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Calculus 2 Formula Sheet

Complete Calculus 2 formula sheet: ODEs, curves, multivariable calculus, vector functions, multiple integrals, vector fields, Fourier. All formulas in PDF.

First-Order Ordinary Differential EquationsTheory
What a differential equation is
F(x,y,y′)=0  ⟶  y′=f(x,y)F(x,y,y') = 0 \;\longrightarrow\; y' = f(x,y)
y(x0)=y0(initial value problem)y(x_0) = y_0 \quad (\text{initial value problem})
y′=f(x,y)⇒slope f(x,y) at each pointy' = f(x,y) \Rightarrow \text{slope } f(x,y) \text{ at each point}
Separable variables
∫dyh(y)=∫g(x) dx+C\int \frac{dy}{h(y)} = \int g(x)\,dx + C
h(y0)=0⇒y≡y0 constant solutionh(y_0)=0 \Rightarrow y\equiv y_0 \text{ constant solution}
y′=ky  ⟹  y=y0 ek(x−x0)y' = ky \implies y = y_0\,e^{k(x-x_0)}
First-order linear equations
μ(x)=e∫P(x) dx\mu(x) = e^{\int P(x)\,dx}
(μy)′=μQ(\mu y)' = \mu Q
y=1μ ⁣(∫μQ dx+C)y = \frac{1}{\mu}\!\left(\int \mu Q\,dx + C\right)
Bernoulli equation
z=y1−nz = y^{1-n}
z′+(1−n)P(x) z=(1−n)Q(x)z' + (1-n)P(x)\,z = (1-n)Q(x)
y=z1/(1−n)y = z^{1/(1-n)}
Homogeneous equations and y'=f(y/x)
v=yx,y′=v+xv′v = \frac{y}{x},\quad y' = v + x v'
x dvdx=F(v)−vx\,\frac{dv}{dx} = F(v) - v
∫dvF(v)−v=ln⁡∣x∣+C\int \frac{dv}{F(v)-v} = \ln|x| + C
Exact equations and integrating factors
∂yM=∂xN  ⇔  exact\partial_y M = \partial_x N \;\Leftrightarrow\; \text{exact}
Fx=M,  Fy=N  ⇒  F(x,y)=CF_x = M,\; F_y = N \;\Rightarrow\; F(x,y)=C
μx=e∫My−NxN dx\mu_x = e^{\int \frac{M_y-N_x}{N}\,dx}
Existence and uniqueness: the Cauchy theorems
f continuous⇒∃ solution (Peano)f \text{ continuous} \Rightarrow \exists \text{ solution (Peano)}
∣f(x,y1)−f(x,y2)∣≤L∣y1−y2∣⇒∃!  (Picard)|f(x,y_1)-f(x,y_2)| \leq L|y_1-y_2| \Rightarrow \exists! \;(\text{Picard})
y′=∣y∣, y(0)=0: y≡0 and y=x2/4y'=\sqrt{|y|},\,y(0)=0:\ y\equiv0 \text{ and } y=x^2/4
Second-Order ODEs and Linear SystemsTheory
What a second-order ODE is
ay′′+by′+cy=f(x)a y'' + b y' + c y = f(x)
y(x0)=y0,y′(x0)=y0′y(x_0)=y_0,\quad y'(x_0)=y_0'
F=m x′′  (Newton’s 2nd law)F = m\,x'' \;(\text{Newton's 2nd law})
Structure of the general solution and the Wronskian
W(y1,y2)=y1y2′−y1′y2W(y_1,y_2) = y_1 y_2' - y_1' y_2
W≠0  ⟺  y1,y2 independentW \neq 0 \iff y_1,y_2 \text{ independent}
y=C1y1+C2y2+yPy = C_1 y_1 + C_2 y_2 + y_P
Characteristic equation (constant coefficients)
