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BYU | ET 502
52 sample documents related to ET 502
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BYU ET 502
Origin of the Name Mathematics Anatolius: Why is mathematics so named? The Peripatetics say that rhetoric and poetry and the whole of popular music can be understood without any course of instructions. But no one can acquire knowledge of the subject -
BYU ET 502
Real Functions REAL FUNCTIONS After the completion of this section the student 1. 2. 3. 4. 5. 6. 7. 8. 9. should recall the definition of the basic algebraic and transcendental functions should be able to determine the main properties of the functi -
BYU ET 502
differentiation derivative f(x) = lim h 0 f(x + h) - f(x) h f (x) f (x + h) equation of tangent line y = f(x 0 )(x - x0 ) + f(x 0 ) if limit exists then function f(x) is differentiable and f\'(x) is a derivative of function f(x) f ( x) = m = -
BYU ET 502
grading symbol out of 100 out of 20 out of 10 out of 5 grade out of 4 excellent perfect + 96-100 20 10 5 almost perfect almost correct answer some details of solution are missing A 4 90-95 very good 18 9 A- 3.7 + small mistake small -
BYU ET 502
integration indefinite integral definite integral partition norm of partition if properties linearity b b F(x) is antiderivative of f ( x) if F( x) = f ( x) then s = {a = x0 ,x1,., xk ,., xn-1 ,x n = b} s = max Dx k k F( x) is antiderivativ -
BYU ET 502
F( x) is antiderivative of f ( x) if indefinite integral b definite integral u - substitution integration by parts F( x) = f ( x) f (x) x n f ( x)dx = F( x) + c F ( x) f (x)dx = F(b) -F( a) a [ f u(x ) u(x)dx = ] f(u)du 2 udv = uv -
BYU ET 502
limits definition of limit uniqueness for any e > 0 there exists if L\'Hospital\'s Rule then lim f(x) = L x c d>0 such that if lim f(x) = L xc 0 < x -c < d then f(x) - L < e L =M Bc o or ) o = open neighborhood of point c (indeterminate ar -
BYU ET 502
table of laplace transforms f ( t) 1 t0 f( s) f (t) t0 f( s) maple 1 s 1 s2 n! s n+1 s>0 e at 1 s -a 1 s>a For calcualtaion of Laplace transform or inverse Laplace transform the package with integral transforms has to be downloaded: t -
BYU ET 502
EngT502 Fall 2005 PELE - POISSON EQUATION-LAPLACE EQUATION 07- DNRN Jing\'ang Li The Laplace Equation: 07 DNRN (Dirichlet- Neumann-Robin- Neumann) u( x, y ) = f 2( x ) x Neumann M Robin u( x, y ) + H3 u = f 3( y ) x 2u = 0 Neumann u( x, -
BYU ET 502
sturm-liouville problem H1 = h1 k1 H2 = h2 k2 X - mX = 0 eigenfunctions norm X ( x) x OE [0, L] 0 L x L kernel 2 n boundary conditions eigenvalues m n = -l2n Xn Xn 2 = X (x)dx 0 K n (x) = Xn (x) Xn Dirichlet Dirichlet X (0 ) = 0 -
BYU ET 502
Journal of Applied Engineering Mathematics Volume 2 MODELING PIEZORESISTIVITY IN SILICON AND POLYSILICON Gary K. Johns Mechanical Engineering Department Brigham Young University Provo,Utah 84602 ABSTRACT The piezoresistive effect of silicon is oft -
BYU ET 502
Journal of Applied Engineering Mathematics Volume 2 1D MOMENTUM INTEGERAL METHOD (THWAITES) VS. 2D FINITE ELEMENT NAVIER-STOKES SOLUTION: COMPUTING FLOW SEPERATION, FLOW RATE AND PRESSURE DROP Gifford Zach Decker Mechanical Engineering Department Br -
BYU ET 502
