Cheat Sheet Multivariate Calculus

Cheat Sheet Multivariate Calculus Limits and continuity for functions of several variables • Limits (f : Rn → R): lim f (x) = A x→a ⇐⇒ |x−a|→0 lim |f (x) − A| = 0 • A function f is continuous at a point a if lim f (x) = f (a). A function f ∈ C(Ω) if f is continuous at x→a every point of the set Ω. Derivatives, differentiability, tangent planes and normal lines, the chain rule • Partial derivatives for a function of 2 variables at a point (a, b): f (a + h, b) − f (a, b) ∂f ′ (a, b) = Dx f (a, b) = D1 f (a, b) = fx (a, b) = lim h→0 ∂x h ∂f f (a, b + k) − f (a, b) ′ (a, b) = Dy f (a, b) = D2 f (a, b) = fy (a, b) = lim k→0 ∂y k Partial derivatives for functions of more than 2 variables are defined in a similar way. • The gradient of f : grad f (x) = ∇f (x) = ( ∂f ∂f (x), . . . , (x))T . ∂x1 ∂xn • A function f : Rn → R is differentiable at a point a = (a1 , · · · , an ) if f (a + h) = f (a) + ∇f (a) · h + |h|ρ(h) in a neighbourhood of a, so that ρ(h) → 0, as h → 0. A differentiable function of two variables: ′ ′ f (a + h, b + k) = f (a, b) + fx (a, b)(x − a) + fy (a, b)(y − b) + h2 + k 2 ρ(h, k). • Tangent plane Π and normal line N to the surface z = f (x, y) at the point (a, b):      ′  x a fx (a, b) ′ ′ ′ Π : z = f (a, b) + fx (a, b)(x − a) + fy (a, b)(y − b); N :  y  =  b  + t  fy (a, b)  , t ∈ R. z f (a, b) −1 • Tangent plane Π and normal line N to the surface F (x, y, z) = 0 at the point (a, b, c):       x−a x a Π : ∇F (a, b, c) ·  y − b  = 0 N :  y  =  b  + t∇F (a, b, c), t ∈ R. z−c z c • The linearization and total derivative of f : Rn → R at a point a: ′ ′ ∆f (a) = f (a + ∆x) − f (a) = ∇f (a) · ∆x = fx1 (a)∆x1 + · · · + fxn (a)∆xn ′ ′ df (a) = ∇f (a) · dx = fx1 (a)dx1 + · · · + fxn (a)dxn ′ ′′ ′ ′′ • Higher-order derivatives are defined recursively, e.g., fxx = (fx )′ , fxy = (fx )′ , etc. A f-n f ∈ C n (Ω) y x if f and all derivatives up to order n are continuous functions at every point of Ω. def def • The mixed derivatives theorem gives a sufficient condition for equality of the mixed derivatives, e.g.: f : R2 → R, f ∈ C(Ω) =⇒ ′′ ′′ fxy = fyx • The chain rule. If f : Rn → R, r = r(t) = (x1 (t), . . . , xn (t)), then: d ′ ′ f (r(t)) = fx1 (r(t))x′ (t) + · · · fxn (r(t))x′ (t) = ∇f (r(t)) · r′ (t) n 1 dt Taylor formula of order n for a smooth function of two variables n f (x0 + h, y0 + k) = k=0 1 (hDx + kDy )k f (x0 , y0 ) + O(|(h, k)|n+1 ). k! May 27, 2008 file: formelblad-fv.pdf Extremes of functions • A point x0 is critical (stationary) if ∇f (x0 ) = 0. A point x0 is singular if ∇f (x0 ) does not exist. • Local extremes: x0 is a loc max pt of f if f (x0 ) > f (x) in some nb-hood of x0 ; loc min, if f (x0 ) < f (x). If f has a local extreme and is differentiable at x0 , then x0 is a critical point of f , i.e ∇f (x0 ) = 0. • Global extremes: x0 is a global max pt of f if f (x0 ) ≥ f (x) (glob min, if f (x0 ) ≤ f (x)) for all x ∈ Df . • If f ∈ C(Df ) and D is compact, f reaches both it’s global max