aλ2+bλ+c=0,Δ=b2−4aca\lambda^2+b\lambda+c=0,\quad \Delta=b^2-4ac
Δ>0:  C1eλ1x+C2eλ2x\Delta>0:\; C_1e^{\lambda_1x}+C_2e^{\lambda_2x}
Δ=0:  (C1+C2x)eλx\Delta=0:\; (C_1+C_2x)e^{\lambda x}
Δ<0:  eαx(C1cos⁡βx+C2sin⁡βx)\Delta<0:\; e^{\alpha x}(C_1\cos\beta x+C_2\sin\beta x)
Particular solution: undetermined coefficients
y′′+py′+qy=ekx⇒yP=ekxk2+pk+q  (if k2+pk+q≠0)y'' + py' + qy = e^{kx} \Rightarrow y_P = \tfrac{e^{kx}}{k^2+pk+q}\;(\text{if } k^2+pk+q\neq0)
resonance⇒multiply by x\text{resonance} \Rightarrow \text{multiply by } x
Variation of parameters
u1′=−y2faW,u2′=y1faWu_1' = -\frac{y_2 f}{aW},\quad u_2' = \frac{y_1 f}{aW}
yP=u1y1+u2y2y_P = u_1 y_1 + u_2 y_2
Mechanical vibrations, damping and resonance
mx′′+γx′+kx=F0cos⁡ωtm x'' + \gamma x' + k x = F_0\cos\omega t
ω0=k/m\omega_0 = \sqrt{k/m}
xP=F02mω0 tsin⁡(ω0t)  (resonance)x_P = \frac{F_0}{2m\omega_0}\,t\sin(\omega_0 t)\;(\text{resonance})
Linear differential systems
x′=Ax\mathbf{x}' = A\mathbf{x}
det⁡(A−λI)=λ2−tr⁡(A)λ+det⁡(A)=0\det(A-\lambda I)=\lambda^2 - \operatorname{tr}(A)\lambda + \det(A)=0
x=C1eλ1tv1+C2eλ2tv2\mathbf{x}=C_1e^{\lambda_1 t}\mathbf{v}_1+C_2e^{\lambda_2 t}\mathbf{v}_2
Classifying equilibria (trace–determinant plane)
τ=tr⁡(A),δ=det⁡(A)\tau=\operatorname{tr}(A),\quad \delta=\det(A)
stable  ⟺  τ<0 and δ>0\text{stable} \iff \tau<0 \text{ and } \delta>0
saddle  ⟺  δ<0,centre  ⟺  τ=0,δ>0\text{saddle} \iff \delta<0,\quad \text{centre} \iff \tau=0,\delta>0
Calculus on CurvesTheory
Parametric curves
r(t)=(x(t),y(t),z(t))\mathbf{r}(t)=(x(t),y(t),z(t))
T^(t)=r′(t)∣r′(t)∣\hat{T}(t) = \frac{\mathbf{r}'(t)}{|\mathbf{r}'(t)|}
regular  ⟺  r′(t)≠0  ∀t\text{regular} \iff \mathbf{r}'(t)\neq\mathbf{0}\;\forall t
Arc length
L=∫abx′2+y′2+z′2 dtL = \int_a^b \sqrt{x'^2+y'^2+z'^2}\,dt
ds=∣r′(t)∣ dtds = |\mathbf{r}'(t)|\,dt
L=∫αβr2+r′2 dθ  (polar)L = \int_\alpha^\beta \sqrt{r^2+r'^2}\,d\theta \;\text{(polar)}
Curvature and the osculating circle
κ=∣dT^ds∣=∣r′×r′′∣∣r′∣3\kappa = \left|\frac{d\hat T}{ds}\right| = \frac{|\mathbf{r}'\times\mathbf{r}''|}{|\mathbf{r}'|^3}
κ=∣y′′∣(1+y′2)3/2\kappa = \frac{|y''|}{(1+y'^2)^{3/2}}
ρ=1κ  (radius of curvature)\rho = \frac{1}{\kappa}\;(\text{radius of curvature})
Line integral of the 1st kind (scalar function)
∫γf ds=∫abf(r(t)) ∣r′(t)∣ dt\int_\gamma f\,ds = \int_a^b f(\mathbf{r}(t))\,|\mathbf{r}'(t)|\,dt
