Rectangular Polar/Cylindrical Spherical Parametric Vector Matrix

Transcript

1 Hrold s Clculus 3 ulti-cordinte System Chet Sheet 15 Octoer 017 Point Rectngulr Polr/Cylindricl Sphericl Prmetric Vector trix -D f(x) y (x, y) (, ) 3-D f(x, y) z (x, y, z) 4-D f(x, y, z) w (x, y, z, w) Slope-Intercept Form: y mx + (r, θ) or r θ x r cos θ y r sin θ z z r x + y r ± x + y tn θ ( y x ) θ tn 1 ( y x ) (ρ, θ, φ) x ρ sin φ cos θ y ρ sin φ sin θ z ρ cos φ ρ r + z ρ x + y + z tn θ ( y x ) φ cos 1 z ( x + y + z ) φ cos 1 ( z ρ ) Point (,) in Rectngulr : x(t) y(t) t 3 rd vrile, usully time, with 1 degree of freedom (df) r x 0, y 0, z 0 [] [x] [] Point-Slope Form: y y 0 m (x x 0 ) Line Generl Form: Ax + By + C 0 where A nd B 0 Clculus Form: f(x) f () x + f(0) where m f () x x 0 3-D: y y 0 z z 0 c < x, y > < x 0, y 0 > + t <, > < x, y > < x 0 + t, y 0 + t > where <, > < x x 1, y y 1 > x(t) x 0 + t y(t) y 0 + t m y x y y 1 x x 1 r r 0 + t v x 0, y 0, z 0 + t,, c [ ] [ x y ] [c] [ c d ] [x y ] [e f ] Copyright y Hrold Toomey, WyzAnt Tutor 1

2 Rectngulr Polr/Cylindricl Sphericl Prmetric Vector trix Plne (x x 0 ) + (y y 0 ) + c(z z 0 ) 0 x + y + cz d where d x 0 + y 0 + cz 0 f(x, y) Ax + By + C (vrile, constnt, constnt) where ρ tkes on ll vlues in the domin (0 ρ < ) r r 0 + sv + tw where: s nd t rnge over ll rel numers v nd w re given vectors defining the plne r 0 is the vector representing the position of n ritrry (ut fixed) point on the plne n (r r 0 ) 0 Conics Generl Eqution for All Conics: Ax + Bxy + Cy + Dx + Ey + F 0 where Line: A B C 0 Circle: A C nd B 0 Ellipse: AC > 0 or B 4AC < 0 Prol: AC 0 or B 4AC 0 Hyperol: AC < 0 or B 4AC > 0 ote: If A + C 0, squre hyperol Rottion: If B 0, then rotte coordinte system: cot θ A C B x x cos θ y sin θ y y cos θ + x sin θ ew (x, y ), Old (x, y) rottes through ngle θ from x- xis Generl Eqution for All Conics: Verticl Axis of Symmetry: p r 1 e cos θ Horizontl Axis of Symmetry: p r 1 e sin θ (1 e ) 0 e < 1 where p { d for { e 1 (e 1) e > 1 p semi-ltus rectum or the line segment running from the focus to the curve in direction prllel to the directrix Eccentricity: Circle e 0 Ellipse 0 e < 1 Prol e 1 Hyperol e > 1 Copyright y Hrold Toomey, WyzAnt Tutor

3 Rectngulr Polr/Cylindricl Sphericl Prmetric Vector trix x + y r (x h) + (y k) r Centered t Origin: r (constnt) θ θ [0, π] or [0, 360 ] Circle Generl Form: Ax + Bxy + Cy + Dx + Ey + F 0 where A C nd B 0 Focus nd Center: (h, k) Centered t (r 0, φ): r + r 0 rr 0 cos(θ φ) R Hint: Lw of Cosines or r r 0 cos(θ φ) + r 0 sin (θ φ) ρ constnt θ θ [0, π] φ constnt 0 x(t) r cos(t) + h y(t) r sin(t) + k [t min, t mx ] [0, π) (h, k) center of circle (h, k) Sphere x + y + z r (x h) + (y k) + (z l) r Focus nd center: (h, k, l) Generl Form: Ax + By + Cz + Dxy + Eyz + Fxz + Gx + Hy + Iz + J 0 where A B C > 0 Cylindricl to Rectngulr: x r cos (θ) y r sin (θ) z z Sphericl to Rectngulr: x r sin θ cos φ y r sin θ sin φ z r cos θ Rectngulr to Cylindricl: r x + y Sphericl to Cylindricl: ρ r sin (θ) φ φ z r cos (θ) ρ constnt θ θ [0, π] φ φ [0, π] Rectngulr to Sphericl: r x + y + z θ rccos ( z r ) φ rctn ( y x ) Cylindricl to Sphericl: r ρ + z θ rctn ( ρ z ) rccos (z r ) φ φ Rectngulr: x r [ y] z Cylindricl: r cos (θ) r [ r sin (θ)] z Sphericl: r sin θ cos φ r [ r sin θ sin φ] r cos θ Copyright y Hrold Toomey, WyzAnt Tutor 3

