MATERIAL THAT MAY BE COVERED IN QUIZ #1, 18 Feb 2003. ------------------------------------------------------- AND REVIEW OF IT ---------------- 1. VECTORS: length, angle between, area, perpendicular and parallelness, understand geometrical and algebraic properties of cross and dot products and vectors. Understand planes, their normal vectors, and the angles between planes and other planes (or planes and lines, if I asked you that). Undertand how to algebraically manipulate formulas involving vectors including differentiate and integrate. In many cases this is just doing the scalar thing 3 times (in 3D), nothing more, but in some other cases there is a bit more to it than that. 2. VECTOR-VALUED FNS OF 1 VARIABLE (also called: PARAMETRICALLY DESCRIBED CURVES): understand how to get parametic curve representations and how to use them to figure out tangent vectors, curvature (for curves in the plane), velocity vectors (if the parameter is "time"), acceleration vectors. 3. REAL-VALUED FNS OF SEVERAL VARIABLES understand both geometrically and algebraically what partial derivatives are, gradient, directional derivative, and how to find them. Understand how to optimize a function by solving for the "flat spot" where gradient is the 0 vector. NOW FOR QUICK REVIEW OF A LOT OF THAT: -------------------------------------- VECTORS: -------- Let V = (V1, V2, V3) be a vector. V should be written with a little arrow on top (book sometimes uses bold print instead of arrow) so you know which letters mean vectors and which mean scalars. In this webpage I will make capital letters C be vectors and uncapital c be scalars (since have no little arrows). Vector may be thought of as xyz coordinates of a point in 3-space, or as a direction (with a magnitude, its length). Length(V) = |V| = squareroot( V1*V1 + V2*V2 + V3*V3 ). dot product A.B = A1*B1 + A2*B2 + A3*B3 = scalar. cross product A X B = (A2*B3-A3*B2, A3*B1-A1*B3, A1*B2-A1*B1) = vector. usual commutative, associative, and distributive properties allow manipulation like usual...: A.B = B.A (kA).B = k(A.B) (kA) X B = k (A X B) A X (B X C)= (A X B) X C A X (B+C) = A X B + A X C A . (B+C) = A . B + A . C EXCEPT that A X B = - B X A (ANTIcommutativity of cross product) A X A = 0. Where 0 here means the 0 VECTOR and should have arrow above. Length(V) = |V| = squareroot( V1*1 + V2*V2 + V3*V3 ) = scalar. cos angle(A,B) = A.B / (|A| |B|) = scalar. distance(A,B) = |A-B| = scalar. |V| * |V| = V.V = scalar. A/|A| is the normalized unit-length version of A; useful for describing direction alone without a magnitude. A.B = 0 if perpendicular (or if one of A=0, B=0). A X B = 0 vector, if parallel (or if one of A=0, B=0). |A X B| = area of parallelogram with 2 sides A, B starting from common point. = |A| |B| sin(angle). Compare A.B = |A| |B| cos(angle). A X B is perpendicular to A and also to B and A,B,C form a right-handed vector triple (as do the X,Y,Z positive-axis directions) thumb, fingers, palm. "VECTOR TRIPLE PRODUCT" A . (B X C) = B . (C X A) = C . (A X B) = -A . (C X B) = -B . (A X C) = -C . (B X A) = 3x3 determinant with rows A,B,C. PLANES - see my math127 web subpage on "planes" (now new & improved!). DETERMINANTS AND MATRICES AND COMPLEX NUMBERS: see my "prerequisites/background" math127 web subpage (now new & improved!). PARAMETRICALLY DESCRIBED CURVES: -------------------------------- F(t) = (X(t), Y(z), Z(t)). If t means time, this describes motion of a point along a curve. If t does not mean time, it is just a curve described by "parameter" t which as you vary it, moves (X,Y,Z) along the curve. F'(t) = (X'(t), Y'(t), Z'(t)) = velocity vector (if t=time) otherwise tangent vector. Sometimes interested in unit-normalized version of tangent vector. F(3t) same curve but moving along it 3 times faster. F(-5t) same curve but moving along it 5 times faster in opposite direction. F"(t) = (X"(t), Y"(t), Z"(t)) = acceleration vector. 