Study Guide # Vectors in R 2 and R 3. (a) v = a, b, c = a i + b j + c k; vector addition and subtraction geometrically using parallelograms

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1 Study Guide # 1 MA Fll Vectors in R 2 nd R 3 () v =, b, c = i + b j + c k; vector ddition nd subtrction geometriclly using prllelogrms spnned by u nd v; length or mgnitude of v =, b, c, v = 2 + b 2 + c 2 ; directed vector from P 0 (x 0, y 0, z 0 ) to P 1 (x 1, y 1, z 1 ) given by v = P 0 P 1 = P 1 P 0 = x 1 x 0, y 1 y 0, z 1 z 0. (b) Dot (or inner) product of = 1, 2, 3 nd b = b 1, b 2, b 3 : b = 1 b b 3 ; properties of dot product; useful identity: = 2 ; ngle between two vectors nd b: b cos θ = b ; b if nd only if b = 0; the vector in R 2 with length r with ngle θ is v = r cos θ, r sin θ : (c) Cross product (only for vectors in R 3 ): i j k b = b 1 b 2 b 3 = 2 3 b 2 b 3 i 1 3 b 1 b 3 j b 1 b 2 k properties of cross products; b is perpendiculr (orthogonl or norml) to both nd b; re of prllelogrm spnned by nd b is A = b : the re of the tringle spnned is A = 1 2 b :

2 Volume of the prllelopiped spnned by, b, c is V = ( b c) : 2. Eqution of line L through P 0 (x 0, y 0, z 0 ) with direction vector d =, b, c : Vector Form: r(t) = x 0, y 0, z 0 + t d. Prmetric Form: x = x 0 + t y = y 0 + b t z = z 0 + c t Symmetric Form: x x 0 = y y 0 b = z z 0 c. (If sy b = 0, then x x 0 = z z 0, y = y 0.) c 3. Eqution of the plne through the point P 0 (x 0, y 0, z 0 ) nd perpendiculr to the vector n =, b, c ( n is norml vector to the plne) is (x x 0 ), (y y 0 ), (z z 0 ) n = 0; Sketching plnes (consider x, y, z intercepts). 4. Qudric surfces (cn sketch them by considering vrious trces, i.e., curves resulting from the intersection of the surfce with plnes x = k, y = k nd/or z = k); some generic equtions hve the form: () Ellipsoid: (b) Elliptic Prboloid: x y2 b 2 + z2 z + y2 (c) Hyperbolic Prboloid (Sddle): (d) Cone: z y2 (e) Hyperboloid of One Sheet: z y2 x y2 b 2 z2 (f) Hyperboloid of Two Sheets: x2 2 y2 b 2 + z2

3 5. Vector-vlued functions r(t) = f(t), g(t), h(t) ; tngent vector r (t) for smooth curves, unit tngent vector T(t) = r (t) r (t) ; principl unit norml vector N(t) T = (t) T ; differentition rules for (t) vector functions, including: (i) {φ(t) v(t)} = φ(t) v (t) + φ (t) v(t), where φ(t) is rel-vlued 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 rel-vlued function 6. Integrls of vector functions prmeterized by r(t) is L = b r(t) dt = f(t) dt, g(t) dt, r (t) dt; rc length function s(t) = h(t) dt ; rc length of curve t r (u) du; reprmeterize by rc length: σ(s) = r(t(s)), where t(s) is the inverse of the rc length function s(t); the curvture of curve prmeterized by r(t) is κ = T (t) r (t). Note: α 2 = α. 7. r(t) = position of prticle, r (t) = v(t) = velocity; (t) = v (t) = r (t) = ccelertion; r (t) = v(t) = speed; Newton s 2 nd Lw: F = m. 8. Domin nd rnge of function f(x, y) nd f(x, y, z); level curves (or contour curves) of f(x, y) re the curves f(x, y) = k; using level curves to sketch surfces; level surfces of f(x, y, z) re the surfces f(x, y, z) = k. 9. Limits of functions f(x, y) nd f(x, y, z); limit of f(x, y) does not exist if different pproches to (, b) yield different limits; continuity. 10. Prtil derivtives (x, y) = f f(x + h, y) f(x, y) x(x, y) = lim, h 0 h (x, y) = f f(x, y + h) f(x, y) y(x, y) = lim ; higher order derivtives: f xy = 2 f h 0 h, f yy = 2 f, f 2 yx = 2 f, etc; mixed prtils. 11. Eqution of the tngent plne to the grph of z = f(x, y) t (x 0, y 0, z 0 ) is given by z z 0 = f x (x 0, y 0 )(x x 0 ) + f y (x 0, y 0 )(y y 0 ). 12. Totl differentil for z = f(x, y) is dz = df = dx + dy; totl differentil for w = f(x, y, z) is dw = df = dx + dy + dz; liner pproximtion for z = f(x, y) is given by z dz, i.e., f(x + x, y + y) f(x, y) dx + dy, where x = dx, y = dy ; Lineriztion of f(x, y) t (, b) is given by L(x, y) = f(, b) + f x (, b)(x ) + f y (, b)(y b); L(x, y) f(x, y) ner (, b).

4 13. CHAIN RULE; different forms of the Chin Rule: Form 1, Form 2; CHAIN RULE (Generl Form): Tree digrms. For exmple: { x = x(t) () If z = f(x, y) nd y = y(t), then df dt = dx dt + dy dt : (b) If z = f(x, y) nd { x = x(s, t) y = y(s, t), then s = s + s nd t = t + t : etc Implicit Differentition: Prt I: If F (x, y) = 0 defines y s function of x (i.e., y = y(x)), then to compute dy dx, differentite both sides of the eqution F (x, y) = 0 w.r.t. x nd solve for dy dx. If F (x, y, z) = 0 defines z s function of x nd y (i.e. z = z(x, y)), then to compute, differentite the eqution F (x, y, z) = 0 w.r.t. x (hold y fixed) nd solve for. differentite the eqution F (x, y, z) = 0 w.r.t. y (hold x fixed) nd solve for. Prt II: If F (x, y) = 0 defines y s function of x = dy dx = while if F (x, y, z) = 0 defines z s function of x nd y = = ; nd = For,.

5 15. Grdient vector for f(x, y): f(x, y) =,, properties of grdients; grdient points in direction of mximum rte of increse of f, mximum rte of increse is f ; f(x 0, y 0 ) level curve f(x, y) = k nd, in the cse of 3 vribles, f(x 0, y 0, z 0 ) level surfce f(x, y, z) = k: 16. Directionl derivtive of f(x, y) t (x 0, y 0 ) in the direction u : D u f(x 0, y 0 ) = f(x 0, y 0 ) u, where u must be unit vector; tngent plnes to level surfces f(x, y, z) = k ( norml vector t (x 0, y 0, z 0 ) is n = f(x 0, y 0, z 0 )). 17. Reltive/locl extrem; criticl points (points where f = 0 or f does not exist) nd Derivtives Test: Suppose the 2 nd prtils of f(x, y) re continuous in disk with center (, b) nd f(, b) = 0. Let D = f xx () If D > 0 nd f xx (, b) > 0 (b) If D > 0 nd f xx (, b) < 0 f yx f xy f yy. (,b) = f(, b) is locl minimum vlue. = f(, b) is locl mximum vlue. (c) If D < 0 = f(, b) is not locl min or locl mx vlue. So (, b) is sddle point of f. If D = 0 (or if f(, b) does not exist or f hs more thn 2 vribles) the test gives no informtion nd you need to do something else to determine if reltive extremum exists.