r t t x t y t z t, y t are zero, then construct a table for all four functions. dy dx 0 and 0 dt dt horizontal tangent vertical tangent

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1 3. 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.. Parametric and Polar ures Distance r r t t x t y t z t o sketch rt xt ˆ i ytˆj: Find all alues of t at which any of x t, y t are zero, then construct a table for all four functions. dy dx 0 and 0 horizontal tangent dy dx 0 and 0 ertical tangent Polar coordinates: x r y r r x y cos, sin ;, tan xt, y x r,. r, n and r, n n are the same point as o sketch r f, sketch artesian y f x with y r, x a polar sketch. r 0 and r 0 is a tangent at the pole. o o o. Vectors yt,, then transfer onto he component of ector u in the direction of ector is u u ˆ ucos d d u d u u and d d u d d u d u u u he distance along a cure between two points whose parameter alues are t 0 and t is tt r 0 t0 t0 t0 t t t t t ds d dx dy dz L ds

2 ENGI 4430 Formula heets Page 3-0 he distance along a polar cure r f is L dr r d d he unit tangent at any point on a cure is dr dr dr ds he unit principal normal at any point on a cure is he unit binormal is d d d N, where radius of curature ds B N Velocity is dr t, speed is t t dr ds and he acceleration [ector] is d d r d x ˆ d y ˆ d z a t i j kˆ a ann d where a ˆ a a and ˆ an a a an a he surface of reolution of he cured surface area from x y f x around y c is y c z f x c a to x b b is A f x c f x dx dx he area between a cure and the x axis is A y t t a he area swept out by a polar cure is A r d omponents of elocity: radial r, transerse r,, N 0 omponents of acceleration: d d aradial r r, atranserse r r r, a, a a a r Line parallel to abc x xo y yo z zo through xo, yo, z o is a b c (except where any of a, b, c is zero) Plane normal to A B containing x, y, z is Ax + By + z + D = 0, where D Ax By z o o o o o o tb a N

3 ENGI 4430 Formula heets Page Multiple Integrals b hx d q y f x, y da f x, y dy dx f x, y dx dy a g x c p y D r or f x, y da f r, r dr d r D entre of mass: Moment of inertia M r da da or dv dv m r r D D V V I y da, I x da, I I I r da x y x y R R R Parallel axis theorem, second moment Ix of mass m about axis y y0 a distance b from the axis y y through the centre of mass: Ix Ix mb 4. treamlines (lines of force) treamlines to F f f f3 are the solutions of d r k ds F dx dy dz (except that fi 0 that component is constant). f f f 3 5. Numerical Integration b a ab, diided into n equal interals. h n b h rapezoidal rule: f xdx f0 f f fn fn a impson s rule: b h f x dx f0 f f f3 f4 fn fn f a 3 f xn Newton s method to sole f x 0 : xn xn f x n 6. he Gradient Vector he directional deriatie of f in the direction of the unit ector û is n D f uˆ f uˆ r dx dx dxn, where f and n df dr f f f d f x x x angent plane to f x, y, z c at,, P xo yo z o is, where o f P n r n r n.

4 ENGI 4430 Formula heets Page 3-04 If x, y ux, y ˆ i x, yˆ u j and di 0, then the stream function x y xy, exists such that and u. treamlines are x, y c. x y Laplacian of f curl grad f 0 F di curl F 0 f g f g f g V V V di grad V 7. onersions between oordinate ystems o conert a ector expressed in artesian components ˆi ˆj k ˆ into the x y z equialent ector expressed in cylindrical polar coordinates ˆ ˆ ˆ zk, express in terms of,,z the artesian components x, y, z x cos, y sin, z z ; then ealuate using cos sin 0 x sin cos 0 y 0 0 z z Use the inerse matrix [= transpose] to transform back to artesian coordinates. o conert a ector expressed in artesian components ˆi ˆj k ˆ into the x y z equialent ector expressed in spherical polar coordinates ˆ ˆ ˆ r r, express the artesian components x, y, z r,, using x r sin cos, y r sin sin, z r cos ; then ealuate in terms of r sin cos sin sin cos x cos cos cos sin sin y sin cos 0 z Use the inerse matrix [= transpose] to transform back to artesian coordinates.

