^{n}a_k=a_m+a_{m+1}+\cdots+a_n\]

Binomial coefficient

Binomial Coefficients

\[\binom{n}{r}=\frac{n!}{r!(n-r)!}\]

Binomial theorem

Binomial Theorem

\[(a+b)^n=\sum_{r=0}^{n}\binom{n}{r}a^{n-r}b^r\]

Functions

4 equations

Function rule

Functions

\[f:A\to B\]

Domain condition

Functions

\[x\in\operatorname{dom}f\]

Inverse relations

Inverse Functions

\[f^{-1}(f(x))=x,\qquad f(f^{-1}(y))=y\]

One-to-one condition

Inverse Functions

\[f(x_1)=f(x_2)\Rightarrow x_1=x_2\]

Trigonometry

7 equations

Pythagoras theorem

Pythagoras Theorem

\[a^2+b^2=c^2\]

Pythagorean identity

Trig Functions

\[\sin^2\theta+\cos^2\theta=1\]

Tangent identity

Trig Functions

\[\tan\theta=\frac{\sin\theta}{\cos\theta}\]

Sine addition

Angle Addition Formulae

\[\sin(A+B)=\sin A\cos B+\cos A\sin B\]

Cosine addition

Angle Addition Formulae

\[\cos(A+B)=\cos A\cos B-\sin A\sin B\]

Double angle identities

Angle Addition Formulae

\[\sin2A=2\sin A\cos A,\qquad \cos2A=\cos^2A-\sin^2A\]

Arctangent angle

Inverse Trig Functions

\[\theta=\arctan\left(\frac{y}{x}\right)\]

Limits

6 equations

Limit definition

Formal Limits

\[\lim_{x\to a}f(x)=L\]

One-sided limits

Limit Variations

\[\lim_{x\to a^-}f(x),\qquad \lim_{x\to a^+}f(x)\]

Sum law

Limit Laws

\[\lim_{x\to a}(f(x)+g(x))=\lim_{x\to a}f(x)+\lim_{x\to a}g(x)\]

Product law

Limit Laws

\[\lim_{x\to a}f(x)g(x)=\left(\lim_{x\to a}f(x)\right)\left(\lim_{x\to a}g(x)\right)\]

Sine limit

Sine Limit

\[\lim_{x\to0}\frac{\sin x}{x}=1\]

Continuity at a point

Continuity

\[\lim_{x\to a}f(x)=f(a)\]

Differentiation

11 equations

Derivative notation

Derivatives

\[f'(x)=\frac{df}{dx}\]

First-principles derivative

First Principles

\[f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}\]

Power rule

Standard Derivatives

\[\frac{d}{dx}x^n=nx^{n-1}\]

Exponential derivative

Standard Derivatives

\[\frac{d}{dx}e^x=e^x\]

Trigonometric derivatives

Standard Derivatives

\[\frac{d}{dx}\sin x=\cos x,\qquad \frac{d}{dx}\cos x=-\sin x\]

Product rule

Product Rule

\[(uv)'=u'v+uv'\]

Quotient rule

Quotient Rule

\[\left(\frac{u}{v}\right)'=\frac{u'v-uv'}{v^2}\]

Chain rule

Chain Rule

\[\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}\]

Second derivative

Higher Order Derivatives

\[f''(x)=\frac{d^2f}{dx^2}\]

Inverse derivative

Inverse Derivatives

\[\frac{d}{dy}f^{-1}(y)=\frac{1}{f'(x)},\qquad y=f(x)\]

L'Hopital rule

L'Hopital Rule

\[\lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x\to a}\frac{f'(x)}{g'(x)}\]

Integration

9 equations

Antiderivative

Antiderivatives

\[F'(x)=f(x)\]

Power integral

Standard Integrals

\[\int x^n\,dx=\frac{x^{n+1}}{n+1}+C,\qquad n\ne -1\]