mass=∫γρ ds,fˉ=1L∫γf ds\text{mass} = \int_\gamma \rho\,ds,\quad \bar f = \frac{1}{L}\int_\gamma f\,ds
Line integral of the 2nd kind (work)
∫γF⋅dr=∫ab(Px′+Qy′+Rz′) dt\int_\gamma \mathbf{F}\cdot d\mathbf{r} = \int_a^b (Px'+Qy'+Rz')\,dt
∫−γF⋅dr=−∫γF⋅dr\int_{-\gamma} \mathbf{F}\cdot d\mathbf{r} = -\int_{\gamma} \mathbf{F}\cdot d\mathbf{r}
Conservative fields and potential
F=∇U  ⟹  ∫γF⋅dr=U(B)−U(A)\mathbf{F}=\nabla U \implies \int_\gamma \mathbf{F}\cdot d\mathbf{r} = U(B)-U(A)
∮γF⋅dr=0  (conservative field)\oint_\gamma \mathbf{F}\cdot d\mathbf{r} = 0 \;(\text{conservative field})
∂yP=∂xQ  (on a simply connected domain)\partial_y P = \partial_x Q \;(\text{on a simply connected domain})
Polar coordinates and notable curves
x=rcos⁡θ,y=rsin⁡θx=r\cos\theta,\quad y=r\sin\theta
A=12∫αβr2 dθA = \tfrac12\int_\alpha^\beta r^2\,d\theta
r=a(1+cos⁡θ)  (cardioid)r=a(1+\cos\theta)\;(\text{cardioid})
The Frenet frame (outline)
N^=T^′/∣T^′∣,B^=T^×N^\hat N = \hat T'/|\hat T'|,\quad \hat B=\hat T\times\hat N
T^′=κN^,  B^′=−τN^\hat T'=\kappa\hat N,\;\hat B'=-\tau\hat N
τ=0  ⟺  plane curve\tau=0 \iff \text{plane curve}
Multivariable Differential CalculusTheory
Functions of several variables: domain and level curves
f:R2→R,(x,y)↦f(x,y)f:\mathbb{R}^2\to\mathbb{R},\quad (x,y)\mapsto f(x,y)
Lc={x∈Rn:f(x)=c}L_c = \{\mathbf{x}\in\mathbb{R}^n : f(\mathbf{x})=c\}
Limits and continuity in ℝⁿ
∀ε>0  ∃δ>0:  ∣x−x0∣<δ  ⟹  ∣f(x)−L∣<ε\forall\varepsilon>0\;\exists\delta>0:\; |\mathbf{x}-\mathbf{x}_0|<\delta \implies |f(\mathbf{x})-L|<\varepsilon
different limits on two paths⇒∄lim⁡\text{different limits on two paths} \Rightarrow \nexists\lim
Partial derivatives and differentiability
∂xf=lim⁡h→0f(x0+h,y0)−f(x0,y0)h\partial_x f = \lim_{h\to0}\frac{f(x_0+h,y_0)-f(x_0,y_0)}{h}
f(x0+h)=f(x0)+∇f⋅h+o(∣h∣)f(\mathbf{x}_0+\mathbf{h})=f(\mathbf{x}_0)+\nabla f\cdot\mathbf{h}+o(|\mathbf{h}|)
f∈C1⇒f differentiable⇒f continuousf\in C^1 \Rightarrow f \text{ differentiable} \Rightarrow f \text{ continuous}
Gradient and directional derivative
∇f=(∂xf, ∂yf)\nabla f = (\partial_x f,\,\partial_y f)
Dv^f=∇f⋅v^≤∣∇f∣D_{\hat v}f = \nabla f\cdot\hat v \leq |\nabla f|
∇f⊥Lc  (level curves)\nabla f \perp L_c \;(\text{level curves})
Tangent plane and Taylor formula
z=f0+fx(x−x0)+fy(y−y0)z = f_0+f_x(x-x_0)+f_y(y-y_0)
f(x0+h)≈f0+∇f⋅h+12hTHf hf(\mathbf{x}_0+\mathbf{h})\approx f_0+\nabla f\cdot\mathbf{h}+\tfrac12\mathbf{h}^T H_f\,\mathbf{h}