and min in D. Sufficient conditions for local extremes of smooth functions ′′ • Hessian: the square matrix of the second partial derivatives H(x0 ) = [fxi xj (x0 )], 1 ≤ i, j ≤ n, computed at the pt x0 . If the mixed derivatives are equal (e.g. if f is C 2 at x0 ), then H(x0 ) is symmetric. n n ′′ ′′ ′′ fxi xj (x0 )hi hj = hT H(x0 )h. For n = 2, and fxy (x0 ) = fyx (x0 ), then i=1 j=1 ′′ ′′ ′′ Q(h, k) = fxx (x0 )h2 + 2fxy (x0 )hk + fyy (x0 )k 2 = (h, k) ′′ ′′ fxx (x0 ) fxy (x0 ) ′′ ′′ fxy (x0 ) fyy (x0 ) • Qadratic form Q(h) = h k . In a neighburhood of a critical pt x0 one has: f (x0 + h) − f (x) = Q(h, k) + O(|(h, k)|3 ). • Type of the crtical point x0 (i.e. ∇f (x0 ) = 0) depending on the sign of Q(h) at x0 for h = 0: Q(h) Q’s type The pt x0 is changes sign indefinite a saddle pt Q(h) > 0 positive definite a local min Q(h) < 0 negative definite a local max Q(h) ≤ 0 or Q(h) ≥ 0 postive/negative semi–definite further investigation needed Optimization under constraints. Lagrange multipliers Necessary conditions for local extremes when looking for • • max/min f (x, y) g(x, y) = 0 . max/min f (x, y, z) In the crtitical points ∇f = λ∇g or ∇g = 0, i.e. −∇f − = 0. −∇g− g(x, y, z) = 0 . In the crtitical points ∇f = λ∇g or ∇g = 0, i.e. ∇f × ∇g = 0. −∇f − In the crtitical points ∇f = λ∇g1 + µ∇g2 , i.e. −∇g1 − = 0. −∇g2 − Two variants of the implicit function theorem • Let γ be the curve γ : F (x, y) = 0 and (a, b) ∈ γ. Then, if F ∈ C 1 in a neighbourhood of (a, b) and ′ Fy (a, b) = 0, then there exists a C 1 -function y = y(x), such that F (x, y(x)) ≡ 0 in some neighbourhood ′ ′ of x = a. Moreover, y ′ (a) = −Fx (a, b)/Fy (a, b). • Let S be the surface S : F (x, y, z) = 0 and (a, b, c) ∈ S. If F ∈ C 1 in a nb-hood of (a, b, c) and ′ Fz (a, b, c) = 0, then there exists a C 1 -function z(x, y), such that F (x, y, z(x, y)) ≡ 0 in some nb-hood ′ ′ ′ ′ ′ ′ of (x, y) = (a, b). Moreover, zx (a, b) = −Fx (a, b, c)/Fz (a, b, c) and zy (a, b) = −Fy (a, b, c)/Fz (a, b, c).   max/min f (x, y, z) g1 (x, y, z) = 0 •  g2 (x, y, z) = 0 . Gradient, Curl and Divergence ∂ ∂ Notation: Φ : Rn → R is a scalar field, F : Rn → Rn is a vector field, ∇ = ( ∂x1 , . . . , ∂xn ) • Gradient: grad Φ = ∇Φ = (Φ′ 1 , . . . , Φ′ n ) x x def • Divergence: div F = ∇ · F = def ∂F1 ∂x1 + ···+ def ∂Fn ∂xn . F is solenoidal if div F = 0. ′ ′ ′ ′ • Curl (in R3 , F = (P, Q, R)): curl F = ∇ × F = (Ry − Q′ , Pz − Rx , Q′ − Py ). F is curl-free if ∇ × F = 0 z x • F is conservative if F = grad Φ, Φ is called a potential for F. Every conservative C 2 -field is curl-free • The Laplacian: ∆ = ∇ · ∇ = def ∂2 ∂2 n 2 + · · · + ∂x2 . If f : R → R satisfies ∆f = 0, then f is harmonic ∂x1 n s. 2 av 4 Cheatsheet Multivariate Calculus Double integrals • Iterated integration (Fubini theorem): b ψ(x) f (x, y) dxdy = D a ϕ(x) f (x, y) dy dx, D = {ϕ(x) ≤ y ≤ ψ(x), a ≤ x ≤ b} • Change of variables, D ↔ D′ one-to-one by f (x, y) dxdy = D D′ x = x(u, v) ⇐⇒ y = y(u, v) u = u(x, y) v = v(x, y) J= d(x, y) = d(u, v) 