Equations. BSU Math 275 sec 002,003 Fall 2018 (Ultman) Final Exam Notes 1. du dv. FTLI : f (B) f (A) = f dr. F dr = Green s Theorem : y da

Transcript

1 BSU Math 275 sec 002,003 Fall 2018 (Ultman) Final Exam Notes 1 Equations r(t) = x(t) î + y(t) ĵ + z(t) k r = r (t) t s = r = r (t) t r(u, v) = x(u, v) î + y(u, v) ĵ + z(u, v) k S = ( ( ) r r u r v = u r ) u v v S = S = r u r v u v F (x, y, z) = P (x, y, z) î + Q(x, y, z) ĵ + R(x, y, z) k = x î + y ĵ + z k graf = f curlf = F ivf = F FTLI : f (B) f (A) = f r C Green s Theorem : Stokes Theorem : Divergence Theorem : C C F r = F r = S D S F S = Q x P y A curlf S W ivf V

2 BSU Math 275 sec 002,003 Fall 2018 (Ultman) Final Exam Notes 2 Common Parameterizations for Some Important Curves Curves in R 2 Line passing through the point P = (x 0, y 0 ), parallel to v = v 1 î + v 2 ĵ : r(t) = OP + tv ( ) = x 0 î + y 0 ĵ ( ) + t v 1 î + v 2 ĵ = (x 0 + tv 1 ) î + (y 0 + tv 2 ) ĵ Graphs of Functions y = f (x) (y is a function of x) : x = g(y) (x is a function of y) : r(t) = t î + f (t) ĵ r(t) = g(t) î + t ĵ Circle x 2 + y 2 = R 2 centere at the origin; raius R; R, ω constant: r(t) = R cos(ωt) î + R sin(ωt) ĵ Circle (x a) 2 + (y b) 2 = R 2, centere at the point P = (a, b); raius R; R, a, b, ω constant: ( ) ( ) r(t) = R cos(ωt) + a î + R sin(ωt) + b ĵ Ellipse ( x ) 2 ( y ) 2 + = 1 in stanar position; a, b, ω constant: a b r(t) = a cos(ωt) î + b sin(ωt) ĵ Curves in R 3 Line passing through the point P = (x 0, y 0, z 0 ), parallel to v = v 1 î + v 2 ĵ + v 3 k : r(t) = OP + tv ( ) = x 0 î + y 0 ĵ + z 0 k ( ) + t v 1 î + v 2 ĵ + v 3 k = (x 0 + tv 1 ) î + (y 0 + tv 2 ) ĵ + (z 0 + tv 3 ) k Helix centere about z-axis; raius r; r, b, ω constant: r(t) = r cos(ωt) î + r sin(ωt) ĵ + bt k

3 BSU Math 275 sec 002,003 Fall 2018 (Ultman) Final Exam Notes 3 Common Parameterizations for Some Important Surfaces an Corresponing Surface Elements Graphs of Functions (Cartesian) (Can solve for z an get the entire surface) General Equation: z = f (x, y) Cartesian parameterization: r(x, y) = x î + y ĵ + f (x, y) k S + k = S k = S = You can use this to fin S an S for any surface that can be written in the form z = f (x, y). This inclues: cones z = m x 2 + y 2 (m a constant) hemispheres z = ± P 2 (x 2 + y 2 ) (P a constant) planes z = Ax + By + D (A, B, D constants) parabolois z = x 2 + y 2 etc. General Equation (Cartesian): z = x 2 + y 2 Cone (cylinrical/polar) cylinrical/polar parameterization: r(r, θ) = r cos θ î + r sin θ ĵ + r k S + k = S k = S =

4 BSU Math 275 sec 002,003 Fall 2018 (Ultman) Final Exam Notes 4 Special Planes: Planes Parallel to Coorinate Planes Parallel to yz-plane (Cartesian): x = c (c a constant) Cartesian parameterization: r(y, z) = c î + y ĵ + z k S + î = S î = S = Parallel to xz-plane (Cartesian): y = c (c a constant) Cartesian parameterization: r(x, z) = x î + c ĵ + z k S + ĵ = S ĵ = S = Parallel to xy-plane (Cartesian & cylinrical/polar): z = c (c a constant) Cartesian parameterization: r(x, y) = x î + y ĵ + c k S + k = S k = S = cylinrical/polar parameterization: r(r, θ) = r cos θ î + r sin θ ĵ + c k S + k = S k = S =

5 BSU Math 275 sec 002,003 Fall 2018 (Ultman) Final Exam Notes 5 Cyliners Centere About z-axis (cylinrical/polar): General Equation (Cartesian): x 2 + y 2 = R 2 (Cartesian) cylinrical parameterization: r(θ, z) = R cos θ î + R sin θ ĵ + z k S in = S out = S = Spheres (cylinrical/polar & spherical) General Equation (Cartesian): x 2 + y 2 + z 2 = P 2 (P a constant) cylinrical parameterization (upper hemisphere): r(r, θ) = r cos θ î +r sin θ ĵ + P 2 r 2 k S in = S out = S = cylinrical parameterization (lower hemisphere): r(r, θ) = r cos θ î +r sin θ ĵ P 2 r 2 k S in = S out = S = spherical parameterization: r(φ, θ) = P sin φ cos θ î + P sin φ sin θ ĵ + P cos φ k S in = S out = S =

6 BSU Math 275 sec 002,003 Fall 2018 (Ultman) Final Exam Notes 6 Derivatives & Integrals x c = 0 x x n = nx n 1 x ex = e x x ln x = 1 x x ax = a x ln a scalar multiple rule: sum rule: x prouct rule: sin x = cos x x cos x = sin x x x tan x = sec2 x x cot x = csc2 x sec x = sec x tan x x csc x = csc x cot x x [ ] cf (x) = cf (x) x [ ] f (x) + g(x) = f (x) + g (x) [ ] f (x)g(x) = f (x)g(x) + f (x)g (x) x quotient rule: x chain rule: [ f (x) g(x) x ] n an c are constants = f (x)g(x) f (x)g (x) ( g(x) ) 2 [ f ( g(x) )] = f (g(x))g (x) x n x = x n+1 + C (n 1) n x = ln x + C x e x x = e x + C ln x x = x ln x x + C sin x x = cos x + C cos x x = sin x + C sec x x = ln sec x + tan x + C csc x x = ln csc x + cot x + C sin 2 x x = 1 2 x 1 4 sin(2x) + C cos 2 x x = 1 2 x sin(2x) + C 1 x = arctan x + C 1 + x 2 scalar multiple rule: cf (x) x = c sum rule: f (x) + g(x) x = f (x) x f (x) x + u-substitution: f ( g(x) ) g (x) x = u = g(x) integration by parts: f (x)g(x) x = uv u = g (x) x v u f (u) u g(x) x u = f (x) v = g(x) x u = f (x) x v = g(x) x c, n, an C are constants

7 BSU Math 275 sec 002,003 Fall 2018 (Ultman) Final Exam Notes 7 Notes If stuent evaluation response rate reaches 55%, you will be allowe to use this entire page for notes. 55% 55% 55% 55% 55% 55% 55% 55% 55% 55% 55% 55% 55% 55% 55%

8 BSU Math 275 sec 002,003 Fall 2018 (Ultman) Final Exam Notes 8 Notes If stuent evaluation response rate reaches 60%, you will be allowe to use this entire page for notes. If stuent evaluation response rate reaches 65%, you will be allowe to use calculators on the final exam.