5. Suggestions for the Formula Sheets

Transcription

1 5. uggestions for the Formula heets Below are some suggestions for many more formulae than can be placed easily on both sides of the two standard 8½" " sheets of paper for the final examination. It is strongly recommended that students compose their own formula sheets, to suit each indiidual s needs.. Vectors The component of ector u in the direction of ector d du ( ) u i i + ui d and d du d d ( ) u u + u + u d is u u i ˆ ucosθ The distance along a cure between two points whose parameter alues are t and t is t t t t t ds dr dx dy dz L ds + + t t t t t The unit tangent at any point on a cure is dr dr dr T ds The unit principal normal at any point on a cure is dt dt dt N ρ ds The unit binormal is B T N Velocity is () t dr, speed is () t () t dr ds and T The acceleration [ector] is d d r () d x d y d z a t,, a TT + ann d where a ˆ T a i i at and ˆ an κ a at a an i

2 ENGI Formula heets Page. Lines of Force The lines of force r associated with a ector field F are gien by ˆ dr dr dx dy dz T F kf( s) ( k ds ) ds ds f f f. The Gradient Vector The directional deriatie of f in the direction of the unit ector û is D ˆ uˆ f f iu df dr f f f dr dx dx dxn fi, where f,,, and,,, Plane with normal ector n +A, B,, passing through point r o +x o, y o, z o, : nr i nr i o Ax + By + z + D, where D ( Axo + Byo + zo ) Line with direction ector +,,, passing through point r o +x o, y o, z o, : x xo y yo z zo r ro + t ( except when a i ) Tangent plane to f (x, y, z) c at P(x o, y o, z o ) is nr nr, where i i o n f. P n.4 Diergence and url f curl grad f i F di curl F Laplacian of i di grad V ( ) V V V If ( x, y) u( x, y) ˆ i + ( x, y) ˆ j and di +, then the stream function ψ ψ ψ ( x, y) exists such that and u. treamlines are ψ ( x, y) c..5 onersions between oordinate ystems To conert a ector expressed in artesian components ˆ ˆ ˆ x i + yj + zk into the equialent ector expressed in cylindrical polar coordinates ˆ ˆ ρ ˆ ρ + φ φ + zk, express the artesian components x, y, z in terms of ( ρφ,,z) using x ρ cos φ, y ρ sin φ, z z ; then ealuate

3 ENGI Formula heets Page.5 onersions between oordinate ystems (continued) ρ cosφ sinφ x φ sinφ cosφ y z z Use the inerse matrix [ transpose] to transform back to artesian coordinates. To conert a ector expressed in artesian components ˆ ˆ ˆ x i + yj + zk into the equialent ector expressed in spherical polar coordinates rˆ + ˆ θ + ˆ φ, express the artesian components x, y, z in terms of ( r, θφ, ) using x rsinθ cos φ, y rsinθ sin φ, z rcosθ ; then ealuate r θ φ r sinθ cosφ sinθ sinφ cosθ x cos cos cos sin sin θ θ φ θ φ θ y φ sinφ cosφ z Use the inerse matrix [ transpose] to transform back to artesian coordinates..6 Basis Vectors in Other oordinate ystems ylindrical Polar: d dφ ˆ ˆ ρ φ r ρ ˆ ρ + z kˆ d ˆ dφ φ ˆ ρ ρ ˆ ρ + ρφ ˆ φ + z e d kˆ pherical Polar: d rˆ d θ ˆ sin d φ θ + θ ˆ φ d ˆ θ d θ ˆ cos d φ r + θ ˆ φ d ˆ φ ( sin ˆ cos ˆ dφ θ r + θ θ ) r r r ˆ r rˆ + r θ ˆ θ + rsinθφ ˆ φ ˆz