1 d(u,v) d(x,y) f (x(u, v), y(u, v)) J dudv, • Polar coordinates: x = r cos ϕ d(x, y) y = r sin ϕ , dxdy = d(r, ϕ) = r drdϕ Triple integrals • Iterated integration (Fubini theorem): ψ(x,y) f (x, y, z) dxdydz = Ω D ϕ(x,y) f (x, y, z) dz f (x, y, z) dxdy, Ω= ϕ(x, y) ≤ z ≤ ψ(x, y) (x, y) ∈ D    x = x(u, v, w)  u = u(x, y, z) • Change of variables, Ω ↔ Ω′ one-to-one by y = y(u, v, w) ⇐⇒ v = v(x, y, z)   z = z(u, v, w) w = w(x, y, z) f (x, y, z) dV = f (x(u, v, w), y(u, v, w), z(u, v, w)) J dV ′ , J= Ω Ω′ d(x, y, z) = d(u, v, w) 1 d(u,v,w) d(x,y,z) • Cylindrical coordinates: x = r cos ϕ d(x, y, z) y = r sin ϕ , dxdydz = = r drdϕdz d(r, ϕ, z) z=z x = r sin ϑ cos ϕ d(x, y, z) = r2 sin ϑ drdϑdϕ y = r sin ϑ sin ϕ , dxdydz = d(r, ϑ, ϕ) z = r cos ϑ • Spherical coordinates: Line integrals over scalar fields • If f : Rn → R and γ is a C 1 -curve with parametrization γ : r = r(t), a < t < b, then b f |dr| = γ a f (r(t)) |r′ (t)| dt, |r′ (t)| = (x′ (t))2 + · · · (x′ (t))2 . n 1 • The length of γ s given by γ |dr|. Line integrals over vector fields ˆ • F : Rn → Rn and T is the unit tangent of the oriented C 1 -curve γ : r = r(t), t : a → b, then ˆ F · T |dr| = γ γ b F · dr = a F(r(t)) · r′ (t) dt • Common notation for line integrals in R2 and R3 when F(x, y) = (P, Q) and F(x, y, z) = (P, Q, R): P dx + Q dy + R dz γ and γ P dx + Q dy Cheatsheet Multivariate Calculus s. 3 av 4 • Common notation for line integrals along closed paths (circulations): γ F · dr. Positive direction of transversal is counterclockwise. • Independence of path: If F is conservative with a potential Φ in an open simply connected domain Ω and γ ⊂ Ω is an arbitrary piecewise C 1 -curve from the point A to B, then F · dr = γ γ ∇Φ · dr = Φ(B) − Φ(A). • The Green formula: If Ω ⊂ R2 is an open connected domain with positively oriented piecewise C 1 boundary ∂Ω and F(x, y) = (P, Q) ∈ C 1 (Ω ∪ ∂Ω), then F · dr = ∂Ω ∂Ω P dx + Q dy = Ω ′ (Q′ − Py ) dxdy x Surface integrals • If f : R3 → R and S is a C 1 -surface with parametrization S : r = r(u, v), (u, v) ∈ Ω ⊂ R2 , then f dS = S Ω f (r(u, v)) |r′ × r′ | dudv v u • If S is the functional surface S : z = z(x, y), (x, y) ∈ Ω ⊂ R2 , then f dS = S Ω ′ ′ f (x, y, z(x, y)) 1 + (zx )2 + (zy )2 dxdy • If f ≡ 1, then S dS gives the area of S Flux integrals Notation: F : R3 → R3 , S is a piecewise orientable C 1 -surface, N is unit normal to S. • If S : r = r(u, v), (u, v) ∈ Ω ⊂ R2 , then F · N dS = ± S Ω F(r(u, v)) · (r′ × r′ ) dudv u v • If S is the functional surface S : z = z(x, y), (x, y) ∈ Ω ⊂ R2 , then  ′ −zx ′ F · N dS = ± F(x, y, z(x, y)) ·  −zy  dxdy 1 S Ω • The Gauss theorem (the divergence theorem): If S is a closed surface, boundary to a domain V ⊂ R3 , with an outer unit normal N then F · N dS = S V div F dV = V ∇ · F dV = V ′ ′ (Px + Q′ + Rz ) dV y • The Stokes theorem: If S has boundary ∂S, then the circulation (counterclockwise seen from N) F · dr = ∂S S curl F · N dS = S (∇ × F) · N dS. Cheatsheet Multivariate Calculus s. 4 av 4