Reciprocal integral

Standard Integrals

\[\int\frac{1}{x}\,dx=\ln|x|+C\]

Definite integral

Definite Integrals

\[\int_a^b f(x)\,dx\]

Fundamental theorem

Fundamental Theorem

\[\int_a^b f(x)\,dx=F(b)-F(a)\]

Substitution

\[\int f(g(x))g'(x)\,dx=\int f(u)\,du\]

Integration by parts

Integration by Parts

\[\int u\,dv=uv-\int v\,du\]

Linear partial fractions

Partial Fractions

\[\frac{P(x)}{(x-a)(x-b)}=\frac{A}{x-a}+\frac{B}{x-b}\]

Trig square reduction

Trig Power Integrals

\[\sin^2x=\frac{1-\cos2x}{2},\qquad \cos^2x=\frac{1+\cos2x}{2}\]

Complex Arithmetic

6 equations

Complex number

Complex Numbers

\[z=a+bi\]

Real part

Real Part

\[\operatorname{Re}(a+bi)=a\]

Imaginary part

Imaginary Part

\[\operatorname{Im}(a+bi)=b\]

Conjugate

\[\overline{a+bi}=a-bi\]

Modulus

\[|a+bi|=\sqrt{a^2+b^2}\]

Division by conjugate

Complex Division

\[\frac{z}{w}=\frac{z\overline w}{|w|^2},\qquad w\ne0\]

Complex Form

6 equations

Polar form

Polar Form

\[z=r(\cos\theta+i\sin\theta)\]

Argument

\[\theta=\arg z\]

Euler formula

Euler Formula

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

Exponential form

Complex Exponential

\[z=re^{i\theta}\]

De Moivre theorem

De Moivre Theorem

\[(re^{i\theta})^n=r^ne^{in\theta}\]

Multiplication in polar form

Complex Multiplication Geometry

\[r_1e^{i\theta_1}r_2e^{i\theta_2}=r_1r_2e^{i(\theta_1+\theta_2)}\]

Complex Equations

7 equations

Linear equation

Linear Complex Equations

\[az+b=0\Rightarrow z=-\frac{b}{a},\qquad a\ne0\]

Quadratic formula

Quadratic Complex Equations

\[z=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\]

Roots of unity

Roots of Unity

\[z_k=e^{2\pi ik/n},\qquad k=0,1,\ldots,n-1\]

General complex roots

General Complex Roots

\[z_k=\rho^{1/n}e^{i(\chi/n+2\pi k/n)},\qquad k=0,1,\ldots,n-1\]

Fundamental theorem of algebra

Fundamental Theorem

\[p(z)=a_n\prod_{k=1}^{n}(z-z_k)\]

Complex equation function

Complex Functions

\[f:\mathbb C\to\mathbb C,\qquad f(z)=0\]

Complex exponential periodicity

Transcendental Equations in a Complex Variable

\[e^{z+2\pi i}=e^z\]

Groups

10 equations

Group operation

Group Definition

\[*:G\times G\to G\]

Associativity

Group Axioms

\[a*(b*c)=(a*b)*c\]

Identity

Group Axioms

\[e*a=a*e=a\]

Inverse

Group Axioms

\[a*a^{-1}=a^{-1}*a=e\]

Subgroup test

Subgroups

\[a,b\in H\Rightarrow ab^{-1}\in H\]

Homomorphism

Group Maps

\[\phi(a*b)=\phi(a)\diamond\phi(b)\]

Kernel

Group Maps

\[\ker\phi=\{g\in G:\phi(g)=e_K\}\]

Cyclic subgroup

Cyclic Groups

\[\langle g\rangle=\{g^n:n\in\mathbb Z\}\]

Dihedral group order

Polygon Symmetries

\[|D_n|=2n\]

General linear group

Matrix Groups

\[\mathrm{GL}_n(\mathbb R)=\{A\in\mathrm{Mat}_n(\mathbb R):\det A\ne0\}\]

