AcademyEquation Sheet

Academy

Equation Sheet

A shared Level 1 - Math I and Math II (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) and Level 1 - Math II (Physics) topic content. Reusable definitions, identities, theorems, and computational formulas are included; derivation-only steps, narrow examples, duplicate formulae, and repeated rearrangements are intentionally omitted.

25 sections216 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

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

Probability

11 equations

Complement rule

Probability Axioms

\[P(A^c)=1-P(A)\]

Addition rule

Events

\[P(A\cup B)=P(A)+P(B)-P(A\cap B)\]

Conditional probability

Conditional Probability

\[P(A\mid B)=\frac{P(A\cap B)}{P(B)}\]

Bayes theorem

Bayes Theorem

\[P(A_i\mid B)=\frac{P(B\mid A_i)P(A_i)}{\sum_j P(B\mid A_j)P(A_j)}\]

Discrete expectation

Expectation

\[\operatorname E[X]=\sum_x xP(X=x)\]

Variance identity

Variance

\[\operatorname{Var}(X)=\operatorname E[X^2]-(\operatorname E[X])^2\]

Binomial probability

Binomial Distribution

\[P(X=k)=\binom nk p^k(1-p)^{n-k}\]

Poisson probability

Poisson Distribution

\[P(X=k)=e^{-\lambda}\frac{\lambda^k}{k!}\]

Normal density

Normal Distribution

\[f(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-(x-\mu)^2/(2\sigma^2)}\]

Standardisation

Standard Normal

\[Z=\frac{X-\mu}{\sigma}\]

Sample mean spread

Sample Mean

\[\operatorname E[\bar X]=\mu,\qquad \operatorname{Var}(\bar X)=\frac{\sigma^2}{n}\]

Vectors

9 equations

Component vector

Vectors

\[\mathbf a=(a_1,a_2,a_3)\]

Vector addition

Vector Addition

\[\mathbf a+\mathbf b=(a_1+b_1,a_2+b_2,a_3+b_3)\]

Scalar multiplication

Scalar Multiplication

\[\lambda\mathbf a=(\lambda a_1,\lambda a_2,\lambda a_3)\]

Dot product

Dot Product

\[\mathbf a\cdot\mathbf b=|\mathbf a||\mathbf b|\cos\theta\]

Vector norm

Dot Product

\[|\mathbf a|=\sqrt{\mathbf a\cdot\mathbf a}\]

Cross product magnitude

Cross Product

\[|\mathbf a\times\mathbf b|=|\mathbf a||\mathbf b|\sin\theta\]

Scalar triple product

Scalar Triple Product

\[[\mathbf a,\mathbf b,\mathbf c]=\mathbf a\cdot(\mathbf b\times\mathbf c)\]

Vector line

Lines

\[\mathbf r=\mathbf a+t\mathbf v\]

Vector plane

Planes

\[\mathbf n\cdot(\mathbf r-\mathbf a)=0\]

Kinematics

9 equations

Position vector

Position

\[\mathbf r(t)=x(t)\mathbf i+y(t)\mathbf j+z(t)\mathbf k\]

Velocity

\[\mathbf v=\frac{d\mathbf r}{dt}\]

Acceleration

\[\mathbf a=\frac{d\mathbf v}{dt}=\frac{d^2\mathbf r}{dt^2}\]

Newton second law

Forces

\[\mathbf F=m\mathbf a\]

Kinetic energy

Energy

\[T=\frac12 m|\mathbf v|^2\]

Work line integral

Work Done

\[W=\int_C\mathbf F\cdot d\mathbf r\]

Angular momentum

Angular Momentum

\[\mathbf L=\mathbf r\times\mathbf p\]

Polar velocity

Polar Coordinates

\[\mathbf v=\dot r\,\mathbf e_r+r\dot\theta\,\mathbf e_\theta\]

Cylindrical position

Cylindrical Polars

\[\mathbf r=\rho\mathbf e_\rho+z\mathbf k\]

Ordinary Differential Equations

9 equations

First-order ODE

ODE Introduction

\[\frac{dy}{dx}=f(x,y)\]

nth-order ODE

Equation Order

\[F(x,y,y',\ldots,y^{(n)})=0\]

Initial condition

Initial Conditions

\[y(x_0)=y_0\]

Separable equation

Separable Equations

\[\frac{dy}{dx}=g(x)h(y)\quad\Rightarrow\quad \int\frac{dy}{h(y)}=\int g(x)\,dx\]

First-order linear solution

Linear First-Order Equations

\[y\mu=\int Q(x)\mu(x)\,dx+C,\qquad \mu=e^{\int P(x)\,dx}\]

Second-order linear equation

Second-Order Equations

\[ay''+by'+cy=f(x)\]

Characteristic equation

Homogeneous Constant-Coefficient Equations

\[ar^2+br+c=0\]

Complementary plus particular

Inhomogeneous Constant-Coefficient Equations

\[y=y_c+y_p\]

Differential operator

Operator Methods

\[D=\frac{d}{dx}\]

Fourier Analysis

9 equations

Periodicity

Periodic Functions

\[f(x+T)=f(x)\]

Function inner product

Function Inner Product

\[\langle f,g\rangle=\int_a^b f(x)g(x)\,dx\]

Function norm

Function Norm

\[\|f\|=\sqrt{\langle f,f\rangle}\]

