First-Order Ordinary Differential Equations
What a differential equation is
F ( x , y , y ′ ) = 0 ⟶ y ′ = f ( x , y ) F(x,y,y') = 0 \;\longrightarrow\; y' = f(x,y) F ( x , y , y ′ ) = 0 ⟶ y ′ = f ( x , y ) y ( x 0 ) = y 0 ( initial value problem ) y(x_0) = y_0 \quad (\text{initial value problem}) y ( x 0 ) = y 0 ( initial value problem ) y ′ = f ( x , y ) ⇒ slope f ( x , y ) at each point y' = f(x,y) \Rightarrow \text{slope } f(x,y) \text{ at each point} y ′ = f ( x , y ) ⇒ slope f ( x , y ) at each point Separable variables
∫ d y h ( y ) = ∫ g ( x ) d x + C \int \frac{dy}{h(y)} = \int g(x)\,dx + C ∫ h ( y ) d y = ∫ g ( x ) d x + C h ( y 0 ) = 0 ⇒ y ≡ y 0 constant solution h(y_0)=0 \Rightarrow y\equiv y_0 \text{ constant solution} h ( y 0 ) = 0 ⇒ y ≡ y 0 constant solution y ′ = k y ⟹ y = y 0 e k ( x − x 0 ) y' = ky \implies y = y_0\,e^{k(x-x_0)} y ′ = k y ⟹ y = y 0 e k ( x − x 0 ) First-order linear equations
μ ( x ) = e ∫ P ( x ) d x \mu(x) = e^{\int P(x)\,dx} μ ( x ) = e ∫ P ( x ) d x ( μ y ) ′ = μ Q (\mu y)' = \mu Q ( μ y ) ′ = μ Q y = 1 μ ( ∫ μ Q d x + C ) y = \frac{1}{\mu}\!\left(\int \mu Q\,dx + C\right) y = μ 1 ( ∫ μ Q d x + C ) Bernoulli equation
z ′ + ( 1 − n ) P ( x ) z = ( 1 − n ) Q ( x ) z' + (1-n)P(x)\,z = (1-n)Q(x) z ′ + ( 1 − n ) P ( x ) z = ( 1 − n ) Q ( x ) y = z 1 / ( 1 − n ) y = z^{1/(1-n)} y = z 1/ ( 1 − n ) Homogeneous equations and y'=f(y/x)
v = y x , y ′ = v + x v ′ v = \frac{y}{x},\quad y' = v + x v' v = x y , y ′ = v + x v ′ x d v d x = F ( v ) − v x\,\frac{dv}{dx} = F(v) - v x d x d v = F ( v ) − v ∫ d v F ( v ) − v = ln ∣ x ∣ + C \int \frac{dv}{F(v)-v} = \ln|x| + C ∫ F ( v ) − v d v = ln ∣ x ∣ + C Exact equations and integrating factors
∂ y M = ∂ x N ⇔ exact \partial_y M = \partial_x N \;\Leftrightarrow\; \text{exact} ∂ y M = ∂ x N ⇔ exact F x = M , F y = N ⇒ F ( x , y ) = C F_x = M,\; F_y = N \;\Rightarrow\; F(x,y)=C F x = M , F y = N ⇒ F ( x , y ) = C μ x = e ∫ M y − N x N d x \mu_x = e^{\int \frac{M_y-N_x}{N}\,dx} μ x = e ∫ N M y − N x d x Existence and uniqueness: the Cauchy theorems
f continuous ⇒ ∃ solution (Peano) f \text{ continuous} \Rightarrow \exists \text{ solution (Peano)} f continuous ⇒ ∃ solution (Peano) ∣ f ( x , y 1 ) − f ( x , y 2 ) ∣ ≤ L ∣ y 1 − y 2 ∣ ⇒ ∃ ! ( Picard ) |f(x,y_1)-f(x,y_2)| \leq L|y_1-y_2| \Rightarrow \exists! \;(\text{Picard}) ∣ f ( x , y 1 ) − f ( x , y 2 ) ∣ ≤ L ∣ y 1 − y 2 ∣ ⇒ ∃ ! ( Picard ) y ′ = ∣ y ∣ , y ( 0 ) = 0 : y ≡ 0 and y = x 2 / 4 y'=\sqrt{|y|},\,y(0)=0:\ y\equiv0 \text{ and } y=x^2/4 y ′ = ∣ y ∣ , y ( 0 ) = 0 : y ≡ 0 and y = x 2 /4 Second-Order ODEs and Linear Systems
What a second-order ODE is
a y ′ ′ + b y ′ + c y = f ( x ) a y'' + b y' + c y = f(x) a y ′′ + b y ′ + cy = f ( x ) y ( x 0 ) = y 0 , y ′ ( x 0 ) = y 0 ′ y(x_0)=y_0,\quad y'(x_0)=y_0' y ( x 0 ) = y 0 , y ′ ( x 0 ) = y 0 ′ F = m x ′ ′ ( Newton’s 2nd law ) F = m\,x'' \;(\text{Newton's 2nd law}) F = m x ′′ ( Newton’s 2nd law ) Structure of the general solution and the Wronskian