Series

13 equations

Infinite series

Series Basics

\[\sum_{n=1}^{\infty}a_n=a_1+a_2+a_3+\cdots\]

Partial sum

Partial Sums

\[S_N=\sum_{n=1}^{N}a_n\]

Finite geometric series

Geometric Series

\[S_N=\frac{a(1-r^N)}{1-r},\qquad r\ne1\]

Infinite geometric series

Geometric Series

\[\sum_{n=0}^{\infty}ar^n=\frac{a}{1-r},\qquad |r|\lt1\]

p-series test

Positive Series

\[\sum_{n=1}^{\infty}\frac{1}{n^p}\text{ converges for }p\gt1\]

Ratio test

Convergence Tests

\[L=\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|\]

Root test

Convergence Tests

\[L=\lim_{n\to\infty}\sqrt[n]{|a_n|}\]

Absolute convergence

Absolute Convergence

\[\sum |a_n|\text{ converges }\Rightarrow\sum a_n\text{ converges}\]

Series convergence by partial sums

Partial Sums

\[\sum_{n=1}^{\infty}a_n=S\quad\Longleftrightarrow\quad \lim_{N\to\infty}S_N=S\]

Linearity of finite sums

Series Sums

\[\sum_{k=1}^{n}(ca_k+b_k)=c\sum_{k=1}^{n}a_k+\sum_{k=1}^{n}b_k\]

Telescoping sum

Series Sums

\[\sum_{k=1}^{n}(b_{k+1}-b_k)=b_{n+1}-b_1\]

Alternating sign test

Conditional Convergence

\[\sum a_n\text{ alternating},\quad |a_n|\to0,\quad |a_{n+1}|\le |a_n|\Rightarrow\sum a_n\text{ converges}\]

Conditional convergence

Conditional Convergence

\[\sum a_n\text{ converges but }\sum |a_n|\text{ diverges}\]

Power Series

12 equations

Power series

Power Series

\[\sum_{n=0}^{\infty}a_n(x-c)^n\]

Taylor coefficient

Coefficients

\[a_n=\frac{f^{(n)}(c)}{n!}\]

Radius from ratio test

Radius of Convergence

\[R=\frac{1}{\lim_{n\to\infty}|a_{n+1}/a_n|}\]

Interior convergence

Interval of Convergence

\[|x-c|\lt R\]

Taylor polynomial

Taylor Polynomials

\[T_N(x)=\sum_{n=0}^{N}\frac{f^{(n)}(c)}{n!}(x-c)^n\]

Taylor series

Taylor Series

\[f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(c)}{n!}(x-c)^n\]

Lagrange remainder bound

Remainders

\[|R_N(x)|\le\frac{M|x-c|^{N+1}}{(N+1)!}\]

Small-angle leading terms

Taylor Limits

\[\sin x=x+O(x^3),\qquad \cos x=1-\frac{x^2}{2}+O(x^4)\]

Termwise derivative

Power Series

\[\frac{d}{dx}\sum_{n=0}^{\infty}a_n(x-c)^n=\sum_{n=1}^{\infty}na_n(x-c)^{n-1}\]

Termwise integral

Power Series

\[\int\sum_{n=0}^{\infty}a_n(x-c)^n\,dx=C+\sum_{n=0}^{\infty}\frac{a_n}{n+1}(x-c)^{n+1}\]

Exponential series

Taylor Series

\[e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!}\]

Sine and cosine series

Taylor Series

\[\sin x=\sum_{n=0}^{\infty}\frac{(-1)^nx^{2n+1}}{(2n+1)!},\qquad \cos x=\sum_{n=0}^{\infty}\frac{(-1)^nx^{2n}}{(2n)!}\]

Matrices

12 equations

Matrix system

Linear Systems

\[A\mathbf x=\mathbf b\]

Matrix entries

Matrix Notation

\[A=(a_{ij})\]