EngT502 Fall 2005 PELE - POISSON EQUATION-LAPLACE EQUATION 01- RRRR Matt Spencer The Laplace Equation: 01 RRRR Robin- Robin - Robin - Robin) Robin - M Robin u + H 2u y = f 2 ( x) y=M - u + H 3u x Robin = f 3 ( y) x =0 2u = 0 Robin L - -
BYU ET 502
EngT502 Fall 2005 PELE - POISSON EQUATION-LAPLACE EQUATION 01- DDDD Vladimir Solovjov The Laplace Equation: 01 DDDD (Dirichlet- Dirichlet-Dirichlet-Dirichlet) Dirichlet u = f2 ( x ) Dirichlet M Dirichlet u = f3 ( y ) 2u = 0 0 u = f4 ( y ) -
BYU ET 502
TABLES first order o.d.e. LINEAR O.D.E. LINEAR O.D.E. WITH CONSTANT COEFFICIENTS y + P( x)y = Q(x ) general solution integrating factor m ( x ) = e P(x)dx y + ay = Q(x ) general solution integrating factor m (x ) = e ax y (x) = cm -1 (x ) -
BYU ET 502
first order o.d.e. LINEAR O.D.E. LINEAR O.D.E. WITH CONSTANT COEFFICIENTS y + P( x)y = Q(x ) general solution integrating factor m ( x ) = e P(x)dx y + ay = Q(x ) general solution integrating factor m (x ) = e ax y (x) = cm -1 (x ) + m-1 ( -
BYU ET 502
linear o.d.e. LINEAR O.D.E. nth ORDER a i (x ),f( x) OEC (D) n n-1) D R L ny a 0 (x)y ( ) + a1 (x)y ( a 0 ( x) 0 for all +L + a n-1 (x)y + an (x )y = f(x ) linear o.d.e. is normal in D if initial value problem x OED Theorem if equation is no -
BYU ET 502
power-series solution POWER SERIES y (x) = a 0 + a1 (x - x 0 ) + a2 (x - x 0 ) +K = a n (x - x0 ) n=0 2 HOMOGENEOUS LINEAR SECOND ORDER O.D.E. n initial conditions: radius of convergence R diverges converges diverges TAYLOR SERIES x -R 0 -
BYU ET 502
power-series solution POWER SERIES y (x) = a 0 + a1 (x - x 0 ) + a2 (x - x 0 ) +K = a n (x - x0 ) n=0 2 HOMOGENEOUS LINEAR SECOND ORDER O.D.E. n initial conditions: radius of convergence R diverges converges diverges TAYLOR SERIES x -R 0 -
BYU ET 502
linear systems of first order o.d.e. STANDARD FORM FUNDAMENTAL SET OF SYSTEM WITH CONSTANT COEFFICIENTS x 1 x 2 M x n = a11x1 + a12 x 2 +Ka1n xn + f1 = a 21x1 + a 22 x 2 +Ka2 n xn + f2 = a n1x1 + an2 x2 +Ka nn xn + fn x = Ax k 1 k2 x = e lt = -
BYU ET 502
fourier series standard form basic case complex exponential forms (d,d + 2p) f(t) = a0 np np + an cos t + b n sin t 2 n=1 p p = c e n n=- nip t p = c n=- -n e -nip t p relations a 0 = 2c 0 Fourier coefficients a n = cn + c- -
BYU ET 502
hyperbolic functions sinhx = e x - e -x 2 coshx = e x + e -x 2 4 definition derivative sinh x = coshx cosh x = sinhx 3 coshx 2 integration sinhxdx = coshx sinh(-x) = -sinhx coshxdx = sinhx cosh(-x) = coshx 1 0 -1 symmetry valu -
BYU ET 502
bessel functions bessel equation (BE) solutions of BE are bessel function of the 1st kind of order n when x 2y + xy + ( x 2 - n2 ) y = 0 , when n = 1 2,. BE can be obtained from eqn by the change of variable 1 y(x) = R(r ) x = lr J 0 r 2R +rR + -
BYU ET 502
laplace transform Laplace transform L{f( t)} = f(s) = f( t)e 0 - st dt inverse Laplace transform f( t) = L-1 {f(s)} f (t ) if is of exponential order if f( t) Me at \"t 0 for some a,M > 0 existance of Laplace transform f (t ) i -
BYU ET 502
convergence of infinite series infinite series partial sums a k= 0 k = a 0 + a1 + a 2 +L Definition infinite series is convergent if and only if the sequence of partial sums is convergent sn = a 0 + a1 + a 2 +L + a n a k= 0 k =L if and o -
BYU ET 502
complex numbers complex number z OEC we need complex numbers to be able to solve algebraic equations such as x +1=0 which has no solution in real numbers 2 complex plane any point in a plane with coordinates (a,b) is associated with a complex number -