Critical points and classification by the Hessian
∇f(x0)=0  (critical point)\nabla f(\mathbf{x}_0)=\mathbf{0}\;(\text{critical point})
Hf=(fxxfxyfyxfyy),det⁡Hf=fxxfyy−fxy2H_f = \begin{pmatrix}f_{xx}&f_{xy}\\f_{yx}&f_{yy}\end{pmatrix},\quad \det H_f=f_{xx}f_{yy}-f_{xy}^2
det⁡H>0, fxx>0⇒min;  det⁡H<0⇒saddle\det H>0,\,f_{xx}>0 \Rightarrow \text{min};\;\det H<0\Rightarrow\text{saddle}
Chain rule and the implicit function theorem
ddtf(r(t))=∇f⋅r′(t)\frac{d}{dt}f(\mathbf{r}(t)) = \nabla f\cdot\mathbf{r}'(t)
Fx+Fy φ′=0⇒φ′(x)=−FxFyF_x + F_y\,\varphi' = 0 \Rightarrow \varphi'(x) = -\frac{F_x}{F_y}
Constrained optimisation: Lagrange multipliers
∇f=λ∇g,g(x,y)=0\nabla f = \lambda\nabla g,\quad g(x,y)=0
{fx=λgxfy=λgyg=0\begin{cases}f_x=\lambda g_x\\ f_y=\lambda g_y\\ g=0\end{cases}
Vector-Valued Functions and ManifoldsTheory
Vector functions: fields, transformations, surfaces
f:Rn→Rm,f=(f1,…,fm)f:\mathbb{R}^n\to\mathbb{R}^m,\quad f=(f_1,\dots,f_m)
r(u,v):R2→R3  (surface)\mathbf{r}(u,v):\mathbb{R}^2\to\mathbb{R}^3\;(\text{surface})
The Jacobian matrix
Jf=(∂1f1⋯∂nf1⋮⋮∂1fm⋯∂nfm)J_f = \begin{pmatrix}\partial_1 f_1&\cdots&\partial_n f_1\\\vdots&&\vdots\\\partial_1 f_m&\cdots&\partial_n f_m\end{pmatrix}
f(x0+h)≈f(x0)+Jf hf(\mathbf{x}_0+\mathbf{h})\approx f(\mathbf{x}_0)+J_f\,\mathbf{h}
∣det⁡Jf∣=volume-scaling factor|\det J_f| = \text{volume-scaling factor}
Vector chain rule
Jf∘g(x)=Jf(g(x))⋅Jg(x)J_{f\circ g}(\mathbf{x}) = J_f(g(\mathbf{x}))\cdot J_g(\mathbf{x})
(m×n)⋅(n×p)=m×p(m\times n)\cdot(n\times p) = m\times p
The inverse function theorem
det⁡Jf(x0)≠0  ⟹  f locally invertible\det J_f(\mathbf{x}_0)\neq0 \implies f \text{ locally invertible}
Jf−1=[Jf]−1J_{f^{-1}}=[J_f]^{-1}
The implicit function theorem (vector form)
det⁡ ⁣(∂F/∂y)≠0⇒y=g(x)\det\!\left(\partial F/\partial\mathbf{y}\right)\neq0 \Rightarrow \mathbf{y}=\mathbf{g}(\mathbf{x})
Jg=− ⁣(∂F∂y)−1 ⁣∂F∂xJ_g = -\!\left(\frac{\partial F}{\partial\mathbf{y}}\right)^{-1}\!\frac{\partial F}{\partial\mathbf{x}}
Parametric surfaces, normal and area
n=ru×rv\mathbf{n} = \mathbf{r}_u\times\mathbf{r}_v
dS=∣ru×rv∣ du dvdS = |\mathbf{r}_u\times\mathbf{r}_v|\,du\,dv
A=∬D∣ru×rv∣ du dvA = \iint_D|\mathbf{r}_u\times\mathbf{r}_v|\,du\,dv
Lagrange multipliers (several constraints)
∇f=λ∇g,g=0\nabla f = \lambda\nabla g,\quad g=0