4 Ellipse Rectngulr Polr/Cylindricl Sphericl Prmetric Vector trix (x h) (y k) + 1 Generl Form: Ax + Bxy + Cy + Dx + Ey +F 0 where B 4AC < 0 or AC > 0 Center: (h, k) Vertices: (h ±, k) nd (h, k ± ) Foci: (h ± c, k) Focus length, c, from center: c Eccentricity: e c If B 0, then rotte coordinte system: A C cot θ B x x cos θ y sin θ y y cos θ + x sin θ Verticl Axis of Symmetry: r (1 e ) 1 ± e cos θ Horizontl Axis of Symmetry: r (1 e ) 1 ± e sin θ Eccentricity: 0 < e < 1 r(θ) ( cos θ) + ( sin θ) reltive to center (h,k) Interesting ote: The sum of the distnces from ech focus to point on the curve is constnt. d 1 + d k x(t) cos(t) + h y(t) sin(t) + k [t min, t mx ] [0, π] (h, k) center of ellipse Rotted Ellipse: x(t) cos t cos θ sin t sin θ + h y(t) cos t sin θ + sin t cos θ + k θ the ngle etween the x-xis nd the mjor xis of the ellipse Ellipsoid ew (x, y ), Old (x, y) rottes through ngle θ from x- xis (x h) (y k) + 1 (z l) + c r cos θ + r sin θ + z c 1 r cos θ sin φ + r sin θ sin φ + r cos φ c 1 x(t, u) cos(t) cos(u) + h y(t, u) cos(t) sin(u) + k z(t, u) c sin(t) + l [t min, t mx ] [ π, π ] [u min, u mx ] [ π, π] (h, k, l) center of ellipsoid (x v) T A 1 (x v) 1 Centered t vector v Copyright y Hrold Toomey, WyzAnt Tutor 4

5 Prol ose Cone Rectngulr Polr/Cylindricl Sphericl Prmetric Vector trix Verticl Axis of Symmetry: x 4 py (x h) 4p(y k) Vertex: (h, k) Focus: (h, k + p) Directrix: y k p Horizontl Axis of Symmetry: y 4 px (y k) 4p(x h) Vertex: (h, k) Focus: (h + p, k) Directrix: x h p Generl Form: Ax + Bxy + Cy + Dx + Ey + F 0 where B 4AC 0 or AC 0 If B 0, then rotte coordinte system: A C cot θ B x x cos θ y sin θ y y cos θ + x sin θ ew (x, y ), Old (x, y) rottes through ngle θ from x- xis (x h) (y k) (z l) + c Verticl Axis of Symmetry: ed r 1 ± e cos θ Horizontl Axis of Symmetry: ed r 1 ± e sin θ Eccentricity: e 1 where d p Verticl xis of symmetry: x(t) pt + h y(t) pt + k (opens upwrds) or y(t) pt + k (opens downwrds) [t min, t mx ] [ c, c] (h, k) vertex of prol Horizontl xis of symmetry: x(t) pt + h y(t) pt + k (opens right) or y(t) pt + k (opens left) [t min, t mx ] [ c, c] Projectile otion: x(t) x 0 + v x t y(t) y 0 + v y t 16t feet y(t) y 0 + v y t 4.9t meters v x v cos θ v y v sin θ Generl Form: x At + Bt + C y Lt + t + where A nd L hve the sme sign Copyright y Hrold Toomey, WyzAnt Tutor 5

6 Hyperol Rectngulr Polr/Cylindricl Sphericl Prmetric Vector trix (x h) (y k) 1 Generl Form: Ax + Bxy + Cy + Dx + Ey + F 0 where B 4AC > 0 or AC < 0 If A + C 0, squre hyperol Center: (h, k) Vertices: (h ±, k) Foci: (h ± c, k) Focus length, c, from center: c + Eccentricity: e c + sec θ 3-D (x h) (y k) + (x h) 1 (y k) 1 (z l) c (z l) + c If B 0, then rotte coordinte system: A C cot θ B x x cos θ y sin θ y y cos θ + x sin θ ew (x, y ), Old (x, y) rottes through ngle θ from x- xis Verticl Axis of Symmetry: r (e 1) 1 ± e cos θ Horizontl Axis of Symmetry: r (e 1) 1 ± e sin θ Eccentricity: e > 1 where e c + sec θ > 1 reltive to center (h,k) cos 1 ( 1 e ) < θ < cos 1 ( 1 e ) p semi-ltus rectum or the line segment running from the focus to the curve in the directions θ ± π Interesting ote: The difference etween the distnces from ech focus to point on the curve is constnt. d 1 d k Left-Right Opening Hyperol: x(t) sec(t) + h y(t) tn(t) + k [t min, t mx ] [ c, c] (h, k) vertex of hyperol Alternte Form: x(t) ± cosh(t) + h y(t) sinh(t) + k Up-Down Opening Hyperol: x(t) tn(t) + h y(t) sec(t) + k Alternte Form: x(t) sinh(t) + h y(t) ± cosh(t) + k Generl Form: x(t) At + Bt + C y(t) Dt + Et + F where A nd D hve different signs Copyright y Hrold Toomey, WyzAnt Tutor 6