Fourier series

Fourier Series

\[f(x)\sim\frac{a_0}{2}+\sum_{n=1}^{\infty}(a_n\cos nx+b_n\sin nx)\]

Cosine coefficient

Fourier Coefficients

\[a_n=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\cos nx\,dx\]

Sine coefficient

Fourier Coefficients

\[b_n=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\sin nx\,dx\]

Complex Fourier series

Complex Fourier Form

\[f(x)\sim\sum_{n=-\infty}^{\infty}c_ne^{inx}\]

Complex coefficient

Complex Fourier Form

\[c_n=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-inx}\,dx\]

Parseval theorem

Parseval Theorem

\[\frac{1}{\pi}\int_{-\pi}^{\pi}|f(x)|^2\,dx=\frac{a_0^2}{2}+\sum_{n=1}^{\infty}(a_n^2+b_n^2)\]

Multivariable Calculus

9 equations

Partial derivative

Partial Derivatives

\[f_x=\frac{\partial f}{\partial x}\]

Second partial derivative

Higher Partial Derivatives

\[f_{xy}=\frac{\partial^2 f}{\partial y\partial x}\]

Clairaut theorem

Clairaut Theorem

\[f_{xy}=f_{yx}\]

Total differential

Differentials

\[df=f_x\,dx+f_y\,dy\]

Gradient

\[\nabla f=\left(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z}\right)\]

Directional derivative

Directional Derivatives

\[D_{\mathbf u}f=\nabla f\cdot\mathbf u\]

Multivariable chain rule

Multivariable Chain Rule

\[\frac{df}{dt}=\frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{dy}{dt}\]

Two-variable Taylor expansion

Taylor Expansions

\[f(\mathbf a+\mathbf h)\approx f(\mathbf a)+\nabla f(\mathbf a)\cdot\mathbf h+\frac12\mathbf h^TH(\mathbf a)\mathbf h\]

Critical point condition

Critical Points

\[\nabla f(\mathbf a)=\mathbf0\]

Partial Differential Equations

6 equations

PDE notation

PDE Introduction

\[F(x,y,u,u_x,u_y,u_{xx},u_{xy},u_{yy})=0\]

Heat equation

Important PDEs

\[u_t=\alpha u_{xx}\]

Wave equation

Important PDEs

\[u_{tt}=c^2u_{xx}\]

Laplace equation

Important PDEs

\[u_{xx}+u_{yy}=0\]

Separated form

Separation of Variables

\[u(x,t)=X(x)T(t)\]

Linear PDE operator form

Linear PDEs

\[L[u]=f\]

Vector Calculus

8 equations

Vector field

Vector Fields

\[\mathbf F(x,y,z)=P\mathbf i+Q\mathbf j+R\mathbf k\]

Del operator

Gradient Operator

\[\nabla=\mathbf i\frac{\partial}{\partial x}+\mathbf j\frac{\partial}{\partial y}+\mathbf k\frac{\partial}{\partial z}\]

Divergence

\[\nabla\cdot\mathbf F=\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}\]

Curl

\[\nabla\times\mathbf F\]

Curl of gradient

Div Grad Curl Identities

\[\nabla\times(\nabla f)=\mathbf0\]

Divergence of curl

Div Grad Curl Identities

\[\nabla\cdot(\nabla\times\mathbf F)=0\]

Gauss law electric

Maxwell Equations

\[\nabla\cdot\mathbf E=\frac{\rho}{\varepsilon_0}\]

Faraday law

Maxwell Equations

\[\nabla\times\mathbf E=-\frac{\partial\mathbf B}{\partial t}\]

Multiple Integrals

8 equations

Double integral

Double Integrals

\[\iint_R f(x,y)\,dA\]

Triple integral

Triple Integrals

\[\iiint_D f(x,y,z)\,dV\]

Polar area element

Polar Integration

\[dA=r\,dr\,d\theta\]

Cylindrical volume element

Cylindrical Coordinates

\[dV=r\,dr\,d\theta\,dz\]

Spherical volume element

Spherical Coordinates

\[dV=\rho^2\sin\phi\,d\rho\,d\phi\,d\theta\]

Jacobian change of variables

Jacobian

\[dA=\left|\frac{\partial(x,y)}{\partial(u,v)}\right|du\,dv\]

Volume by double integral

Volume Interpretation

\[V=\iint_R f(x,y)\,dA\]

Iterated integral region

Order Switching

\[\int_a^b\int_{g_1(x)}^{g_2(x)}f(x,y)\,dy\,dx\]

Complex Analysis

6 equations

Complex derivative

Complex Differentiability

\[f'(z_0)=\lim_{h\to0}\frac{f(z_0+h)-f(z_0)}{h}\]

Cauchy-Riemann equations

Cauchy Riemann Equations

\[u_x=v_y,\qquad u_y=-v_x\]

Power derivative

Complex Derivative Examples

\[\frac{d}{dz}z^n=nz^{n-1}\]

Two-dimensional Laplace equation

Laplace Equation

\[u_{xx}+u_{yy}=0\]

Harmonic condition

Harmonic Functions

\[\nabla^2u=0\]

Analytic function

Complex Differentiability

\[f\text{ analytic on }D\Rightarrow f'\text{ exists throughout }D\]

Flashcards

Real Numbers