W ( y 1 , y 2 ) = y 1 y 2 ′ − y 1 ′ y 2 W(y_1,y_2) = y_1 y_2' - y_1' y_2 W ( y 1 , y 2 ) = y 1 y 2 ′ − y 1 ′ y 2 W ≠ 0 ⟺ y 1 , y 2 independent W \neq 0 \iff y_1,y_2 \text{ independent} W = 0 ⟺ y 1 , y 2 independent y = C 1 y 1 + C 2 y 2 + y P y = C_1 y_1 + C_2 y_2 + y_P y = C 1 y 1 + C 2 y 2 + y P Characteristic equation (constant coefficients)
a λ 2 + b λ + c = 0 , Δ = b 2 − 4 a c a\lambda^2+b\lambda+c=0,\quad \Delta=b^2-4ac a λ 2 + bλ + c = 0 , Δ = b 2 − 4 a c Δ > 0 : C 1 e λ 1 x + C 2 e λ 2 x \Delta>0:\; C_1e^{\lambda_1x}+C_2e^{\lambda_2x} Δ > 0 : C 1 e λ 1 x + C 2 e λ 2 x Δ = 0 : ( C 1 + C 2 x ) e λ x \Delta=0:\; (C_1+C_2x)e^{\lambda x} Δ = 0 : ( C 1 + C 2 x ) e λ x Δ < 0 : e α x ( C 1 cos β x + C 2 sin β x ) \Delta<0:\; e^{\alpha x}(C_1\cos\beta x+C_2\sin\beta x) Δ < 0 : e α x ( C 1 cos β x + C 2 sin β x ) Particular solution: undetermined coefficients
y ′ ′ + p y ′ + q y = e k x ⇒ y P = e k x k 2 + p k + q ( if k 2 + p k + q ≠ 0 ) y'' + py' + qy = e^{kx} \Rightarrow y_P = \tfrac{e^{kx}}{k^2+pk+q}\;(\text{if } k^2+pk+q\neq0) y ′′ + p y ′ + q y = e k x ⇒ y P = k 2 + p k + q e k x ( if k 2 + p k + q = 0 ) resonance ⇒ multiply by x \text{resonance} \Rightarrow \text{multiply by } x resonance ⇒ multiply by x Variation of parameters
u 1 ′ = − y 2 f a W , u 2 ′ = y 1 f a W u_1' = -\frac{y_2 f}{aW},\quad u_2' = \frac{y_1 f}{aW} u 1 ′ = − aW y 2 f , u 2 ′ = aW y 1 f y P = u 1 y 1 + u 2 y 2 y_P = u_1 y_1 + u_2 y_2 y P = u 1 y 1 + u 2 y 2 Mechanical vibrations, damping and resonance
m x ′ ′ + γ x ′ + k x = F 0 cos ω t m x'' + \gamma x' + k x = F_0\cos\omega t m x ′′ + γ x ′ + k x = F 0 cos ω t ω 0 = k / m \omega_0 = \sqrt{k/m} ω 0 = k / m x P = F 0 2 m ω 0 t sin ( ω 0 t ) ( resonance ) x_P = \frac{F_0}{2m\omega_0}\,t\sin(\omega_0 t)\;(\text{resonance}) x P = 2 m ω 0 F 0 t sin ( ω 0 t ) ( resonance ) Linear differential systems
x ′ = A x \mathbf{x}' = A\mathbf{x} x ′ = A x det ( A − λ I ) = λ 2 − tr ( A ) λ + det ( A ) = 0 \det(A-\lambda I)=\lambda^2 - \operatorname{tr}(A)\lambda + \det(A)=0 det ( A − λ I ) = λ 2 − tr ( A ) λ + det ( A ) = 0 x = C 1 e λ 1 t v 1 + C 2 e λ 2 t v 2 \mathbf{x}=C_1e^{\lambda_1 t}\mathbf{v}_1+C_2e^{\lambda_2 t}\mathbf{v}_2 x = C 1 e λ 1 t v 1 + C 2 e λ 2 t v 2 Classifying equilibria (trace–determinant plane)
τ = tr ( A ) , δ = det ( A ) \tau=\operatorname{tr}(A),\quad \delta=\det(A) τ = tr ( A ) , δ = det ( A ) stable ⟺ τ < 0 and δ > 0 \text{stable} \iff \tau<0 \text{ and } \delta>0 stable ⟺ τ < 0 and δ > 0 saddle ⟺ δ < 0 , centre ⟺ τ = 0 , δ > 0 \text{saddle} \iff \delta<0,\quad \text{centre} \iff \tau=0,\delta>0 saddle ⟺ δ < 0 , centre ⟺ τ = 0 , δ > 0 Calculus on Curves
Parametric curves
r ( t ) = ( x ( t ) , y ( t ) , z ( t ) ) \mathbf{r}(t)=(x(t),y(t),z(t)) r ( t ) = ( x ( t ) , y ( t ) , z ( t )) T ^ ( t ) = r ′ ( t ) ∣ r ′ ( t ) ∣ \hat{T}(t) = \frac{\mathbf{r}'(t)}{|\mathbf{r}'(t)|} T ^ ( t ) = ∣ r ′ ( t ) ∣ r ′ ( t ) regular ⟺ r ′ ( t ) ≠ 0 ∀ t \text{regular} \iff \mathbf{r}'(t)\neq\mathbf{0}\;\forall t regular ⟺ r ′ ( t ) = 0 ∀ t Arc length