7 Rectngulr Polr/Cylindricl Sphericl Prmetric Vector trix Limit lim f(x) L x c 1 st Derivtive nd Derivtive f f(x + h) f(x) (x) lim h 0 h f f(x) f(c) (c) lim x c x c f (x) dy y D x f (x) d (dy ) d y y dy dy dr sin θ + r cos θ dr cos θ r sin θ Hint: Use Product Rule for y r sin θ x r cos θ d y d (dy ) d (dy ) dy dy, provided 0 d y d (dy ) Riemnn Sum: n d (dy ) S f(y i )(x i x i 1 ) i 1 d (r ) r Unit tngent vector T (t) r (t) r (t) where r (t) 0 Unit norml vector (t) T (t) T (t) where T (t) 0 Left Sum: Integrl F(x) f(x) F() F() S ( 1 n ) [ f() + f ( + 1 n ) + f ( + n ) + + f( 1 n ) ] iddle Sum: S ( ) [f ( + ) + f ( + n n n ) + + f( 1 n )] r (t) f(t), g(t), h(t) Right Sum: S ( 1 n ) [f ( + 1 n ) + f ( + n ) + + f()] Doule Integrl d(y) f(x, y) dy c(y) Sme s rectngulr, ut f(x, y) f(ρ cos φ, ρ sin φ) Copyright y Hrold Toomey, WyzAnt Tutor 7

8 Triple Integrl Inverse Functions Arc Length Curvture Perimeter d(z) Rectngulr Polr/Cylindricl Sphericl Prmetric Vector trix g(y,z) f(x, y, z) dy dz c(z) e(y,z) If f(x) y, then f 1 (y) x Inverse Function Theorem: f 1 1 () f () where f () L 1 + [f (x)] Proof: s (x x 0 ) + (y y 0 ) s ( x) + ( y) ds + dy ds + dy ( ) ds + ( dy ) ds (1 + ( dy ) ) ds 1 + ( dy ) κ L ds Sme s rectngulr, ut f(x, y, z) f(ρ cos φ, ρ sin φ, z) if y sin θ if y cos θ if y tn θ if y csc θ if y sec θ if y cot θ then θ sin 1 y then θ cos 1 y then θ tn 1 y then θ csc 1 y then θ sec 1 y then θ cot 1 y Polr: L r + ( dr ) Where r f(θ) Circle: L s rθ Proof: L (frction of circumference) π (dimeter) L ( θ ) π (r) rθ π y κ(θ) r + r rr (1 + y ) 3 (r + r ) 3 Squre: P 4s Rectngle: P l + w Tringle : P + + c for r(θ) Circle: C πd πr Ellipse: C π( + ) π Ellipse: C 4 1 ( c ) sin θ 0 Sme s rectngulr, ut f(x, y, z) f(ρ cos θ sin φ, ρ sin θ sin φ, ρ cos φ) θ rcsin y θ rccos y θ rctn y θ rccsc y θ rcsec y θ rccot y C πd πr Rectngulr D: L ( L ( t L ( dr t t 1 ( dρ ) t 1 κ + ( dy Rectngulr 3D: + ( dy + ( dz Cylindricl: + r ( + ( dz Sphericl: L + ρ sin φ ( + ρ ( dφ (z y y z ) + (x z z x ) + (y x x y ) (x + y + z ) 3 where f(t) (x(t), y(t), z(t)) L r (t) t s(t) r (u) 0 κ d T ds κ T (t) r (t) du κ r (t) r (t) r (t) 3 (See Wikipedi : Curvture) Circle: C πr Copyright y Hrold Toomey, WyzAnt Tutor 8