5 ENGI 4430 Formula heets Page 3-05 Basis Vectors ylindrical Polar: d d ˆ ˆ r ˆ zkˆ d ˆ d ˆ ˆ ˆ zkˆ d kˆ 0 pherical Polar: d rˆ d ˆ sin d ˆ d ˆ d ˆ cos d r ˆ d ˆ sin ˆ cos ˆ d r r r r ˆ r rˆ r ˆ r sin ˆ Gradient operator in any orthonormal coordinate system eˆ eˆ eˆ 3 Gradient operator h u h u h3 u3 eˆ V eˆ ˆ V e3 V Gradient V h u h u h u 3 3 Diergence h h f h h f h h f F h h h 3 u u u url F h h h h eˆ h f u h eˆ h f 3 u h eˆ h f u3 Laplacian h h3 V h3 h V hh V V h h h 3 u h u u h u u3 h3 u 3

6 ENGI 4430 Formula heets Page 3-06 dv h h h 3 du du du 3 x, y, z dududu3 u, u, u 3 x x x u u u 3 y y y u u u 3 z z z u u u 3 artesian: hx hy hz ylindrical polar: h hz, h pherical polar: hr, h r, h r sin du du du 3 8. Line Integrals Work done by F along cure is W he location r of the centre of mass of a wire is r M, where m t t t0 t0 F dr F ds ds ds dr dx dy dz M r, m and. If a potential function V for F exists, then W = (potential difference) V end start dr Green s heorem For a simple closed cure enclosing a finite region D of f and for any ector function F f that is differentiable eerywhere on and eerywhere in D, f f F dr x y D da

7 ENGI 4430 Formula heets Page 3-07 Path Independence When a ector field F is defined on a simply connected domain, these statements are all equialent (that is, all of them are true or all of them are false): F for some scalar field that is differentiable eerywhere in ; F is conseratie; F dr is path-independent (has the same alue no matter which path within is chosen between the two endpoints, for any two endpoints in ); end start F dr (for any two endpoints in ); F dr 0 for all closed cures lying entirely in ; f x f y eerywhere in ; and F 0 eerywhere in (so that the ector field F is irrotational). here must be no singularities anywhere in the domain in order for the aboe set of equialencies to be alid. 9. urface Integrals - Projection Method For surfaces z f x, y, f f N x y and z z g r d g r da (where da dx dy ) x y D urface Integrals - urface Method With a coordinate grid (u, ) on the surface, r r he total flux of a ector field F through a surface is ˆ r r F d F N d F du d u ome common parametric nets are listed on pages 9.9 and 9.0. r r g d g du d u

8 ENGI 4430 Formula heets Page heorems of Gauss and tokes; Potential Functions Gauss diergence theorem: F d V F dv on a simply-connected domain. 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: E d 0 otherwise tokes theorem: d curl F dr F N F d On a simply-connected domain the following statements are either all true or all false: F is conseratie. F F 0 P P F dr - independent of the path between the two points. end F dr 0 start x y z is the potential function for F, so that F F F 3.. Major lassifications of ommon PDEs u u u u u Ax, y B x, y x, y f x, y, u,, x x y y x y D = B 4A Hyperbolic, whereer (x, y) is such that D > 0; Parabolic, whereer (x, y) is such that D = 0; Elliptic, whereer (x, y) is such that D < 0.

9 ENGI 4430 Formula heets Page 3-09 d Alembert olution A.E.: u u u A B r x, y x x y y A B.F.: u x y f y x g y x 0,, [except when D = 0] where B D and B D and D = B 4A A A When D = 0, u x, y f y x hx, yg y x, where hx, y is a linear function that is neither zero nor a multiple of y x. P..: if RH = n th order polynomial in x and y, then try an (n+) th order polynomial.. he Wae Equation Vibrating Finite tring y y c for 0 x L and t 0 t x y x,0 f x for 0 x L and ubstitute yx, t X x t y t x,0 y 0, t y L, t 0 for t 0, with g x for 0 x L into the PDE.... leads, ia Fourier series, to L n u n x n ct y x, t f usin du sin cos L n 0 L L L L n u n x n ct g usin du sin sin c n n 0 L L L he Heat Equation If the temperature u x, t in a bar satisfies u t u k x together with the boundary conditions u0, t and ul, t and the initial condition ux,0 f x, X x t... leads to,, u x t L u x t x t x L where, then n z n x n kt x, t f z z sin dz sin exp L n 0 L L L L

10 ENGI 4430 Formula heets Page 3-0 [pace for Additional Notes]