L = ∫ a b x ′ 2 + y ′ 2 + z ′ 2 d t L = \int_a^b \sqrt{x'^2+y'^2+z'^2}\,dt L = ∫ a b x ′2 + y ′2 + z ′2 d t d s = ∣ r ′ ( t ) ∣ d t ds = |\mathbf{r}'(t)|\,dt d s = ∣ r ′ ( t ) ∣ d t L = ∫ α β r 2 + r ′ 2 d θ (polar) L = \int_\alpha^\beta \sqrt{r^2+r'^2}\,d\theta \;\text{(polar)} L = ∫ α β r 2 + r ′2 d θ (polar) Curvature and the osculating circle
κ = ∣ d T ^ d s ∣ = ∣ r ′ × r ′ ′ ∣ ∣ r ′ ∣ 3 \kappa = \left|\frac{d\hat T}{ds}\right| = \frac{|\mathbf{r}'\times\mathbf{r}''|}{|\mathbf{r}'|^3} κ = d s d T ^ = ∣ r ′ ∣ 3 ∣ r ′ × r ′′ ∣ κ = ∣ y ′ ′ ∣ ( 1 + y ′ 2 ) 3 / 2 \kappa = \frac{|y''|}{(1+y'^2)^{3/2}} κ = ( 1 + y ′2 ) 3/2 ∣ y ′′ ∣ ρ = 1 κ ( radius of curvature ) \rho = \frac{1}{\kappa}\;(\text{radius of curvature}) ρ = κ 1 ( radius of curvature ) Line integral of the 1st kind (scalar function)
∫ γ f d s = ∫ a b f ( r ( t ) ) ∣ r ′ ( t ) ∣ d t \int_\gamma f\,ds = \int_a^b f(\mathbf{r}(t))\,|\mathbf{r}'(t)|\,dt ∫ γ f d s = ∫ a b f ( r ( t )) ∣ r ′ ( t ) ∣ d t mass = ∫ γ ρ d s , f ˉ = 1 L ∫ γ f d s \text{mass} = \int_\gamma \rho\,ds,\quad \bar f = \frac{1}{L}\int_\gamma f\,ds mass = ∫ γ ρ d s , f ˉ = L 1 ∫ γ f d s Line integral of the 2nd kind (work)
∫ γ F ⋅ d r = ∫ a b ( P x ′ + Q y ′ + R z ′ ) d t \int_\gamma \mathbf{F}\cdot d\mathbf{r} = \int_a^b (Px'+Qy'+Rz')\,dt ∫ γ F ⋅ d r = ∫ a b ( P x ′ + Q y ′ + R z ′ ) d t ∫ − γ F ⋅ d r = − ∫ γ F ⋅ d r \int_{-\gamma} \mathbf{F}\cdot d\mathbf{r} = -\int_{\gamma} \mathbf{F}\cdot d\mathbf{r} ∫ − γ F ⋅ d r = − ∫ γ F ⋅ d r Conservative fields and potential
F = ∇ U ⟹ ∫ γ F ⋅ d r = U ( B ) − U ( A ) \mathbf{F}=\nabla U \implies \int_\gamma \mathbf{F}\cdot d\mathbf{r} = U(B)-U(A) F = ∇ U ⟹ ∫ γ F ⋅ d r = U ( B ) − U ( A ) ∮ γ F ⋅ d r = 0 ( conservative field ) \oint_\gamma \mathbf{F}\cdot d\mathbf{r} = 0 \;(\text{conservative field}) ∮ γ F ⋅ d r = 0 ( conservative field ) ∂ y P = ∂ x Q ( on a simply connected domain ) \partial_y P = \partial_x Q \;(\text{on a simply connected domain}) ∂ y P = ∂ x Q ( on a simply connected domain ) Polar coordinates and notable curves
x = r cos θ , y = r sin θ x=r\cos\theta,\quad y=r\sin\theta x = r cos θ , y = r sin θ A = 1 2 ∫ α β r 2 d θ A = \tfrac12\int_\alpha^\beta r^2\,d\theta A = 2 1 ∫ α β r 2 d θ r = a ( 1 + cos θ ) ( cardioid ) r=a(1+\cos\theta)\;(\text{cardioid}) r = a ( 1 + cos θ ) ( 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 N ^ = T ^ ′ /∣ T ^ ′ ∣ , B ^ = T ^ × N ^ T ^ ′ = κ N ^ , B ^ ′ = − τ N ^ \hat T'=\kappa\hat N,\;\hat B'=-\tau\hat N T ^ ′ = κ N ^ , B ^ ′ = − τ N ^ τ = 0 ⟺ plane curve \tau=0 \iff \text{plane curve} τ = 0 ⟺ plane curve Multivariable Differential Calculus
Functions of several variables: domain and level curves
f : R 2 → R , ( x , y ) ↦ f ( x , y ) f:\mathbb{R}^2\to\mathbb{R},\quad (x,y)\mapsto f(x,y) f : R 2 → R , ( x , y ) ↦ f ( x , y ) L c = { x ∈ R n : f ( x ) = c } L_c = \{\mathbf{x}\in\mathbb{R}^n : f(\mathbf{x})=c\} L c = { x ∈ R n : f ( x ) = c } Limits and continuity in ℝⁿ