2 F"(t) - F'(t) (F"(t) . F'(t)) / |F'(t)| = the portion of acceleration that is perpendicular to velocity. This points in the direction of curvature of the curve. It is perpendicular to the curve at t. Arc-length s = integral ds ds = |(dx,dy,dz)| ARC LENGTH EXAMPLE: for curve F(t) = (5t, 6t+7, 1/t), we have 2 dx/dt = 5, dy/dt = 6, dz/dt = -1/t 2 2 So ds = squareroot( 5*5 + 6*6 + (-1/t ) ) dt So s = arclength from 0 to t 2 2 = integral(from 0 to t)of squareroot( 5*5 +6*6 + (-1/t ) ) dt. End of example. >3 ways to describe curves in the plane: parametric description based on arclength s: give (X(s), Y(s)) where s is arclength along curve. parametric description based on t: give (X(t), Y(t)) where t maybe not arclength. If t=x then this is y=f(x)type description. arclength-angle description: give theta (=angle of curve above X axis) as a function of s (=arclength along the curve). arclength-curvature description: give (signed) curvature (=dtheta/ds) as a function of s (=arclength along the curve). (Unsigned) Curvature = 1/RadiusOfOsculatingCircle Sign of (signed) curvature = + if curve bending up (or to the left as s increases), - if down (" right). Ways to convert between the 3 descriptions of curves in plane: 2 2 tan(theta) = dy/dx, theta = arctan(dy/dx), ds = squareroot(dx + dy ) curvature = dtheta/ds. If curve is y=f(x) then theta = arctan( f'(x) ). 2 3/2 curvature = f"(x) / [ 1 + f'(x) ]. If curve is theta(s) then (X(s), Y(s)) = integral(from t=0 to t=s)of (cos(theta(t)), sin(theta(t)) dt. curvature(s) = theta'(s), theta(s) = integral(from t=0 to s)of curvature(t) dt. REAL-VALUED FUNCTIONS OF VECTORS f(V), PARTIAL DERIVATIVES ETC. --------------------------------------------------------------- Let V=(x,y,z). The partial derivative of f(V) with respect to x is you take deriv with respect to x while regarding y,z as CONSTANTS. This is a SCALAR. geometrical meaning: rate of change in f as x moves, but y,z stay fixed. The GRADIENT of f(V) is the VECTOR of partial derivatives with respect to x,y,z in that order. geometrical meaning: grad f is the direction of fastest increase (and magnitude gives rate of increase) of f. Points "up the mountain." directional derivative in direction D (usually one demands D be a unit vector) at V = d f( V + k D ) / dk = [ grad f(V) ] . D (this is a vector dot product) = scalar. geometrical meaning: rate of increase of F as you walk from V in direction D. Maxima & Minima: If f is smooth, then any (local) max or min must be a "flat spot" at which all directional derivatives in any direction all are 0. (Since: If not 0: then could walk in D or in -D direction to increase F, which would contradict maximality.) Thus to maximize f, we set grad f(V) = 0 as a vector eqaution (this is d scalar equations in d-dimensional space) and solve for V (which is d unknowns). Ability to solve systems of n equations for n unknowns by "elimination" technique is useful here... (eliminate 1 variable at time, reducing to simpler system of 1 fewer number of eqns in 1 fewer number of unknowns). 2 2 EXAMPLE: f(x,y,z) = x + y - x + xy - 3 y. Maximize over all x,y,z. first we find grad f = ( 1-2x+y, 1+x-6y, 0 ). The vector equations grad f = 0 is (as 3 scalar equations) 1-2x+y=0, 1+x-6y=0, 0=0, unknowns x,y,z. Answer: z=anything, x=7/11, y=3/11. f(x,y,z)=5/11 is the max (actually since this does not depend on z, have here a whole line of equi-maxes parallel to the z-axis at this x,y). Warning: This could have ben a flat spot which was not a max (and not a min either). grad f = 0 is necessary (if f smooth) but not sufficient to have a max or min. In this case it really is a max though. End of example. EQUALITY OF MIXED PARTIALS: 2 2 d f d f ----- = ----- dx dy dy dx this kind of obvious from the commutativity of multiplication (of infinitesimals)... geometrical meaning: if you walk tiny amounts in x direction and then in y direction, or instead you perform those two subwalks in the other order, you get to the same place (the far corner of a little rectangle of sides dx, dy) and hence f(x,y)'s infinitesimal value-change is the same either way. =============end of review====================