∇f=∑i=1kλi∇gi,g1=⋯=gk=0\nabla f = \sum_{i=1}^k\lambda_i\nabla g_i,\quad g_1=\cdots=g_k=0
Change of coordinates and the Jacobian in integrals
∬T(D)f dx dy=∬Df(T) ∣det⁡JT∣ du dv\iint_{T(D)} f\,dx\,dy = \iint_D f(T)\,|\det J_T|\,du\,dv
dx dy=r dr dθ  (polar)dx\,dy = r\,dr\,d\theta\;(\text{polar})
dV=ρ2sin⁡ϕ dρ dϕ dθ  (spherical)dV = \rho^2\sin\phi\,d\rho\,d\phi\,d\theta\;(\text{spherical})
Multiple IntegralsTheory
From the single to the double integral
∬Df dA=lim⁡ΔA→0∑if(xi,yi) ΔA\iint_D f\,dA = \lim_{\Delta A\to0}\sum_i f(x_i,y_i)\,\Delta A
∬D1 dA=area(D)\iint_D 1\,dA = \text{area}(D)
Fubini's theorem (iterated integration)
∬Rf dA=∫ab ⁣∫cdf dy dx=∫cd ⁣∫abf dx dy\iint_R f\,dA = \int_a^b\!\int_c^d f\,dy\,dx = \int_c^d\!\int_a^b f\,dx\,dy
f continuous⇒order swappablef \text{ continuous} \Rightarrow \text{order swappable}
Normal domains and swapping the order
Dy-norm={a≤x≤b, g1(x)≤y≤g2(x)}D_{y\text{-norm}}=\{a\le x\le b,\ g_1(x)\le y\le g_2(x)\}
∬Df=∫ab ⁣∫g1(x)g2(x)f dy dx\iint_D f = \int_a^b\!\int_{g_1(x)}^{g_2(x)} f\,dy\,dx
Polar coordinates in the plane
x=rcos⁡θ, y=rsin⁡θ,dA=r dr dθx=r\cos\theta,\ y=r\sin\theta,\quad dA = r\,dr\,d\theta
∬Df dA=∫αβ ⁣∫0R(θ)f r dr dθ\iint_D f\,dA = \int_\alpha^\beta\!\int_0^{R(\theta)} f\,r\,dr\,d\theta
The triple integral
∭Vf dV=∫ab ⁣∫g1g2 ⁣∫h1h2f dz dy dx\iiint_V f\,dV = \int_a^b\!\int_{g_1}^{g_2}\!\int_{h_1}^{h_2} f\,dz\,dy\,dx
∭V1 dV=volume(V)\iiint_V 1\,dV = \text{volume}(V)
Cylindrical and spherical coordinates
dVcyl=r dr dθ dzdV_{\text{cyl}} = r\,dr\,d\theta\,dz
dVsph=ρ2sin⁡ϕ dρ dϕ dθdV_{\text{sph}} = \rho^2\sin\phi\,d\rho\,d\phi\,d\theta
x2+y2+z2=ρ2x^2+y^2+z^2 = \rho^2
The general change-of-variables formula
∬Df dx dy=∬D∗f(Φ) ∣det⁡JΦ∣ du dv\iint_D f\,dx\,dy = \iint_{D^*} f(\Phi)\,|\det J_\Phi|\,du\,dv
JΦ=∂(x,y)∂(u,v)J_\Phi = \frac{\partial(x,y)}{\partial(u,v)}
Applications: mass, centroid, moment of inertia
m=∭Vρ dVm = \iiint_V \rho\,dV
xˉ=1m∭Vx ρ dV\bar x = \frac{1}{m}\iiint_V x\,\rho\,dV
Iz=∭V(x2+y2) ρ dVI_z = \iiint_V (x^2+y^2)\,\rho\,dV
Vector Fields and Integral TheoremsTheory
What a vector field is
F:Rn→Rn,x↦F(x)\mathbf{F}:\mathbb{R}^n\to\mathbb{R}^n,\quad \mathbf{x}\mapsto\mathbf{F}(\mathbf{x})
r′(t)=F(r(t))  (flow lines)\mathbf{r}'(t) = \mathbf{F}(\mathbf{r}(t))\;(\text{flow lines})
Divergence and curl
∇⋅F=∂xP+∂yQ+∂zR\nabla\cdot\mathbf{F} = \partial_x P+\partial_y Q+\partial_z R