∀ ε > 0 ∃ δ > 0 : ∣ x − x 0 ∣ < δ ⟹ ∣ f ( x ) − L ∣ < ε \forall\varepsilon>0\;\exists\delta>0:\; |\mathbf{x}-\mathbf{x}_0|<\delta \implies |f(\mathbf{x})-L|<\varepsilon ∀ ε > 0 ∃ δ > 0 : ∣ x − x 0 ∣ < δ ⟹ ∣ f ( x ) − L ∣ < ε different limits on two paths ⇒ ∄ lim \text{different limits on two paths} \Rightarrow \nexists\lim different limits on two paths ⇒ ∄ lim Partial derivatives and differentiability
∂ x f = lim h → 0 f ( x 0 + h , y 0 ) − f ( x 0 , y 0 ) h \partial_x f = \lim_{h\to0}\frac{f(x_0+h,y_0)-f(x_0,y_0)}{h} ∂ x f = h → 0 lim h f ( x 0 + h , y 0 ) − f ( x 0 , y 0 ) f ( x 0 + h ) = f ( x 0 ) + ∇ f ⋅ h + o ( ∣ h ∣ ) f(\mathbf{x}_0+\mathbf{h})=f(\mathbf{x}_0)+\nabla f\cdot\mathbf{h}+o(|\mathbf{h}|) f ( x 0 + h ) = f ( x 0 ) + ∇ f ⋅ h + o ( ∣ h ∣ ) f ∈ C 1 ⇒ f differentiable ⇒ f continuous f\in C^1 \Rightarrow f \text{ differentiable} \Rightarrow f \text{ continuous} f ∈ C 1 ⇒ f differentiable ⇒ f continuous Gradient and directional derivative
∇ f = ( ∂ x f , ∂ y f ) \nabla f = (\partial_x f,\,\partial_y f) ∇ f = ( ∂ x f , ∂ y f ) D v ^ f = ∇ f ⋅ v ^ ≤ ∣ ∇ f ∣ D_{\hat v}f = \nabla f\cdot\hat v \leq |\nabla f| D v ^ f = ∇ f ⋅ v ^ ≤ ∣∇ f ∣ ∇ f ⊥ L c ( level curves ) \nabla f \perp L_c \;(\text{level curves}) ∇ f ⊥ L c ( level curves ) Tangent plane and Taylor formula
z = f 0 + f x ( x − x 0 ) + f y ( y − y 0 ) z = f_0+f_x(x-x_0)+f_y(y-y_0) z = f 0 + f x ( x − x 0 ) + f y ( y − y 0 ) f ( x 0 + h ) ≈ f 0 + ∇ f ⋅ h + 1 2 h T H f h f(\mathbf{x}_0+\mathbf{h})\approx f_0+\nabla f\cdot\mathbf{h}+\tfrac12\mathbf{h}^T H_f\,\mathbf{h} f ( x 0 + h ) ≈ f 0 + ∇ f ⋅ h + 2 1 h T H f h Critical points and classification by the Hessian
∇ f ( x 0 ) = 0 ( critical point ) \nabla f(\mathbf{x}_0)=\mathbf{0}\;(\text{critical point}) ∇ f ( x 0 ) = 0 ( critical point ) H f = ( f x x f x y f y x f y y ) , det H f = f x x f y y − f x y 2 H_f = \begin{pmatrix}f_{xx}&f_{xy}\\f_{yx}&f_{yy}\end{pmatrix},\quad \det H_f=f_{xx}f_{yy}-f_{xy}^2 H f = ( f xx f y x f x y f y y ) , det H f = f xx f y y − f x y 2 det H > 0 , f x x > 0 ⇒ min ; det H < 0 ⇒ saddle \det H>0,\,f_{xx}>0 \Rightarrow \text{min};\;\det H<0\Rightarrow\text{saddle} det H > 0 , f xx > 0 ⇒ min ; det H < 0 ⇒ saddle Chain rule and the implicit function theorem
d d t f ( r ( t ) ) = ∇ f ⋅ r ′ ( t ) \frac{d}{dt}f(\mathbf{r}(t)) = \nabla f\cdot\mathbf{r}'(t) d t d f ( r ( t )) = ∇ f ⋅ r ′ ( t ) F x + F y φ ′ = 0 ⇒ φ ′ ( x ) = − F x F y F_x + F_y\,\varphi' = 0 \Rightarrow \varphi'(x) = -\frac{F_x}{F_y} F x + F y φ ′ = 0 ⇒ φ ′ ( x ) = − F y F x Constrained optimisation: Lagrange multipliers
∇ f = λ ∇ g , g ( x , y ) = 0 \nabla f = \lambda\nabla g,\quad g(x,y)=0 ∇ f = λ ∇ g , g ( x , y ) = 0 { f x = λ g x f y = λ g y g = 0 \begin{cases}f_x=\lambda g_x\\ f_y=\lambda g_y\\ g=0\end{cases} ⎩ ⎨ ⎧ f x = λ g x f y = λ g y g = 0 Vector-Valued Functions and Manifolds
Vector functions: fields, transformations, surfaces