BYU ET 502
Fourier\'s law heat conduction in continuous medium k W m K W 2 m K m2 s W 3 m coefficient of thermal conductivity coefficient of convective heat transfer thermal diffusivity heat generation per unit volume rate of heat genera -
BYU ET 502
Vectors in Euclidian Space free vector a directed segment x position vector z directed segment with the fixed initial point P(x1,y1,z1) y1 y coordinate vector triple of real numbers 1 order tensor ai with index convention: i =1 ,2 ,3 st z1 OP -
BYU ET 502
TABLES first order o.d.e. LINEAR O.D.E. LINEAR O.D.E. WITH CONSTANT COEFFICIENTS y + P( x)y = Q(x ) general solution integrating factor m ( x ) = e P(x)dx y + ay = Q(x ) general solution integrating factor m (x ) = e ax y (x) = cm -1 (x ) -
BYU ET 502
EngT502 Fall 2005 PELE - POISSON EQUATION-LAPLACE EQUATION 01- DDDD Vladimir Solovjov The Laplace Equation: 01 DDDD (Dirichlet- Dirichlet-Dirichlet-Dirichlet) Dirichlet u = f2 ( x ) Dirichlet M Dirichlet u = f3 ( y ) 0 u =0 2 u = f4 ( y ) -
BYU ET 502
EngT502 Fall 2005 PELE - POISSON EQUATION-LAPLACE EQUATION 02- NDDR Jared Crosby The Laplace Equation: 02 NDDR (Neumann-Dirichlet-Dirichlet-Robin) Dirichlet u = f2 ( x ) Robin M Dirichlet u = f3 ( y ) 2u = 0 L u + H 4u x = f4 (y) x=L 0 -
BYU ET 502
EngT502 Fall 2005 PELE - POISSON EQUATION-LAPLACE EQUATION 03- NNRD Russ Burnett The Laplace Equation: 03 NNRD (Neumann-Neumann-Robin-Dirichlet) Neumann u = f 2 ( x) x y = M M Robin Dirichlet u x + H 3u = f 3 ( x) y =0 2u = 0 u = f4 -
BYU ET 502
EngT502 Fall 2005 PELE - POISSON EQUATION-LAPLACE EQUATION 03- NNRD Russ Burnett The Laplace Equation: 03 NNRD (Neumann-Neumann-Robin-Dirichlet) Neumann u = f 2 ( x) x y = M M Robin Dirichlet u + H 3u x = f 3 ( x) y =0 2u = 0 u = f4 ( y ) -
BYU ET 502
EngT502 Fall 2005 PELE - POISSON EQUATION-LAPLACE EQUATION 04 - DRND Taylor Harvey The Laplace Equation: 04 DRND (Dirichlet- Robin-Neumann-Dirichlet) Robin du f 2 ( x) = + H 2u dy y =M M Neumann Dirichlet du f3 ( y) = dx x =0 0 2u -
BYU ET 502
EngT502 Fall 2005 PELE - POISSON EQUATION-LAPLACE EQUATION 04 - DRND Taylor Harvey The Laplace Equation: 04 DRND (Dirichlet- Robin-Neumann-Dirichlet) Robin f 2 (x ) = M Neumann du + H 2u dy y=M Dirichlet f3 (y) = du dx 2u = 0 x =0 u = f -
BYU ET 502
EngT502 Fall 2005 PELE - POISSON EQUATION-LAPLACE EQUATION 05 RDDN Mark Woodruff The Laplace Equation: 05 RDDN (Robin- Dirichlet-Dirichlet-Neumann) u = f2 ( x ) = x ( L x ) M Dirichlet Dirichlet Neumann u = f3 ( y ) = y ( M y ) 0 2u = 0 L -
BYU ET 502
EngT502 Fall 2005 PELE - POISSON EQUATION-LAPLACE EQUATION 05 RDDN Mark Woodruff The Laplace Equation: 05 RDDN (Robin- Dirichlet-Dirichlet-Neumann) u = f2 ( x ) = x ( L x ) Dirichlet M Dirichlet Neumann u = f3 ( y ) = y ( M y ) 2u = 0 u -
BYU ET 502
EngT502 Fall 2005 PELE - POISSON EQUATION-LAPLACE EQUATION 06- DDNR Yin Zhang The Laplace Equation: 06 DDNR (Dirichlet- Dirichlet-Neumann-Robin) Dirichlet u = f2 ( x ) Robin M u( x, y ) = f 3( x ) x Neumann u =0 2 u( x, y ) + H u = f ( -
BYU ET 502
EngT502 Fall 2005 PELE - POISSON EQUATION-LAPLACE EQUATION 06- DDNR Yin Zhang The Laplace Equation: 06 DDNR (Dirichlet- Dirichlet-Neumann-Robin) Dirichlet u = f2 ( x ) M u( x, y ) = f3( x ) x Neumann Robin u =0 2 u( x, y ) + H4 u = f4( y -
