Math 32A Review Sheet

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1 Review Sheet Tau Beta Pi - Boelter 6266 Contents 1 Parametric Equation Line Circle Ellipse Cycloi Vectors in the Plane Vectors in the Plane Vector Algebra Vectors in 3D Dot Prouct Angles Between Lines Cross Prouct Planes in 3D Quaratic Surfaces Calculus of Vector-Value Functions Calculation Rules Arc Length, Spee, an Curvature Motion in 3D Differentiation in Several Variables Limits an Continuity Partial Derivatives Graient of a Function Optimization in Several Variables

2 1 Parametric Equation Parametric Equations: escribe a particle s motion over time. If a particle follows a 2-D curve escribe by C(t), its motion can also be escribe by the coorinates x(t) an y(t): C(t) = (x(t), y(t)) When parametrizing any functino or curve, the goal is to isolate a variable an to express the isolate variable in terms of t. 1.1 Line Line can be escribe as: y = mx + b where m stans for the slope an b represents a constant (the calue of y when x=0). If a line passes through a point (a,b) with a slope of m its motion can be escribe as: x(t) = a + rt; y(t) = b + st such that m= s r, r 0 r an s represent how much x an y vary with t; an a an b represent the value of x when y is 0, (x(0)) an the value of y when x is 0, (y(0)). If a line passes through two points M(a,b) an N(c,) with unknow slope, the slope can be foun as: m = s r = b c a Therefore, the escriptions turn to: x(t) = a + (c a)t y(t) = b + (b )t 1.2 Circle A circle can be escribe as: x(t) = a + Rcos(t); y(t) = b + Rsin(t) where R is the raiuse of the circle. 1.3 Ellipse Similar to a circle, an ellipse can be expresse as: x(t) = a + Ccos(t); y(t) = b + Dsin(t) where a an b reoresent the x an y coorinates of the center of the ellipse, an C an D represent the horizontal an verticle istance from the center to the ege repectively. 1.4 Cycloi A cycloi is forme by the motion of a point on a circle as the circle rolls without slipping. x(t) = t(t sin(t)); y(t) = (1 )cos(t)) Page 2

3 2 Vectors in the Plane 2.1 Vectors in the Plane Two imensional vector v is etermine by two points in a plane (an initial point + a erminal point): v = P Q where P = (a1, b1) an Q = (a2, b2) Length of magnitue: v.ancanbecalculateby : v = P Q = a 2 + b 2. Parallal: The lines through v an w are parallel. Translation: When a vector is move to begin at a new point without changing its length or irection. (If a vector is the translation of another vector, aka. have the same components, these two vectors are efine as equivalent to each other). 2.2 Vector Algebra Basic Properties of Vector Algebra For all vectors u, v, w an for all scalars λ Commutative Law: v + w = w + v Associative Law: u + (v + w) = (u + v) + w Distributive Law for Scalars: λ(v + w) = λv + λw Linear Combination: Every vector is a linear combinatio of other vectors. For every vector u, u = λ v + µ w, forming a parallelogram. Unit Vector: e v = 1 v v. e v = 1. Triangle Inequality: + w v + w. 2.3 Vectors in 3D Right Han Rule: Sphere of raius R an center (a, b, c): (x a) 2 + (y b) 2 + (z c) 2 = R 2 Cyliner of raius R with vertical axis through (a, b, 0): Distance Formula: (x a) 2 + (y b) 2 = R 2 P = (a1, b1, c1)anq = (a2, b2, c2) P Q = v = P Q = (a2 a1) 2 + (b2 b1) 2 + (c2 c1) 2 Vector Parametrization: Equations for the line through P 0 = (x 0, y 0, z 0 ) with irection vector v = (a, b, c): Vectorparametrization: r(t) = (x 0, y 0, z 0 ) + t(a, b, c) Parametric equations: x = x 0 + at, y = y 0 + bt, z = z 0 + ct Page 3