f : R n → R m , f = ( f 1 , … , f m ) f:\mathbb{R}^n\to\mathbb{R}^m,\quad f=(f_1,\dots,f_m) f : R n → R m , f = ( f 1 , … , f m ) r ( u , v ) : R 2 → R 3 ( surface ) \mathbf{r}(u,v):\mathbb{R}^2\to\mathbb{R}^3\;(\text{surface}) r ( u , v ) : R 2 → R 3 ( surface ) The Jacobian matrix
J f = ( ∂ 1 f 1 ⋯ ∂ n f 1 ⋮ ⋮ ∂ 1 f m ⋯ ∂ n f m ) 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} J f = ∂ 1 f 1 ⋮ ∂ 1 f m ⋯ ⋯ ∂ n f 1 ⋮ ∂ n f m f ( x 0 + h ) ≈ f ( x 0 ) + J f h f(\mathbf{x}_0+\mathbf{h})\approx f(\mathbf{x}_0)+J_f\,\mathbf{h} f ( x 0 + h ) ≈ f ( x 0 ) + J f h ∣ det J f ∣ = volume-scaling factor |\det J_f| = \text{volume-scaling factor} ∣ det J f ∣ = volume-scaling factor Vector chain rule
J f ∘ g ( x ) = J f ( g ( x ) ) ⋅ J g ( x ) J_{f\circ g}(\mathbf{x}) = J_f(g(\mathbf{x}))\cdot J_g(\mathbf{x}) J f ∘ g ( x ) = J f ( g ( x )) ⋅ J g ( x ) ( m × n ) ⋅ ( n × p ) = m × p (m\times n)\cdot(n\times p) = m\times p ( m × n ) ⋅ ( n × p ) = m × p The inverse function theorem
det J f ( x 0 ) ≠ 0 ⟹ f locally invertible \det J_f(\mathbf{x}_0)\neq0 \implies f \text{ locally invertible} det J f ( x 0 ) = 0 ⟹ f locally invertible J f − 1 = [ J f ] − 1 J_{f^{-1}}=[J_f]^{-1} J 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}) det ( ∂ F / ∂ y ) = 0 ⇒ y = g ( x ) J g = − ( ∂ F ∂ y ) − 1 ∂ F ∂ x J_g = -\!\left(\frac{\partial F}{\partial\mathbf{y}}\right)^{-1}\!\frac{\partial F}{\partial\mathbf{x}} J g = − ( ∂ y ∂ F ) − 1 ∂ x ∂ F Parametric surfaces, normal and area
n = r u × r v \mathbf{n} = \mathbf{r}_u\times\mathbf{r}_v n = r u × r v d S = ∣ r u × r v ∣ d u d v dS = |\mathbf{r}_u\times\mathbf{r}_v|\,du\,dv d S = ∣ r u × r v ∣ d u d v A = ∬ D ∣ r u × r v ∣ d u d v A = \iint_D|\mathbf{r}_u\times\mathbf{r}_v|\,du\,dv A = ∬ D ∣ r u × r v ∣ d u d v Lagrange multipliers (several constraints)
∇ f = λ ∇ g , g = 0 \nabla f = \lambda\nabla g,\quad g=0 ∇ f = λ ∇ g , g = 0 ∇ f = ∑ i = 1 k λ i ∇ g i , g 1 = ⋯ = g k = 0 \nabla f = \sum_{i=1}^k\lambda_i\nabla g_i,\quad g_1=\cdots=g_k=0 ∇ f = i = 1 ∑ k λ i ∇ g i , g 1 = ⋯ = g k = 0 Change of coordinates and the Jacobian in integrals
∬ T ( D ) f d x d y = ∬ D f ( T ) ∣ det J T ∣ d u d v \iint_{T(D)} f\,dx\,dy = \iint_D f(T)\,|\det J_T|\,du\,dv ∬ T ( D ) f d x d y = ∬ D f ( T ) ∣ det J T ∣ d u d v d x d y = r d r d θ ( polar ) dx\,dy = r\,dr\,d\theta\;(\text{polar}) d x d y = r d r d θ ( polar ) d V = ρ 2 sin ϕ d ρ d ϕ d θ ( spherical ) dV = \rho^2\sin\phi\,d\rho\,d\phi\,d\theta\;(\text{spherical}) d V = ρ 2 sin ϕ d ρ d ϕ d θ ( spherical ) Multiple Integrals
From the single to the double integral
∬ D f d A = lim Δ A → 0 ∑ i f ( x i , y i ) Δ A \iint_D f\,dA = \lim_{\Delta A\to0}\sum_i f(x_i,y_i)\,\Delta A ∬ D f d A = Δ A → 0 lim i ∑ f ( x i , y i ) Δ A ∬ D 1 d A = area ( D ) \iint_D 1\,dA = \text{area}(D) ∬ D 1 d A = area ( D ) Fubini's theorem (iterated integration)
∬ R f d A = ∫ a b ∫ c d f d y d x = ∫ c d ∫ a b f d x d y \iint_R f\,dA = \int_a^b\!\int_c^d f\,dy\,dx = \int_c^d\!\int_a^b f\,dx\,dy ∬ R f d A = ∫ a b ∫ c d f d y d x = ∫ c d ∫ a b f d x d y f continuous ⇒ order swappable f \text{ continuous} \Rightarrow \text{order swappable} f continuous ⇒ order swappable Normal domains and swapping the order
