MA 261 - Spring 2010Study Guide # 11. Vectors in R2and R3(a)~v = ha, b, ci = a~i + b~j + c~k; vector addition and subtraction geometrically using paral-lelograms spanned by~u and~v; length or magnitude of~v = ha, b, ci, |~v| =√a2+ b2+ c2;directed vector from P0(x0,y0,z0)toP1(x1,y1,z1)givenby~v = P0P1= P1− P0=hx1− x0,y1− y0,z1− z0i.(b) Dot (or inner) product of~a = ha1,a2,a3i and~b = hb1,b2,b3i:~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 = hr cos θ, r sin θi: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 = ha, b, ci:Vector Form:~r(t)=hx0,y0,z0i + t~d.(x ,y ,z )000dParametric Form:x = x0+ aty = y0+ btz = z0+ ctSymmetric Form:x − x0a=y − y0b=z − z0c.(Ifsayb =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 = ha, b, ci(~n is a normal vector to the plane) is h(x − x0), (y − y0), (z − z0)i·~n = 0; Sketching planes(consider x, y, z intercepts).n(x ,y ,z )0004. 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 genericequations 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)=hf(t),g(t),h(t)i; tangent vector~r0(t) for smooth curves, unit tan-gent vector~T(t)=~r0(t)|~r0(t)|; unit normal vector~N(t)=~T0(t)|~T0(t)|differentiation rules for vectorfunctions, including:(i) {φ(t)~v(t)}0= φ(t)~v0(t)+φ0(t)~v(t), where φ(t) is a real-valued function(ii) (~u ·~v)0=~u ·~v0+~u0·~v(iii) (~u ×~v)0=~u ×~v0+~u0×~v(iv) {~v(φ(t))}0= φ0(t)~v0(φ(t)), where φ(t) is a real-valued function6. Integrals of vector functionsZ~r(t) dt =Zf(t) dt,Zg(t) dt,Zh(t) dt; arc length of curveparameterized by~r(t)isL =Zba|~r0(t)|dt; arc length function s(t)=Zta|~r0(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)isκ =|~T0(t)||~r0(t)|. Note:√α2= |α|.7.~r(t) = position of a particle,~r0(t)=~v(t)=velocity;~a(t)=~v0(t)=~r00(t) = acceleration;|~r0(t)| = |~v(t)| = speed; Newton’s 2ndLaw:~F = m~a.8. Domain and range of a function f(x, y)andf(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.9. Limits of functions f (x, y)andf(x, y, z); limit of f(x, y) does not exist if different approaches to(a, b) yield different limits; continuity.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)isgivenbyz − 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)isgivenby∆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)isgivenby 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)ttssxsxtytsyfyetc.....14. Implicit Differentiation:Part 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.15. 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 ; ∇f (x0,y0) ⊥ level curve f(x, y)=C and, in the caseof3variables,∇f (x0,y0,z0) ⊥ level surface f (x, y, z)=C:0(x ,y )xyf(x,y,z)=Cxy(x ,y ,z )000n = ∆0n = ∆zf(x ,y )00f(x,y)=Cf(x ,y ,z )00016. Directional 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 unitvector; tangent planes to level surfaces f (x, y, z)=C (a normal vectorat (x0,y0,z0)is~n = ∇f