∇×F=(Ry−Qz,  Pz−Rx,  Qx−Py)\nabla\times\mathbf{F} = (R_y-Q_z,\;P_z-R_x,\;Q_x-P_y)
∇⋅F>0:source,  <0:sink\nabla\cdot\mathbf{F}>0:\text{source},\;<0:\text{sink}
Conservative fields and potential
F=∇U  ⟺  ∇×F=0  (s.c. domain)\mathbf{F}=\nabla U \iff \nabla\times\mathbf{F}=\mathbf{0}\;(\text{s.c. domain})
∇×(∇U)=0\nabla\times(\nabla U)=\mathbf{0}
∮γF⋅dr=0 on every closed curve\oint_\gamma\mathbf{F}\cdot d\mathbf{r}=0 \text{ on every closed curve}
Surface integral (flux)
Φ=∬ΣF⋅n^ dS\Phi = \iint_\Sigma\mathbf{F}\cdot\hat n\,dS
=∬DF⋅(ru×rv) du dv= \iint_D\mathbf{F}\cdot(\mathbf{r}_u\times\mathbf{r}_v)\,du\,dv
Green's theorem in the plane
∮∂D(P dx+Q dy)=∬D(Qx−Py) dA\oint_{\partial D}(P\,dx+Q\,dy) = \iint_D(Q_x-P_y)\,dA
A=12∮∂D(x dy−y dx)A = \tfrac12\oint_{\partial D}(x\,dy-y\,dx)
The Divergence theorem (Gauss)
∯∂VF⋅dS=∭V(∇⋅F) dV\oiint_{\partial V}\mathbf{F}\cdot d\mathbf{S} = \iiint_V(\nabla\cdot\mathbf{F})\,dV
∇⋅E=ρ/ε0  (Gauss)\nabla\cdot\mathbf{E} = \rho/\varepsilon_0\;(\text{Gauss})
Stokes' theorem
∮∂ΣF⋅dr=∬Σ(∇×F)⋅dS\oint_{\partial\Sigma}\mathbf{F}\cdot d\mathbf{r} = \iint_\Sigma(\nabla\times\mathbf{F})\cdot d\mathbf{S}
∮B⋅dr=μ0I  (Ampeˋre)\oint\mathbf{B}\cdot d\mathbf{r}=\mu_0 I\;(\text{Ampère})
The three formulas as one idea
∇⋅(∇×F)=0\nabla\cdot(\nabla\times\mathbf{F})=0
∫Ωdω=∫∂Ωω  (general Stokes)\int_\Omega d\omega=\int_{\partial\Omega}\omega\;(\text{general Stokes})
Power Series, Fourier Series, and TransformsTheory
Numerical series — convergence tests
lim⁡∣an+1/an∣=L<1  ⟹  abs. conv.\lim|a_{n+1}/a_n| = L < 1 \implies \text{abs. conv.}
lim sup⁡∣an∣n=L<1  ⟹  abs. conv.\limsup\sqrt[n]{|a_n|} = L < 1 \implies \text{abs. conv.}
∑(−1)nbn,  bn↘0  ⟹  converges (Leibniz)\sum(-1)^n b_n,\; b_n\searrow0 \implies \text{converges (Leibniz)}
∑n=1∞1/np converges   ⟺  p>1\sum_{n=1}^\infty 1/n^p \text{ converges } \iff p>1
Power series and radius of convergence
R=1/lim sup⁡∣cn∣nR = 1/\limsup\sqrt[n]{|c_n|}
∣x−x0∣<R  ⟹  abs. conv.|x-x_0|<R \implies \text{abs. conv.}
f′(x)=∑ncn(x−x0)n−1  (same R)f'(x) = \sum n c_n (x-x_0)^{n-1} \;(\text{same }R)
Taylor series and notable expansions
f(x)=∑n=0∞f(n)(x0)n!(x−x0)nf(x)=\sum_{n=0}^\infty\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n
ex=∑xn/n!,  sin⁡x=∑(−1)nx2n+1/(2n+1)!e^x = \sum x^n/n!,\; \sin x = \sum(-1)^n x^{2n+1}/(2n+1)!