Matrix product

Matrix Operations

\[(AB)_{ij}=\sum_k a_{ik}b_{kj}\]

Two by two determinant

Determinants

\[\det\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc\]

Product determinant

Determinant Properties

\[\det(AB)=\det(A)\det(B)\]

Inverse condition

Inverse Matrices

\[A^{-1}A=AA^{-1}=I\]

Two by two inverse

Inverse Matrices

\[\begin{pmatrix}a&b\\c&d\end{pmatrix}^{-1}=\frac{1}{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}\]

LU factorisation

LU Decomposition

\[A=LU\]

Cofactor expansion

Determinants

\[\det(A)=\sum_{j=1}^{n}(-1)^{i+j}a_{ij}M_{ij}\]

Determinant invertibility test

Inverse Matrices

\[A\text{ is invertible}\quad\Longleftrightarrow\quad \det(A)\ne0\]

Solving with inverse

Inverse Matrices

\[A\mathbf x=\mathbf b\quad\Rightarrow\quad \mathbf x=A^{-1}\mathbf b\]

Transpose of product

Matrix Operations

\[(AB)^T=B^TA^T\]

Vector Spaces

11 equations

Linear combination

Vector Spaces

\[a_1\mathbf v_1+\cdots+a_n\mathbf v_n\]

Subspace closure

Subspaces

\[\mathbf u,\mathbf v\in W\Rightarrow a\mathbf u+b\mathbf v\in W\]

Linear independence

Linear Independence

\[a_1\mathbf v_1+\cdots+a_n\mathbf v_n=\mathbf0\Rightarrow a_1=\cdots=a_n=0\]

Span

Spanning Sets

\[\operatorname{span}\{\mathbf v_1,\ldots,\mathbf v_n\}=\{a_1\mathbf v_1+\cdots+a_n\mathbf v_n\}\]

Basis representation

Bases

\[\mathbf v=a_1\mathbf b_1+\cdots+a_n\mathbf b_n\]

Dimension

\[\dim V=\text{number of vectors in a basis}\]

Rank

\[\operatorname{rank}A=\dim\operatorname{col}(A)\]

Rank-nullity

Nullity

\[\operatorname{rank}A+\operatorname{nullity}A=n\]

Coordinate vector

Coordinates

\[\mathbf v=c_1\mathbf b_1+\cdots+c_n\mathbf b_n\]

Image

Rank

\[\operatorname{Im}(A)=\{\mathbf w:\mathbf w=A\mathbf v\text{ for some }\mathbf v\}\]

Kernel

Nullity

\[\ker(A)=\{\mathbf v:A\mathbf v=\mathbf0\}\]

Linear Maps

12 equations

Linearity

Linear Maps

\[T(a\mathbf u+b\mathbf v)=aT(\mathbf u)+bT(\mathbf v)\]

Matrix representation

Matrix Representations

\[[T(\mathbf v)]_{\beta}=A[\mathbf v]_{\alpha}\]

Orthogonal matrix

Special Matrices

\[Q^TQ=I\]

Eigenvalue equation

Eigenvalues

\[A\mathbf v=\lambda\mathbf v\]

Characteristic equation

Eigenvalues

\[\det(A-\lambda I)=0\]

Eigenspace

Eigenspaces

\[E_{\lambda}=\ker(A-\lambda I)\]

Diagonalisation

\[A=PDP^{-1}\]

Powers by diagonalisation

Diagonalisation Applications

\[A^n=PD^nP^{-1}\]

Hermitian condition

Special Matrices

\[A^{\dagger}=A\]

Unitary condition

Special Matrices

\[A^{\dagger}A=AA^{\dagger}=I\]

Normal matrix

Special Matrices

\[A^{\dagger}A=AA^{\dagger}\]

Matrix exponential by diagonalisation

Diagonalisation Applications

\[A=PDP^{-1}\quad\Rightarrow\quad e^A=Pe^DP^{-1}\]

Flashcards

Real Numbers