D y -norm = { a ≤ x ≤ b , g 1 ( x ) ≤ y ≤ g 2 ( x ) } D_{y\text{-norm}}=\{a\le x\le b,\ g_1(x)\le y\le g_2(x)\} D y -norm = { a ≤ x ≤ b , g 1 ( x ) ≤ y ≤ g 2 ( x )} ∬ D f = ∫ a b ∫ g 1 ( x ) g 2 ( x ) f d y d x \iint_D f = \int_a^b\!\int_{g_1(x)}^{g_2(x)} f\,dy\,dx ∬ D f = ∫ a b ∫ g 1 ( x ) g 2 ( x ) f d y d x Polar coordinates in the plane
x = r cos θ , y = r sin θ , d A = r d r d θ x=r\cos\theta,\ y=r\sin\theta,\quad dA = r\,dr\,d\theta x = r cos θ , y = r sin θ , d A = r d r d θ ∬ D f d A = ∫ α β ∫ 0 R ( θ ) f r d r d θ \iint_D f\,dA = \int_\alpha^\beta\!\int_0^{R(\theta)} f\,r\,dr\,d\theta ∬ D f d A = ∫ α β ∫ 0 R ( θ ) f r d r d θ The triple integral
∭ V f d V = ∫ a b ∫ g 1 g 2 ∫ h 1 h 2 f d z d y d x \iiint_V f\,dV = \int_a^b\!\int_{g_1}^{g_2}\!\int_{h_1}^{h_2} f\,dz\,dy\,dx ∭ V f d V = ∫ a b ∫ g 1 g 2 ∫ h 1 h 2 f d z d y d x ∭ V 1 d V = volume ( V ) \iiint_V 1\,dV = \text{volume}(V) ∭ V 1 d V = volume ( V ) Cylindrical and spherical coordinates
d V cyl = r d r d θ d z dV_{\text{cyl}} = r\,dr\,d\theta\,dz d V cyl = r d r d θ d z d V sph = ρ 2 sin ϕ d ρ d ϕ d θ dV_{\text{sph}} = \rho^2\sin\phi\,d\rho\,d\phi\,d\theta d V sph = ρ 2 sin ϕ d ρ d ϕ d θ x 2 + y 2 + z 2 = ρ 2 x^2+y^2+z^2 = \rho^2 x 2 + y 2 + z 2 = ρ 2 The general change-of-variables formula
∬ D f d x d y = ∬ D ∗ f ( Φ ) ∣ det J Φ ∣ d u d v \iint_D f\,dx\,dy = \iint_{D^*} f(\Phi)\,|\det J_\Phi|\,du\,dv ∬ D f d x d y = ∬ D ∗ f ( Φ ) ∣ det J Φ ∣ d u d v J Φ = ∂ ( x , y ) ∂ ( u , v ) J_\Phi = \frac{\partial(x,y)}{\partial(u,v)} J Φ = ∂ ( u , v ) ∂ ( x , y ) Applications: mass, centroid, moment of inertia
m = ∭ V ρ d V m = \iiint_V \rho\,dV m = ∭ V ρ d V x ˉ = 1 m ∭ V x ρ d V \bar x = \frac{1}{m}\iiint_V x\,\rho\,dV x ˉ = m 1 ∭ V x ρ d V I z = ∭ V ( x 2 + y 2 ) ρ d V I_z = \iiint_V (x^2+y^2)\,\rho\,dV I z = ∭ V ( x 2 + y 2 ) ρ d V Vector Fields and Integral Theorems
What a vector field is
F : R n → R n , x ↦ F ( x ) \mathbf{F}:\mathbb{R}^n\to\mathbb{R}^n,\quad \mathbf{x}\mapsto\mathbf{F}(\mathbf{x}) F : R n → R n , x ↦ F ( x ) r ′ ( t ) = F ( r ( t ) ) ( flow lines ) \mathbf{r}'(t) = \mathbf{F}(\mathbf{r}(t))\;(\text{flow lines}) r ′ ( t ) = F ( r ( t )) ( flow lines ) Divergence and curl
∇ ⋅ F = ∂ x P + ∂ y Q + ∂ z R \nabla\cdot\mathbf{F} = \partial_x P+\partial_y Q+\partial_z R ∇ ⋅ F = ∂ x P + ∂ y Q + ∂ z R ∇ × F = ( R y − Q z , P z − R x , Q x − P y ) \nabla\times\mathbf{F} = (R_y-Q_z,\;P_z-R_x,\;Q_x-P_y) ∇ × 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} ∇ ⋅ F > 0 : source , < 0 : 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}) F = ∇ U ⟺ ∇ × F = 0 ( s.c. domain ) ∇ × ( ∇ U ) = 0 \nabla\times(\nabla U)=\mathbf{0} ∇ × ( ∇ U ) = 0 ∮ γ F ⋅ d r = 0 on every closed curve \oint_\gamma\mathbf{F}\cdot d\mathbf{r}=0 \text{ on every closed curve} ∮ γ F ⋅ d r = 0 on every closed curve Surface integral (flux)