4 2.4 Dot Prouct The ot prouct of v = (a1,b1,c1) an w = (a2,b2,c2) is v w = a1a2 + b1b2 + c1c2 Basic Properties: - Commutativity:v w = w v - Pullingoutscalars: (λv) w = v (λw) = λ(v w) - DistributiveLaw: u (v + w) = u v + u w (v + w) u = v u + w u - v v = v 2 - v w = v w cos(θ), where θ is the angle between v an w Angles Between Lines -Perpenicular: v w=0. -Acute: if v w > 0. -obtuse if v w < 0. -Every vector u has a ecomposition u = u v + u v, where u v is parallel to v, an u v is orthogonal to v. The vector u v is calle the projection of u along v. -Let e v = v v, Then u v = ( u v v v )v = ( u v v )v = (u v 2 v )e v -The coefficient u v v is calle the component of u along v. 2.5 Cross Prouct Determinants: a 11 a 12 a 21 a 22 = a 11a 22 a 12 a 21 a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 = a 11 a 22 a 23 a 32 a 33 a 12 a 21 a 23 a 31 a 33 + a 13 a 21 a 22 a 31 a 32 Cross Prouct: The coross Prouct between v =< v 1, v 2, v 3 > anw =< w 1, w 2, w 3 > is the symbolic eterminant: v w = v 2 v 3 w 2 w 3 i v 1 v 3 w 1 w 3 j + v 1 v 2 w 1 w 2 k Basic Properties: - v w is orthogonal to v an w. - v w has length v w sinθ.(θistheanglebetweenvanw, 0 θ π). - v, w, v w is a right-hane system. - w v=-v w - v w=0 iff w = λv for some scalar or v=0. - (λv) w=v (λw)=λ(v w) - (u+v) w=u w+v w v (u + w) = v u + v w - For stanar basis vectors: i j=k, j k=i, k i=j Geometries: - Parallelogram spanne by v an w has area: v w v w - Triangle spanne by v an w has area: 2 - Parallelepipe spanne by u, v, an w has volume: u (v w) Page 4

5 2.6 Planes in 3D - Equation of plane through P 0 = (x 0, y 0, z 0 ) with normal vector n = (a, b, c): Vector form: n (x, y, z) = Scalar forms: a(x x 0 ) + b(y y 0 ) + c(z z 0 ) = 0 ax + by + cz = where = n (x 0, y 0, z 0 ) = ax 0 + by 0 + cz 0. - For a plane through three points P, Q, R that are not collinear: 2.7 Quaratic Surfaces n = P Q P R, = n (x 0, y 0, z 0 ), wherep = (x 0, y 0, z 0 ) Page 5

6 3 Calculus of Vector-Value Functions Vector-value function: a function of the form 3.1 Calculation Rules Differentiation rules: Sum Rule: (r 1 (t) + r 2 (t)) = r 1(t) + r 2(t) Constant Multiple Rule: (cr(t)) = cr (t) Chain Rule: t r(g(t)) = g (t)r (g(t)) r(t) = (x(t), y(t), z(t)) = x(t)i + y(t)j + z(t)k Prouct Rules: Scalar times vector: t = f (t)r(t) + f(t)r (t) Dot prouct: t = r 1(t) r 2 (t) = r 1(t) r 2 (t) + r 1 (t) r 2(t) Cross prouct: t = r 1(t) r 2 (t) = r 1(t) r 2 (t) + r 1 (t) r 2(t) The tangent vector or velocity vector: erivative r (t 0 ). The Funamental Theorem for vector-value functions: If r(t) is continuous an R(t) is an antierivative of r(t), then: b 3.2 Arc Length, Spee, an Curvature Arc length function: s(t) = b a r (u) u Spee: v(t) = s t = b a r (t) t a r(t)t = R(b) R(a) r(s) is an arc length parametrization if r (s) = 1 for all s. In this case, the length of the path for a s b is b - a. Regular parametrization r(t): r (t) 0 for all t. The unit tangent vector for regular r(t): T (t) = Curvature: k(s) = T s, where r(s) is an arc length parametrization or k(s) = 1 v(t) length parametrization. Formula vali for arbitrary regular parametrizations: k(t) = r (t) r (t) r (t) 3 The curvature at a point on a graph y = f (x) in the plane: T (t) T (t). k(x) = f (x) (1 + f (x) 2 ) 3/2 Unit normal vector N(t) = T (t) = k(t)v(t)n(t) The binormal vector: B = T N. T t r (t) r (t) if r(t) is not an arc 3.3 Motion in 3D For an object whose path is escribe by a vector-value function r(t), v(t) = r (t), v(t) = v(t), a(t) = r (t) The acceleration vector a is the sum of a tangential component (reflecting change in spee) an a normal component (reflecting change in irection): a(t) = a T (t)t (t) + a N (t)n(t) Page 6

