MA 261 - Spring 2011Study Guide # 20. Gradient vector for f(x, y): ∇f(x, y) =⟨∂f∂x,∂f∂y⟩, properties of gradients; gradient points indirection of maximum rate of increase of f; The maximum value of the directional derivative isequal to ∥∇f∥; ∇f(x0, y0) ⊥ level curve f(x, y) = C and, in the case of 3 variables, ∇f (x0, y0, z0) ⊥level surface f(x, y, z) = C:0(x ,y )xyf(x,y,z)=Cxy(x ,y ,z )0 0 0n = ∆0n = ∆zf(x ,y )00f(x,y)=Cf(x ,y ,z )0001. Relative/local extrema; critical points (∇f =0 or ∇f does not exist); 2ndDerivatives Test: Acritical points is a local min if D = fxxfyy− f2xy> 0 and fxx> 0, local max if D > 0 and fxx< 0,saddle if D < 0; absolute extrema; Max-Min Problems; Lagrange Multipliers: Extremize f (x)subject to a constraint g(x) = C, solve the system: ∇f = λ∇g and g(x) = C.2. Double integrals; Midpoint Rule for rectangle :∫∫Rf(x, y) dA ≈m∑i=1n∑j=1f(xi, yj) ∆A;3. Type I region D :{g1(x) ≤ y ≤ g2(x)a ≤ x ≤ b; Type II region D :{h1(y) ≤ x ≤ h2(y)c ≤ y ≤ d;iterated integrals over Type I and II regions:∫∫Df(x, y) dA =∫ba∫g2(x)g1(x)f(x, y) dy dx and∫∫Df(x, y) dA =∫dc∫h2(y)h1(y)f(x, y) dx dy, respectively; Reversing Order of Integration (regionsthat are both Type I and Type II); properties of double integrals.4. Integral inequalities: mA ≤∫∫Df(x, y) dA ≤ MA, where A = area of D and m ≤ f (x, y) ≤ Mon D.5. Change of Variables Formula in Polar Coordinates: if D :{h1(θ) ≤ r ≤ h2(θ)α ≤ θ ≤ β, then∫∫Df(x, y) dA =∫βα∫h2(θ)h1(θ)f(r cos θ, r sin θ) r dr dθ.↑6. Applications of double integrals:(a) Area of region D is A(D) =∫∫DdA(b) Volume of solid under graph of z = f(x, y), where f(x, y) ≥ 0, is V =∫∫Df(x, y) dA(c) Mass of D is m =∫∫Dρ(x, y) dA, where ρ(x, y) = density (per unit area); sometimes writem =∫∫Ddm, where dm = ρ(x, y) dA.(d) Moment ab out the x-axis Mx=∫∫Dy ρ(x, y) dA; moment about the y-axis My=∫∫Dx ρ(x, y) dA.(e) Center of mass (x, y), where x =Mym=∫∫Dx ρ(x, y) dA∫∫Dρ(x, y) dA, y =Mxm=∫∫Dy ρ(x, y) dA∫∫Dρ(x, y) dARemark: centroid = center of mass when density is constant (this is useful).7. Elementary solids E ⊂ R3of Type 1, Type 2, Type 3; triple integrals over solids E:∫∫∫Ef(x, y, z) dV =∫∫D∫v(x,y)u(x,y)f(x, y, z) dz dA for E = {(x, y) ∈ D, u(x, y) ≤ z ≤ v(x, y)};volume of solid E is V (E) =∫∫∫EdV ; applications of triple integrals, mass of a solid, momentsabout the coordinate planes Mxy, Mxz, Myz, center of mass of a solid (x, y, z).8. Cylindrical Coordinates (r, θ, z):From CC to RC :x = r cos θy = r sin θz = zxzzy(x,y,z)0θrGoing from RC to CC use x2+ y2= r2and tan θ =yx(make sure θ is in correct quadrant).9. Spherical Coordinates (ρ, θ, ϕ), where 0 ≤ ϕ ≤ π:From SC to RC :x = (ρ sin ϕ) cos θy = (ρ sin ϕ) sin θz = ρ cos ϕxzy(x,y,z)0θφρρ cos φφsinρGoing from RC to SC use x2+ y2+ z2= ρ2, tan θ =yxand cos ϕ =zρ.10. Triple integrals in Cylindrical Coordinates:x = r cos θy = r sin θz = z, dV = r dz dr dθ∫∫∫Ef(x, y, z) dV =∫∫∫Ef(r cos θ, r sin θ , z) r dz dr dθ↑11. Triple integrals in Spherical Coordinates:x = (ρ sin ϕ) cos θy = (ρ sin ϕ) sin θz = ρ cos ϕ, dV = ρ2sin ϕ dρ dϕ dθ∫∫∫Ef(x, y, z) dV =∫∫∫Ef(ρ sin ϕ cos θ, ρ sin ϕ sin θ, ρ cos ϕ) ρ2sin ϕ dρ dϕ dθ↑12. Vector fields on R2and R3:F(x, y) = ⟨P (x, y), Q(x, y)⟩ andF(x, y, z) = ⟨P (x, y), Q(x, y), R(x, y)⟩;F is a conservative vector field ifF = ∇f, for some real-valued function f