Φ = ∬ Σ F ⋅ n ^ d S \Phi = \iint_\Sigma\mathbf{F}\cdot\hat n\,dS Φ = ∬ Σ F ⋅ n ^ d S = ∬ D F ⋅ ( r u × r v ) d u d v = \iint_D\mathbf{F}\cdot(\mathbf{r}_u\times\mathbf{r}_v)\,du\,dv = ∬ D F ⋅ ( r u × r v ) d u d v Green's theorem in the plane
∮ ∂ D ( P d x + Q d y ) = ∬ D ( Q x − P y ) d A \oint_{\partial D}(P\,dx+Q\,dy) = \iint_D(Q_x-P_y)\,dA ∮ ∂ D ( P d x + Q d y ) = ∬ D ( Q x − P y ) d A A = 1 2 ∮ ∂ D ( x d y − y d x ) A = \tfrac12\oint_{\partial D}(x\,dy-y\,dx) A = 2 1 ∮ ∂ D ( x d y − y d x ) The Divergence theorem (Gauss)
∯ ∂ V F ⋅ d S = ∭ V ( ∇ ⋅ F ) d V \oiint_{\partial V}\mathbf{F}\cdot d\mathbf{S} = \iiint_V(\nabla\cdot\mathbf{F})\,dV ∬ ∂ V F ⋅ d S = ∭ V ( ∇ ⋅ F ) d V ∇ ⋅ E = ρ / ε 0 ( Gauss ) \nabla\cdot\mathbf{E} = \rho/\varepsilon_0\;(\text{Gauss}) ∇ ⋅ E = ρ / ε 0 ( Gauss ) Stokes' theorem
∮ ∂ Σ F ⋅ d r = ∬ Σ ( ∇ × F ) ⋅ d S \oint_{\partial\Sigma}\mathbf{F}\cdot d\mathbf{r} = \iint_\Sigma(\nabla\times\mathbf{F})\cdot d\mathbf{S} ∮ ∂ Σ F ⋅ d r = ∬ Σ ( ∇ × F ) ⋅ d S ∮ B ⋅ d r = μ 0 I ( Amp e ˋ re ) \oint\mathbf{B}\cdot d\mathbf{r}=\mu_0 I\;(\text{Ampère}) ∮ B ⋅ d r = μ 0 I ( Amp e ˋ re ) The three formulas as one idea
∇ ⋅ ( ∇ × F ) = 0 \nabla\cdot(\nabla\times\mathbf{F})=0 ∇ ⋅ ( ∇ × F ) = 0 ∫ Ω d ω = ∫ ∂ Ω ω ( general Stokes ) \int_\Omega d\omega=\int_{\partial\Omega}\omega\;(\text{general Stokes}) ∫ Ω d ω = ∫ ∂ Ω ω ( general Stokes ) Power Series, Fourier Series, and Transforms
Numerical series — convergence tests
lim ∣ a n + 1 / a n ∣ = L < 1 ⟹ abs. conv. \lim|a_{n+1}/a_n| = L < 1 \implies \text{abs. conv.} lim ∣ a n + 1 / a n ∣ = L < 1 ⟹ abs. conv. lim sup ∣ a n ∣ n = L < 1 ⟹ abs. conv. \limsup\sqrt[n]{|a_n|} = L < 1 \implies \text{abs. conv.} lim sup n ∣ a n ∣ = L < 1 ⟹ abs. conv. ∑ ( − 1 ) n b n , b n ↘ 0 ⟹ converges (Leibniz) \sum(-1)^n b_n,\; b_n\searrow0 \implies \text{converges (Leibniz)} ∑ ( − 1 ) n b n , b n ↘ 0 ⟹ converges (Leibniz) ∑ n = 1 ∞ 1 / n p converges ⟺ p > 1 \sum_{n=1}^\infty 1/n^p \text{ converges } \iff p>1 n = 1 ∑ ∞ 1/ n p converges ⟺ p > 1 Power series and radius of convergence
R = 1 / lim sup ∣ c n ∣ n R = 1/\limsup\sqrt[n]{|c_n|} R = 1/ lim sup n ∣ c n ∣ ∣ x − x 0 ∣ < R ⟹ abs. conv. |x-x_0|<R \implies \text{abs. conv.} ∣ x − x 0 ∣ < R ⟹ abs. conv. f ′ ( x ) = ∑ n c n ( x − x 0 ) n − 1 ( same R ) f'(x) = \sum n c_n (x-x_0)^{n-1} \;(\text{same }R) f ′ ( x ) = ∑ n c n ( x − x 0 ) n − 1 ( same R ) Taylor series and notable expansions
f ( x ) = ∑ n = 0 ∞ f ( n ) ( x 0 ) n ! ( x − x 0 ) n f(x)=\sum_{n=0}^\infty\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n f ( x ) = n = 0 ∑ ∞ n ! f ( n ) ( x 0 ) ( x − x 0 ) n e x = ∑ x n / n ! , sin x = ∑ ( − 1 ) n x 2 n + 1 / ( 2 n + 1 ) ! e^x = \sum x^n/n!,\; \sin x = \sum(-1)^n x^{2n+1}/(2n+1)! e x = ∑ x n / n ! , sin x = ∑ ( − 1 ) n x 2 n + 1 / ( 2 n + 1 )! ln ( 1 + x ) = ∑ ( − 1 ) n − 1 x n / n , ∣ x ∣ ≤ 1 \ln(1+x)=\sum(-1)^{n-1}x^n/n,\; |x|\leq1 ln ( 1 + x ) = ∑ ( − 1 ) n − 1 x n / n , ∣ x ∣ ≤ 1 ( α n ) = α ( α − 1 ) ⋯ ( α − n + 1 ) n ! \binom{\alpha}{n} = \frac{\alpha(\alpha-1)\cdots(\alpha-n+1)}{n!} ( n α ) = n ! α ( α − 1 ) ⋯ ( α − n + 1 ) Fourier series — periods and coefficients