BYU ET 502
EngT502 Fall 2005 PELE - POISSON EQUATION-LAPLACE EQUATION 07- DNRN Jingang Li The Laplace Equation: 07 DNRN (Dirichlet- Neumann-Robin- Neumann) u( x, y ) =f 2( x ) x Neumann M Robin u( x, y ) +H u = f ( y ) x 3 3 2u = 0 Neumann -
BYU ET 502
EngT502 Fall 2005 PELE - POISSON EQUATION-LAPLACE EQUATION 09- DDRR R. Scott Larson The Laplace Equation: 09 DDRR (Dirichlet- Dirichlet-Robin-Robin) u = x ( L x) = f 2 ( x ) Dirichlet M Robin Robin u x + H 3 u = f 3 ( y ) x =0 2u = 0 -
BYU ET 502
EngT502 Fall 2005 PELE - POISSON EQUATION-LAPLACE EQUATION 09- DDRR R. Scott Larson The Laplace Equation: 09 DDRR (Dirichlet- Dirichlet-Robin-Robin) u = x( L x) = f 2 ( x) Robin Dirichlet M Robin u + H 3u x = f 3 ( y) x =0 u =0 2 u + H -
BYU ET 502
EngT502 Fall 2005 PELE - POISSON EQUATION-LAPLACE EQUATION 10-DNRR Daniel Reimann The Laplace Equation: 10 DNRR (Dirichlet- Neumann- Robin- Robin) Neumann u f 2 ( x) = y y = M M Robin Robin u f 3 ( y ) = + H 4u x x =0 0 2u = 0 u -
BYU ET 502
EngT502 Fall 2005 PELE - POISSON EQUATION-LAPLACE EQUATION 10-DNRR Daniel Reimann The Laplace Equation: 10 DNRR (Dirichlet- Neumann- Robin- Robin) Neumann f 2 (x ) = u y y=M M Robin Robin f 3 (y) = u + H 4u x 2u = 0 x =0 f 4 (y) = u + -
BYU ET 502
EngT502 Fall 2005 PELE - POISSON EQUATION-LAPLACE EQUATION 11- RRDR Rob Chase The Laplace Equation: 11 RRDR (Robin- Robin-Dirichlet-Robin) Robin u = f2 ( x ) M Dirichlet u = f3 ( y ) Robin u = f4 ( y ) u =0 2 0 u = f1 ( x ) Robin f2 ( x) L -
BYU ET 502
EngT502 Fall 2005 PELE - POISSON EQUATION-LAPLACE EQUATION 11- RRDR Rob Chase The Laplace Equation: 11 RRDR (Robin- Robin-Dirichlet-Robin) Robin u = f2 ( x ) M Dirichlet u = f3 ( y ) Robin u = f4 ( y ) u =0 2 0 u = f1 ( x ) L Supplemental -
BYU ET 502
EngT502 Fall 2005 PELE - POISSON EQUATION-LAPLACE EQUATION 12- RNRR Kevin Jeffs The Laplace Equation: 12 RNRR (Robin-Neumann-Robin-Robin) Neumann u y = f 2 (x ) y=M M Robin Robin u + H 3u x = f3 (y) x =0 u =0 2 u + H 4u x = f 4 (y) x=L -
BYU ET 502
EngT502 Fall 2005 PELE - POISSON EQUATION-LAPLACE EQUATION 13-RRNN Gary Johns The Laplace Equation: 13 RRNN (Robin- Robin- Neumann- Neumann) u = f2 ( x ) M Nuemann Robin u = f3 ( y ) Neumann u =0 2 u = f4 ( y ) 0 u = f1 ( x ) Robin L Su -
BYU ET 502
EngT502 Fall 2005 PELE - POISSON EQUATION-LAPLACE EQUATION 13-RRNN Gary Johns The Laplace Equation: 13 RRNN (Robin- Robin- Neumann- Neumann) u = f2 ( x ) Robin M Nuemann Neumann u = f3 ( y ) u =0 2 u = f4 ( y ) 0 u = f1 ( x ) Robin L Su -
BYU ET 502
EngT502 Fall 2005 PELE - POISSON EQUATION-LAPLACE EQUATION 01- RRRR Matt Spencer The Laplace Equation: 01 RRRR Robin- Robin - Robin - Robin) Robin u = f 2 ( x) y + H 2 u y =M M Robin Robin u x + H 3u = f 3 ( y ) x =0 0 2u = 0 Rob -
BYU ET 502
Journal of Applied Engineering Mathematics Volume 2 1D MOMENTUM INTEGERAL METHOD (THWAITES) VS. 2D FINITE ELEMENT NAVIER-STOKES SOLUTION: COMPUTING FLOW SEPERATION, FLOW RATE AND PRESSURE DROP Gifford Zach Decker Mechanical Engineering Department Brigham -
BYU ET 502
Origin of the Name Mathematics Anatolius: Why is mathematics so named? The Peripatetics say that rhetoric and poetry and the whole of popular music can be understood without any course of instructions. But no one can acquire knowledge of the subjects call
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