MA 261 - Spring 2011Study Guide # 11. Vectors in R2and R3(a)v = ⟨a, b, c⟩ = ai + bj + ck; vector addition and subtraction geometrically using paral-lelograms spanned byu andv; length or magnitude ofv = ⟨a, b, c⟩, |v| =√a2+ b2+ c2;directed vector from P0(x0, y0, z0) to P1(x1, y1, z1) given byv = P0P1= P1− P0=⟨x1− x0, y1− y0, z1− z0⟩.(b) Dot (or inner) product ofa = ⟨a1, a2, a3⟩ andb = ⟨b1, b2, b3⟩:a ·b = a1b1+ a2b2+ a3b3;properties of dot product; useful identity:a ·a = |a|2; angle between two vectorsa andb:cos θ =a ·b|a||b|;a ⊥b if and only ifa ·b = 0; the vector in R2with length r with angle θ isv = ⟨r cos θ, r sin θ⟩:xy0θr(c) Projection ofb alonga: projab ={a ·b|a|}a|a|; Work =F ·D.bprojaprojabbaab(d) Cross product (only for vectors in R3):a ×b =ijka1a2a3b1b2b3=a2a3b2b3i −a1a3b1b3j +a1a2b1b2kproperties of cross products;a ×b is perpendicular (orthogonal or normal) to botha andb; area of parallelogram spanned bya andb is A = |a ×b|:bathe area of the triangle spanned is A =12|a ×b|:baVolume of the parallelopiped spanned bya,b,c is V = |a · (b ×c)|:bac2. Equation of a line L through P0(x0, y0, z0) with direction vectord = ⟨a, b, c⟩:Vector Form:r(t) = ⟨x0, y0, z0⟩ + td.(x ,y ,z )0 0 0dParametric Form:x = x0+ a ty = y0+ b tz = z0+ c tSymmetric Form:x − x0a=y − y0b=z − z0c. (If say b = 0, thenx − x0a=z − z0c, y = y0.)3. Equation of the plane through the point P0(x0, y0, z0) and perpendicular to the vectorn = ⟨a, b, c⟩(n is a normal vector to the plane) is ⟨(x − x0), (y − y0), (z − z0)⟩ ·n = 0; Sketching planes(consider x, y, z intercepts).n(x ,y ,z )0 004. Quadric surfaces (can sketch them by considering various traces, i.e., curves resulting from theintersection of the surface with planes x = k, y = k and/or z = k); some generic equations havethe form:(a) Ellipsoid:x2a2+y2b2+z2c2= 1(b) Elliptic Paraboloid:zc=x2a2+y2b2(c) Hyperbolic Paraboloid (Saddle):zc=x2a2−y2b2(d) Cone:z2c2=x2a2+y2b2(e) Hyperboloid of One Sheet:x2a2+y2b2−z2c2= 1(f) Hyperboloid of Two Sheets: −x2a2−y2b2+z2c2= 15. Vector-valued functionsr(t) = ⟨f(t), g(t), h(t)⟩; tangent vectorr′(t) for smooth curves, unit tan-gent vectorT(t) =r′(t)|r′(t)|; unit normal vectorN(t) =T′(t)|T′(t)|differentiation rules for vectorfunctions, including:(i) {ϕ(t)v(t)}′= ϕ(t)v′(t) + ϕ′(t)v(t), where ϕ(t) is a real-valued function(ii) (u ·v)′=u ·v′+u′·v(iii) (u ×v)′=u ×v′+u′×v(iv) {v(ϕ(t))}′= ϕ′(t)v′(ϕ(t)), where ϕ(t) is a real-valued function6. Integrals of vector functions∫r(t) dt =⟨∫f(t) dt,∫g(t) dt,∫h(t) dt⟩; arc length of curveparameterized byr(t) is L =∫ba|r′(t)|dt; arc length function s(t) =∫ta|r′(u)|du; reparameterizeby arc length: σ(s) =r(t(s)), where t(s) is the inverse of the arc length function s(t); thecurvature of a curve parameterized byr(t) is κ =|T′(t)||r′(t)|. Note:√α2= |α|.7.r(t) = position of a particle,r′(t) =v(t) = velocity;a(t) =v′(t) =r′′(t) = acceleration;|r′(t)| = |v(t)| = speed; Newton’s 2ndLaw:F = ma.8. Domain and range of a function f(x, y) and f (x, y, z); level curves (or contour curves) of f (x, y)are the curves f(x, y) = k; using level curves to sketch surfaces; level surfaces of f(x, y, z) are thesurfaces f(x, y, z) = k.10. Partial derivatives∂f∂x(x, y) = fx(x, y) = limh→0f(x + h, y) − f(x, y)h,∂f∂y(x, y) = fy(x, y) = limh→0f(x, y + h) − f (x, y)h; higher order derivatives: fxy=∂2f∂y ∂x,fyy=∂2f∂y2, fyx=∂2f∂x ∂y, etc; mixed partials.11. Equation of the tangent plane to the graph of z = f (x, y) at (x0, y0, z0) is given byz − z0= fx(x0, y0)(x − x0) + fy(x0, y0)(y − y0).12. Total differential for z = f(x, y) is dz = df =∂f∂xdx +∂f∂ydy; total differential for w = f (x, y, z)is dw = df =∂f∂xdx +∂f∂ydy +∂f∂zdz; linear approximation for z = f(x, y) is given by ∆z ≈ dz,i.e., f(x + ∆x, y + ∆y) − f (x, y) ≈∂f∂xdx +∂f∂ydy , where ∆x = dx, ∆y = dy ;Linearization of f(x, y) at (a, b) is given by L(x, y) = f (a, b) + fx(a, b)(x − a) + fy(a, b)(y − b);L(x, y) ≈ f(x, y) near (a, b).13. Different forms of the Chain Rule: Form 1, Form 2; General Form: Tree diagrams. For example:(a) If z = f (x, y) and{x = x(t)y = y(t), thendfdt=∂f∂xdxdt+∂f∂ydydt:xydxdydtdtz=f(x,y)ttxyff(b) If z = f (x, y) and{x = x(s, t)y = y(s, t), then∂f∂s=∂f∂x∂x∂s+∂f∂y∂y∂sand∂f∂t=∂f∂x∂x∂t+∂f∂y∂y∂t:xyfxz=f(x,y)tts sxsxtytsyfyetc.....14. Implicit Differentiation and Directional Derivative:Implicit DifferentiationPart I: If F (x, y) = 0 defines y as function of x (i.e., y = y(x)), then to computedydx,differentiate both sides of the equation F (x, y) = 0 w.r.t. x and solve fordydx.If F (x, y, z) = 0 defines z as function of x and y (i.e. z = z(x, y)) , then to compute∂z∂x,differentiate the equation F (x, y, z) = 0 w.r.t. x (hold y fixed) and solve for∂z∂x. For∂z∂y,differentiate the equation F (x, y, z) = 0 w.r.t. y (hold x fixed) and solve for∂z∂y.Part II: If F (x, y) = 0 defines y as function of x =⇒dydx= −∂F∂x∂F∂y;while if F (x, y, z) = 0 defines z as function of x and y =⇒∂z∂x= −∂F∂x∂F∂zand∂z∂y= −∂F∂y∂F∂z.Directional derivativeDirectional derivative of f (x, y) at (x0, y0) in the directionu : Duf(x0, y0) = ∇f (x0, y0)·u, whereu must be a unit