f ( x ) ∼ a 0 2 + ∑ n = 1 ∞ ( a n cos n π x L + b n sin n π x L ) 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) f ( x ) ∼ 2 a 0 + n = 1 ∑ ∞ ( a n cos L nπ x + b n sin L nπ x ) a n = 1 L ∫ − L L f ( x ) cos n π x L d x , b n = 1 L ∫ − L L f ( x ) sin n π x L d x a_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 a n = L 1 ∫ − L L f ( x ) cos L nπ x d x , b n = L 1 ∫ − L L f ( x ) sin L nπ x d x c n = 1 2 L ∫ − L L f ( x ) e − i n π x / L d x c_n = \frac1{2L}\int_{-L}^{L} f(x)e^{-i n\pi x/L}\,dx c n = 2 L 1 ∫ − L L f ( x ) e − inπ x / L d x f even ⇒ b n = 0 , f odd ⇒ a n = 0 f\text{ even}\Rightarrow b_n=0,\quad f\text{ odd}\Rightarrow a_n=0 f even ⇒ b n = 0 , f odd ⇒ 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) f ( x ± ) = h → 0 + lim f ( x ± h ) Dirichlet: f ( x + ) + f ( x − ) 2 at discontinuities \text{Dirichlet: } \frac{f(x^+)+f(x^-)}{2} \text{ at discontinuities} Dirichlet: 2 f ( x + ) + f ( x − ) at discontinuities Parseval: 1 2 L ∫ − L L ∣ f ∣ 2 = ∑ − ∞ ∞ ∣ c n ∣ 2 \text{Parseval: } \frac1{2L}\int_{-L}^{L}|f|^2 = \sum_{-\infty}^{\infty}|c_n|^2 Parseval: 2 L 1 ∫ − L L ∣ f ∣ 2 = − ∞ ∑ ∞ ∣ c n ∣ 2 Fourier transform — from discrete to continuous
f ^ ( ξ ) = ∫ − ∞ ∞ f ( x ) e − 2 π i ξ x d x \hat f(\xi)=\int_{-\infty}^{\infty} f(x)e^{-2\pi i\xi x}\,dx f ^ ( ξ ) = ∫ − ∞ ∞ f ( x ) e − 2 π i ξ x d x f ( x ) = ∫ − ∞ ∞ f ^ ( ξ ) e 2 π i ξ x d ξ f(x)=\int_{-\infty}^{\infty} \hat f(\xi)e^{2\pi i\xi x}\,d\xi f ( x ) = ∫ − ∞ ∞ f ^ ( ξ ) e 2 π i ξ x d ξ f ′ ^ = 2 π i ξ f ^ , f ∗ g ^ = f ^ g ^ \widehat{f'}=2\pi i\xi\,\hat f,\quad \widehat{f*g}=\hat f\,\hat g f ′ = 2 π i ξ f ^ , f ∗ g = f ^ g ^ Laplace transform
F ( s ) = ∫ 0 ∞ f ( t ) e − s t d t F(s)=\int_0^\infty f(t)e^{-st}\,dt F ( s ) = ∫ 0 ∞ f ( t ) e − s t d t L { f ′ } = s F ( s ) − f ( 0 ) , L { f ′ ′ } = s 2 F ( s ) − s f ( 0 ) − f ′ ( 0 ) \mathcal{L}\{f'\}=sF(s)-f(0),\; \mathcal{L}\{f''\}=s^2F(s)-s f(0)-f'(0) L { f ′ } = s F ( s ) − f ( 0 ) , L { f ′′ } = s 2 F ( s ) − s f ( 0 ) − f ′ ( 0 ) L { e a t } = 1 / ( s − a ) , L { sin ω t } = ω / ( s 2 + ω 2 ) \mathcal{L}\{e^{at}\}=1/(s-a),\; \mathcal{L}\{\sin\omega t\}=\omega/(s^2+\omega^2) L { e a t } = 1/ ( s − a ) , L { sin ω t } = ω / ( s 2 + ω 2 ) Applications: ODEs, circuits, and control
L { y ( n ) } = s n Y ( s ) − s n − 1 y ( 0 ) − ⋯ − y ( n − 1 ) ( 0 ) \mathcal{L}\{y^{(n)}\} = s^nY(s) - s^{n-1}y(0) - \dots - y^{(n-1)}(0) L { y ( n ) } = s n Y ( s ) − s n − 1 y ( 0 ) − ⋯ − y ( n − 1 ) ( 0 ) lim t → ∞ f ( t ) = lim s → 0 s F ( s ) ( final value ) \lim_{t\to\infty} f(t) = \lim_{s\to0} sF(s) \;(\text{final value}) t → ∞ lim f ( t ) = s → 0 lim s F ( s ) ( final value ) H ( s ) = Y ( s ) / U ( s ) ( transfer function ) H(s) = Y(s)/U(s) \;(\text{transfer function}) H ( s ) = Y ( s ) / U ( s ) ( transfer function ) 