4 ENGI Formula heets Page 4.7 Gradient Operator in Other oordinate ystems Gradient operator eˆ eˆ eˆ + + h h h Gradient eˆ ˆ ˆ V e V e V V + + h h h Diergence ( ) ( ) ( ) h h f h hf h h f F + + hhh url F h eˆ h eˆ h eˆ hhh h f h f h f Laplacian hh V hh V hh V V hhh + + h h h dv h h h du du du ( x, y, z) ( u, u, u ) du du z z z du du du du artesian: h x h y h z. ylindrical polar: h h, h. ρ z φ ρ pherical polar: h, h r, h r sin. r θ φ θ

5 ENGI Formula heet Page 5. Line Integrals The location r of the centre of mass of a wire is M r, where m t t ds ds ds dr dx dy dz M ρr, m ρ and + + t t.. Green s Theorem For a simple closed cure enclosing a finite region D of ú and for any ector function F f, f that is differentiable eerywhere on and eerywhere in D, f f Fdr i da D. Path Independence When a ector field F is defined on a simply connected domain Ω, these statements are all equialent (that is, all of them are true or all of them are false): F φ for some scalar field φ that is differentiable eerywhere in Ω; F is conseratie; Fdr i is path-independent (has the same alue no matter which path within Ω is chosen between the two endpoints, for any two endpoints in Ω); Fdr i φend φstart (for any two endpoints in Ω); Fdr i for all closed cures lying entirely in Ω; f f eerywhere in Ω; and F eerywhere in Ω (so that the ector field F is irrotational). There must be no singularities anywhere in the domain Ω in order for the aboe set of equialencies to be alid..4 urface Integrals - Projection Method f f For surfaces z f (x, y), N,, + and z z g( r) d g( r) + + da (where da dx dy) D

6 ENGI Formula heet Page 6.5 urface Integrals - urface Method With a coordinate grid (u, ) on the surface, g ( r) d g ( r) du d The total flux of a ector field F through a surface is ˆ Φ Fd i FN i d Fi du d ome ommon Parametric Nets ) The circular plate ( x x ) ( ) o + y yo a in the plane z z o. Let the parameters be r, θ where < r a, θ < π x x o + r cos θ, y y o + r sin θ, z z o ˆi ˆj kˆ N ± ± cosθ sinθ θ r sinθ r cosθ ± r kˆ o o ) The circular cylinder ( x x ) ( y y ) + a with z o z z. Let the parameters be z, θ where z o z z, θ < π x a cos θ, y a sin θ, z z ˆi ˆj kˆ N ± ± ± acosθ ˆ asinθ ˆ i j z θ asinθ acosθ Outward normal: N acosθ ˆi + asinθ ˆj ( ) o o o where w w w and wo w. Let the parameters here be r, θ where w wo w w r o, a a θ < π x u uo + r cos θ, y o + rsin θ, z w wo + ar ˆi ˆj kˆ N ± ± cosθ sinθ a θ rsinθ r cosθ ± ( arcosθ) ˆ + ( arsinθ) ˆ + r ˆ i j k Outward normal: N ar cosθ ˆi + ar sinθ ˆj r kˆ ) The frustrum of the circular cone w w a ( u u ) + ( )

7 ENGI Formula heet Page 7 4) The portion of the elliptic paraboloid ( ) ( ) o o o with o z z a x x + b y y z z z z Let the parameters here be r, θ where z zo z zo r, θ < π a cos θ + b sin θ a cos θ + b sin θ x x o + r cos θ, y y o + r sin θ, z z o + r (a cos θ + b sin θ) ˆi ˆj kˆ N ± ± cosθ sinθ r( a cos θ + b sin θ) θ r sinθ r cosθ r b a sinθ cosθ ( ) ± ( ar cosθ) ˆ + ( br sinθ) ˆ + rˆ i j k Outward normal: ( cosθ) + ( sinθ) j N ar ˆi br ˆ rkˆ. 5) The surface of the sphere ( x x ) + ( y y ) + ( z z ) a Let the parameters here be θ, φ where θ π, φ < π x x o + a sin θ cos φ, y y o + a sin θ sin φ, z z o + a cos θ ˆ ˆ ˆ i j k N ± ± acosθ cosφ acosθ sinφ asinθ θ φ asinθ sinφ asinθ cosφ a ( ) i ( ) j ( ) sinθ sinθ cosφ ˆ sinθ sinφ ˆ cosθ ˆ k N a sinθ sinθ cosφ ˆ + sinθ sinφ ˆ + cosθ) ˆ i j k ± + + Outward normal: ( ) ( ) ( 6) The part of the plane ( x x ) + B( y y ) + ( z z ) A in the first octant with A, B, > and Ax o + By o + z o >. Let the parameters be x, y where x Ax By z By ; y Ax By z A B ˆi ˆj kˆ A A ˆ Bˆ ˆ N ± ± ± + + y i j k B