7 - Unit tangent vector: T (t) = v(t) v(t) - Unit normal vector: N(t) = T a T = v (t) = a T = a v v (t) T (t) - Tangential component: - Normal component: a T T = ( a v v v )v a N = k(t)v(t) 2 = a 2 a T 2 a N N = a a T T = a ( a v v v )v 4 Differentiation in Several Variables 4.1 Limits an Continuity The limit of a prouct f(x, y) = h(x)g(y) is a prouct of limits: lim f(x, y) = ( lim h(x))(lim g(y)) (x,y) (a,b) x a y b A function f of two variables is continuous at P = (a, b) if: lim f(x, y) = f(a, b) (x,y) (a,b) To prove that a limit oes not exist, it is enough to show that the limits obtaine along two ifferent paths are not equal. 4.2 Partial Derivatives The partial erivatives of f (x, y) are efine as the limits f x (a, b) = f x = lim f(a + h, b) f(a, b) h 0 h (Compute fx by holing y constant, an compute fy by holing x constant.) For small changes xan y, f(a + x, b) f(a, b) = f x (a, b) x The secon-orer partial erivatives: 2 x 2 f = f xx, f(a, b + y) f(a, b) = f y (a, b) y 2 y x f = f xy, 2 x y f = f yx, 2 y 2 f = f yy Clairaut s Theorem: mixe partials are equal (fxy = fyx)provie that fxy an fyx are continuous. More generally, higher orer partial erivatives may be compute in any orer. For example, fxyyz = fyxzy if f is a function of x, y, z whose fourth-orer partial erivatives are continuous. Page 7

8 4.3 Graient of a Function The graient of a function f is the vector of partial erivatives: f = ( f x, f ) or f = ( f y x, f y, f z ) Chain Rule for Paths: t f(r(t)) = f r(t) r (t) Basic geometric properties of the graient (assume f P 0): f P points in the irection of maximum rate of increase. The maximum rate of increase is f P. f P points in the irection of maximum rate of ecrease. The maximum rate of ecrease is f P. f P is orthogonal to the level curve (or surface) through P. Equation of the tangent plane to the level surface F (x, y, z) = k at P = (a, b, c): f P (x a, y b, z c) = 0 F x (a, b, c)(x a) + F y (a, b, c)(y b) + F z (a, b, c)(z c) = Optimization in Several Variables P = (a,b) is a critical point of f(x,y) if f x (a, b) = 0orf x (a, b) oes not exist, an f y (a, b) = 0orf y (a, b) oes not exist. The local minimum or maximum values of f occur at critical points. The iscriminant of f (x, y) at P = (a, b) is the quantity D(a, b) = f xx (a, b)f yy (a, b) f 2 xy(a, b) Secon Derivative Test: If P = (a, b) is a critical point of f (x, y), then D(a, b) > 0, f xx (a, b) > 0 f(a, b) is a local minimum D(a, b) > 0, f xx (a, b) < 0 f(a, b) is a local maximum D(a, b) < 0 sale point D(a, b) = 0 test inconclusive A point P is an interior point of a omain D if D contains some open isk D(P,r) centere at P. A point P is a bounary point of D if every open isk D(P,r) contains points in D an points not in D. The interior of D is the set of all interior points, an the bounary is the set of all bounary points. A omain is close if it contains all of its bounary points an open if it is equal to its interior. Existence an location of global extrema: If f is continuous an D is close an boune, then f takes on both a minimum an a maximum value on D. The extreme values occur either at critical points in the interior of D or at points on the bounary of D. Page 8