9 Are Lterl Surfce Are Totl Surfce Are Surfce of Revolution Rectngulr Polr/Cylindricl Sphericl Prmetric Vector trix Squre: A s² Rectngle: A lw Rhomus: A ½ Prllelogrm: A h Trpezoid: A ( 1+ ) h Kite: A d 1 d Tringle: A ½ h Tringle: A ½ sin(c) Tringle: A s(s )(s )(s c), where s ++c Equilterl Tringle: A ¼ 3s Frustum: A 1 3 ( 1+ ) h Circle: A πr² Circulr Sector: A ½ r²θ Ellipse: A π Cylinder: S πrh Cone: S πrl S π f(x) 1 + [f (x)] Cue: S 6s² Rectngulr Box: S lw + wh + hl Regulr Tetrhedron: S h Cylinder: S πr (r + h) Cone: S πr² + πrl πr (r + l) Sphere: S 4πr² For revolution out the x-xis: A π f(x) 1 + ( dy ) For revolution out the y-xis: A π x 1 + ( dy ) dy A 1 [f(θ)] where r f(θ) Proof: Are of sector: A s dr r θ dr 1 r θ where rc length s r θ For rottion out the x-xis: S πy ds For rottion out the y-xis: S πx ds ds r + ( dr ) r f(θ), θ Ellipsoid: S 1 p 4π ( p p + p c p + p c p ) 3 Where p , E 1.061% (Knud Thomsen s Formul) For revolution out the x-xis: A π r cos θ r + ( dr ) For revolution out the y-xis: A π r sin θ r + ( dr ) Sphere: S 4πr² Sphere: S 4πr² Ellipsoid: S A g(t) f (t) where f(t) x nd g(t) y or x(t) f(t) nd y(t) g(t) Simplified: A y(t) (t) Proof: f(x) y f(x) g(t) df(t) f (t) For rottion out the x-xis: S πy ds For rottion out the y-xis: S πx ds ds ( ) + ( dy ) if x f(t), y g(t), t where For revolution out the x-xis: A x π y(t) ( + ( dy For revolution out the y-xis: A y π x(t) ( + ( dy A r u r du dv v D Copyright y Hrold Toomey, WyzAnt Tutor 9

10 Volume Rectngulr Polr/Cylindricl Sphericl Prmetric Vector trix Cue: V s³ Rectngulr Prism: V lwh Cylinder: V πr²h Tringulr Prism: V Bh Tetrhedron: V ⅓ h Pyrmid: V ⅓ Bh Sphere: V 4 3 πr3 Ellipsoid: V 4 3 πc Cone: V ⅓ h ⅓ πr²h f(r cos θ, r sin θ, z)r dz dr ρ sin φ cos θ, f ( ρ sin φ sin θ,) ρ cos φ ρ sin φ dρ dφ Ellipsoid: V 4 3 π det(a 1 ) f(x, y, z) dy dz Disc ethod - Rottion out the x-xis: V π [f(x)] Disc ethod: Cylindricl Shell ethod: Wsher ethod - Rottion out the x-xis: Volume of Revolution V π { [f(x)] [g(x)] } Cylinder ethod - Rottion out the y-xis: V πx f(x) (circumference) (hight) oments of Inerti Center of ss I m i r i m r dr 0 R 1 m i r i where m i 1-D for Discrete: x cm m 1x 1 + m x m 1 + m -D for Discrete: y m i x i x m i y i x y, y x 3-D for Discrete: x cm x 1 m i x i y cm y 1 m i y i z cm z 1 m i z i 3-D for Continuous: x 1 x dm 0 y 1 y dm 0 z 1 z dm where dm 0 nd dm ρ dz dy 0 I ρ(r) d(r) dv(r) V R 1 r dm R 1 ρ(r) r dv V Where r is distnce from the xis of rottion, not origin. (see Wikipedi) Copyright y Hrold Toomey, WyzAnt Tutor 10

11 Rectngulr Polr/Cylindricl Sphericl Prmetric Vector trix Grdient ƒ f x f f ƒ(ρ, ϕ, z) i + j + y z k f ρ e ρ + 1 f ρ ϕ e ϕ + f z e z ƒ(r, θ, ϕ) f r e r + 1 f r θ e θ + 1 f r sin θ ϕ e ϕ ( ƒ(x)) v D v f(x) ƒ f i x j e i e j where ƒ (ƒ 1, ƒ, ƒ 3 ) Line Integrl f ds C f(r(t)) r (t) F(r) dr C F(r(t)) r (t) f ds S f(x(s, t)) T x s x ds t v ds S Surfce Integrl where x(s, t) (x(s, t), y(s, t), z(s, t)) nd ( x s x t ) (y, z) (z, x) (x, y) (,, (s, t) (s, t) (s, t) ) (v n) ds S v(x(s, t)) T ( x s x ) ds t Copyright y Hrold Toomey, WyzAnt Tutor 11