8 ENGI Formula heet Page 8.6 Theorems of Gauss and tokes; Potential Functions Gauss diergence theorem: Fd i ifdv on a simply-connected domain. V Gauss law for the net flux through any smooth simple closed surface, in the presence q if encloses O of a point charge q, is: Ed i ε otherwise Fdr FN Fi d tokes theorem: i i d ( curl ) On a simply-connected domain Ω the following statements are either all true or all false: F is conseratie. F φ F Fdr i φ( Pend ) φ( Pstart ) - independent of the path between the two points. Fdr i Ω φ φ φ φ is the potential function for F, so that,, F, F, F z. Fourier eries The Fourier series of f (x) on the interal ( L, L) is where and a nπx nπx f ( x) + ancos bnsin + n L L L nπ x an f x dx n L L L L ( ) cos, (,,,, ) nπ x bn f ( x) sin dx, ( n,,, ) L L L If f (x) is een (f ( x) + f (x) for all x), then b n for all n, leaing a Fourier cosine series (and perhaps a constant term) only for f (x). If f (x) is odd (f ( x) f (x) for all x), then a n for all n, leaing a Fourier sine series only for f (x)..

9 ENGI Formula heet Page 9 4. Major lassifications of ommon PDEs u u u A( x, y) + B ( x, y) + ( x, y) f x, y, u,, D B 4A Hyperbolic, whereer (x, y) is such that D > ; Parabolic, whereer (x, y) is such that D ; Elliptic, whereer (x, y) is such that D <. 4. d Alembert olution A.E.:.F.: u u u A B r x + +, Aλ Bλ + + ( y) ( ) ( λ ) ( λ ) u x, y f y+ x + g y+ x, [except when D ] where B D and B + D λ λ and D B 4A A A u x, y f y+ λx + h x, y g y+ λx, When D, ( ) ( ) ( ) ( ) where h(x, y) is a linear function that is neither zero nor a multiple of ( y λ x) +. P..: if RH n th order polynomial in x and y, then try an (n+) th order polynomial. 4. The Wae Equation Vibrating Finite tring y y c for < x < L and t > with y(, t) y(l, t) for t >, t y(x, ) f (x) for < x < L and t ( x,) ( ) for g x x L ubstitute y(x, t) X(x) T(t) into the PDE.... leads, ia Fourier series, to L y( x, t) nπu nπ x nπct f ( u) sin du sin cos L n L L L + L nπ u nπ x nπct g( u) sin du sin sin π c n n L L L

10 ENGI Formula heet Page 4.4 The Maximum-Minimum Principle Let m and M be the minimum and maximum alues respectiely of u on the boundary of the domain Ω. u r < M u r M r in Ω If u in Ω, then u is subharmonic and ( ) or ( ) u r > m u r m r in Ω If u in Ω, then u is superharmonic and ( ) or ( ) If u in Ω, then u is harmonic (both subharmonic and superharmonic) and u is either constant on Ω or m < u < M eerywhere on Ω. 4.5 The Heat Equation u If the temperature u(x, t) in a bar is k together with the boundary conditions t u(, t) T and u(l, t) T and the initial condition u(x, ) f (x), then T T u x, t x, t + x + T L u(x, t) X(x) T(t)... leads to ( ) ( ) L where T T nπz nπx n π kt ( x, t) f ( z) z T sin dz sin exp L n L L L L