ln⁡(1+x)=∑(−1)n−1xn/n,  ∣x∣≤1\ln(1+x)=\sum(-1)^{n-1}x^n/n,\; |x|\leq1
(αn)=α(α−1)⋯(α−n+1)n!\binom{\alpha}{n} = \frac{\alpha(\alpha-1)\cdots(\alpha-n+1)}{n!}
Fourier series — periods and coefficients
f(x)∼a02+∑n=1∞(ancos⁡nπxL+bnsin⁡nπxL)f(x)\sim\frac{a_0}{2}+\sum_{n=1}^\infty\big(a_n\cos\frac{n\pi x}{L}+b_n\sin\frac{n\pi x}{L}\big)
an=1L∫−LLf(x)cos⁡nπxL dx,  bn=1L∫−LLf(x)sin⁡nπxL dxa_n=\frac1L\int_{-L}^{L}f(x)\cos\frac{n\pi x}{L}\,dx,\; b_n=\frac1L\int_{-L}^{L}f(x)\sin\frac{n\pi x}{L}\,dx
cn=12L∫−LLf(x)e−inπx/L dxc_n = \frac1{2L}\int_{-L}^{L} f(x)e^{-i n\pi x/L}\,dx
f even⇒bn=0,f odd⇒an=0f\text{ even}\Rightarrow b_n=0,\quad f\text{ odd}\Rightarrow a_n=0
Convergence of Fourier series and Gibbs phenomenon
f(x±)=lim⁡h→0+f(x±h)f(x^\pm)=\lim_{h\to0^+}f(x\pm h)
Dirichlet: f(x+)+f(x−)2 at discontinuities\text{Dirichlet: } \frac{f(x^+)+f(x^-)}{2} \text{ at discontinuities}
Parseval: 12L∫−LL∣f∣2=∑−∞∞∣cn∣2\text{Parseval: } \frac1{2L}\int_{-L}^{L}|f|^2 = \sum_{-\infty}^{\infty}|c_n|^2
Fourier transform — from discrete to continuous
f^(ξ)=∫−∞∞f(x)e−2πiξx dx\hat f(\xi)=\int_{-\infty}^{\infty} f(x)e^{-2\pi i\xi x}\,dx
f(x)=∫−∞∞f^(ξ)e2πiξx dξf(x)=\int_{-\infty}^{\infty} \hat f(\xi)e^{2\pi i\xi x}\,d\xi
f′^=2πiξ f^,f∗g^=f^ g^\widehat{f'}=2\pi i\xi\,\hat f,\quad \widehat{f*g}=\hat f\,\hat g
Laplace transform
F(s)=∫0∞f(t)e−st dtF(s)=\int_0^\infty f(t)e^{-st}\,dt
L{f′}=sF(s)−f(0),  L{f′′}=s2F(s)−sf(0)−f′(0)\mathcal{L}\{f'\}=sF(s)-f(0),\; \mathcal{L}\{f''\}=s^2F(s)-s f(0)-f'(0)
L{eat}=1/(s−a),  L{sin⁡ωt}=ω/(s2+ω2)\mathcal{L}\{e^{at}\}=1/(s-a),\; \mathcal{L}\{\sin\omega t\}=\omega/(s^2+\omega^2)
Applications: ODEs, circuits, and control
L{y(n)}=snY(s)−sn−1y(0)−⋯−y(n−1)(0)\mathcal{L}\{y^{(n)}\} = s^nY(s) - s^{n-1}y(0) - \dots - y^{(n-1)}(0)
lim⁡t→∞f(t)=lim⁡s→0sF(s)  (final value)\lim_{t\to\infty} f(t) = \lim_{s\to0} sF(s) \;(\text{final value})
H(s)=Y(s)/U(s)  (transfer function)H(s) = Y(s)/U(s) \;(\text{transfer function})
Tutti i formulari