13.1: Vector-Valued Functions and Motion in Space, 14.1: Functions of Several Variables, and 14.2: Limits and Continuity in Higher Dimensions

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1 13.1: Vector-Value Functions an Motion in Space, 14.1: Functions of Several Variables, an 14.2: Limits an Continuity in Higher Dimensions TA: Sam Fleischer November 3 Section 13.1: Vector-Value Functions an Motion in Space Think of a particle s coorinates as a function of time. x = f(t), y = g(t), z = h(t), t I The points (x, y, z) = (f(t), g(t), h(t)), t I make up the curve in space that we call the particle s path. Above is the parametric form. Here is the vector form: r(t) = OP = f(t)i + g(t)j + h(t)k from the origin to the particle s position P (f(t), g(t), h(t)). We call r a vector-value function or vector function since its output is a vector. where f(t) = cos t, g(t) = sin t, an h(t) = t. Definition - Limit r(t) = (cos t)i + (sin t)j + tk Let r(t) = f(t)i + g(t)j + h(t)k be a vector function with omain D, an L a vector. We say that r has it L as t approaches t 0 an write r(t) = L t t 0 if, for every number ɛ > 0, there exists a corresponing number δ > 0 such that for all t D, Definition - Continuity r(t) L < ɛ whenever 0 < t t 0 < δ A vector function r(t) is continuous at a point t = t 0 in its omain if t t0 function is continuous if it is continuous over its interval omain. = r(t 0 ). The 1

2 Definition - Differentiability The vector function r(t) = f(t)i + g(t)j + h(t)k has a erivative at t if f, g, an h have erivatives at t. The erivative is r (t) = r = r(t + t) r(t) t 0 t Definitions - Velocity an Acceleration Vectors = f (t)i + g (t)j + h (t)k If r is the position vector of a particle moving along a smooth curve in space, then v(t) = r is the particle s velocity vector, tangent to the curve. At any time t, the irection of v is the irection of motion, the magnitue of v is the particle s spee, an the erivative a = v, when it exists, is the particle s acceleration vector. In summary, 1. Velocity is the erivative of position: v = r 2. Spee is the magnitue of velocity: Spee = v 3. Acceleration is the erivative of velocity: a = v = 2 r 2 4. The unit vector v v is the irection of motion at time t. The velocity of a particle whose motion in space is governe by the position vector is r(t) = 2 cos t i + 2 sin t j + 5 cos 2 t k v(t) = r (t) = 2 sin t i + 2 cos t j 10 cos t sin t k = 2 sin t i + 2 cos t j 5 sin 2t k an the acceleration is a(t) = v (t) = r (t) = 2 cos t i 2 sin t j 10 cos 2t k 2

3 Differentiation Rules for Vector Functions Let u an v be vector functions of t, C a constant vector, c any scalar, an f any ifferentiable scalar function. 1. Constant Function Fule: C=0 2. Scalar Multiple Rules: [cu(t)] = cu0 (t) [f (t)u(t)] = f 0 (t)u(t) + f (t)u0 (t) 3. Sum Rule: [u(t) + v(t)] = u0 (t) + v0 (t) 4. Difference Rule: [u(t) v(t)] = u0 (t) v0 (t) 5. Dot Prouct Rule: [u(t) v(t)] = u0 (t) v(t) + u(t) v0 (t) 6. Cross Prouct Rule: [u(t) v(t)] = u0 (t) v(t) + u(t) v0 (t) 7. Chain Rule [u(f (t))] = f 0 (t)u0 (f (t)) Section 14.1: Functions of Several Variables Definition - Real-Value Function Suppose D is a set of n-tuples of real numbers (x1,..., xn ). A real-value function f on D is a rule that assigns a unique (single) real number w = f (x1,..., xn ) to each element in D. Lets start with Two-Dimensional 3

4 s Function p f (x, y) = y x2 g(x, y) = 1 xy h(x, y) = sin(xy) Domain Range y x2 [0, ] xy 6= 0 (, 0) (0, ) Entire Plane [ 1, 1] Definitions - Interior Point, Bounar Point, Open, Close A point (x0, y0 ) in a region (set) R in the xy-plane is an interior point of R if it is the center of a isk of positive raius that lies entirely in R. A point (x0, y0 ) is a bounar point of R if every isk centere at (x0, y0 ) contains points that lie outsie of R as well as points that lie insie of R (the bounar point itself nee not belong to R.) A region is open if it consists entirely of interior points. A region is close if it contains all its bounary points Definition - Boune A region in the plane is boune if it lies insie a isk of fixe raius. A raius is unboune if it is not boune. p The omain D of the function f (x, y) = y x2 is the set of all points in R2 (2-imensional Cartesian coorinate plane) such that y x2. D = {(x, y) R2 such that y x2 } D is a close an unboune set. 4

5 Definitions - Level Curve, Surface The set of points in the plane where a function f (x, y) has a constant value f (x, y) = c is calle a level-curve of f. The set of all points (x, y, f (x, y)) in space, for (x, y) in the omain of f, is calle the surface of z = f (x, y). Section 14.2: Limits an Continuity in Higher Dimensions Definition - Limit We say that a function f (x, y) approahces the it L as (x, y) approaches (x0, y0 ), an write f (x, y) = L if, for every number > 0, there exists a corresponing number δ > 0 such that for all (x, y) in the omain of f, p f (x, y) L < whenever 0 < (x x0 )2 + (y y0 )2 < δ Theorem 1 - Properties of Limits of Functions of Two Variables The following rules hol if L, M, an k are real numbers an f (x, y) = L an 1. (f (x, y) + g(x, y)) = L + M 2. (f (x, y) g(x, y)) = L M 3. kf (x, y) = kl 4. (f (x, y) +g(x, y)) = L +M 5. f (x,y) g(x,y) = L M M 6= 0 5 g(x, y) = M

6 6. (x,y) (x 0,y 0 ) y)]n = L n n Z >0 7. (x,y) (x 0,y 0 ) n f(x, y) = n L Definition - Continuity A function f(x, y) is continuous at the point (x 0, y 0 ) if 1. f is efine at (x 0, y 0 ), 2. f(x, y) exists, (x,y) (x 0,y 0 ) 3. (x,y) (x 0,y 0 ) f(x, y) = f(x 0, y 0 ). A function is continuous if it is continuous at every point of its omain. 2xy, (x, y) (0, 0) f(x, y) = x 2 + y2 0, (x, y) = (0, 0) is continuous at every point except the origin. This is because ifferent paths of approach to the origin can lea to ifferent results. For example, f(x, y) = y=mx 2xy x 2 + y 2 y=mx = 2x(mx) x 2 + (mx) 2 = So now approach (0, 0) along the line y = mx... [ ] = f(x, y) (x,y) (0,0) (x,y) (0,0) y=mx along y=mx 2mx2 x 2 + m 2 x 2 = = 2m 1 + m 2 2m 1 + m 2 This is a ifferent value for ifferent choices of m, so the it oes not exist, violating criteria 2 an 3 of the efinition of continuity. Thus f is not continuous at (0, 0). Two-Path Test for Nonexistence of a Limit If a function f(x, y) has ifferent its along two ifferent paths in the omain of f as (x, y) approaches (x 0, y 0 ), then (x,y) (x0,y 0 ) f(x, y) oes not exist. (These paths o NOT have to be linear!) Continuity of Composites If f is continuous at (x 0, y 0 ) an g is a single-variable function continuous at f(x 0, y 0 ), then the composite function h = g f efine by h(x, y) = g(f(x, y)) is continuous at (x 0, y 0 ). 6