Surface Integrals (6A)
|
|
- Jeffery Waters
- 5 years ago
- Views:
Transcription
1 Surface Integrals (6A) Surface Integral Stokes' Theorem
2 Copright (c) 2012 Young W. Lim. Permission is granted to cop, distribute and/or modif this document under the terms of the GNU Free Documentation License, Version 1.2 or an later version published b the Free Software Foundation; with no Invariant Sections, no Front-Cover Tets, and no Back-Cover Tets. A cop of the license is included in the section entitled "GNU Free Documentation License". Please send corrections (or suggestions) to oungwlim@hotmail.com. This document was produced b using OpenOffice and Octave.
3 Arc Length In the Plane = f () b s = a 1 + ( d ) 2 d d d s d d Surface Integral 3
4 Surface Area In the Space z = f (, ) v Area of the surface over u Δ T A(S) = 1 + [ f (, ) ] 2 + [ f (, )] 2 d A = 1 + ( ) 2 + ( ) 2 d A Δ Δ Δ Differential of surface area ds = 1 + [ f (, )] 2 + [ f (, ) ] 2 d A Δ T Surface Integral 4
5 Differential of Surface Area (1) Differential of surface area ds = 1 + [ f (, )] 2 + [ f (, ) ] 2 d A Slope along direction = 0.4 Slope along direction = 0.2 Δ u Δ T v Δ Δ u v Δ u = Δ i + Δ k Δ Δ Δ i k j v = Δ j + Δ k Surface Integral 5
6 Differential of Surface Area (2) Differential of surface area ds = 1 + [ f (, )] 2 + [ f (, ) ] 2 d A Slope along direction = 0.4 Slope along direction = 0.2 Δ Δ Δ u v Δ Δ u Δ u = Δ i + Δ k i k j Δ Δ v = Δ j + Δ k Surface Integral 6
7 Differential of Surface Area (3) Differential of surface area ds = 1 + [ f (, )] 2 + [ f (, ) ] 2 d A u = Δ i + Δ k v = Δ j + Δ k Δ u v Δ u Δ v Δ u v Δ T i j k Δ 0 u v = Δ 0 Δ Δ = [ i j + k ] Δ Δ i k j u v Δ T u v = ( ) 2 + ( ) Δ Δ Surface Integral 7
8 Differential of Surface Area (4) Differential of surface area ds = 1 + [ f (, )] 2 + [ f (, ) ] 2 d A u = Δ i + Δ k v = Δ j + Δ k Δ u v Δ u Δ v Δ Δ T u v = [ i j + k ] Δ Δ u v = ( ) 2 + ( ) Δ Δ S i k j Δ Δ Δ Δ T = [ f (, )] 2 + [ f (, )] Δ Δ Δ S = [ f (, )] 2 + [ f (, )] Δ A Surface Integral 8
9 Line Integral with an Eplicit Curve Function = f () a b Curve C d d = f '() d = f '() d d s = [d ] 2 + [d ] 2 d s = 1 + [ f '()] 2 d d s d d C G(, ) d = a b G(, f ()) d C G(, ) d = a b G(, f ()) f '() d C G(, ) d s = a b G(, f ()) 1 + [ f '()] 2 d Surface Integral 9
10 Surface Integral with an Eplicit Surface Function z = f (, ) a b c d egion d f d = f (, ) d f d = f (, ) ds = 1 + [ f (, )] 2 + [ f (, ) ] 2 d A S G(,, z) d S = G(,, f (, )) 1 + [ f (, )] 2 + [ f (, )] 2 d A Surface Integral 10
11 Surface Integral Over XY (1) domain z ds 3 range 4 G(,, f (, )) z z = f (, ) G(,, z) G(,, f (, )) a b c d Surface Integral 11
12 Surface Integral Over XY (2) domain z ds 3 range 4 G(,, f (, )) z z = f (, ) S G(,, z) d S = G(,, f (, )) 1 + [ f (, )] 2 + [ f (, )] 2 d A Surface Integral 12
13 Surface Integral Over XZ (1) domain z 3 range 4 G(, g(, z), z) z ds = g(, z) G(,, z) G(, g(, z), z) a b c d Surface Integral 13
14 Surface Integral Over XZ (2) domain z 3 range 4 G(, g(, z), z) z ds = g(, z) S G(,, z) d S = G(, g(, z), z) 1 + [g (, z)] 2 + [g z (, z) ] 2 d A Surface Integral 14
15 Surface Integral with an Eplicit Surface Function z = f (, ) egion S G(,, z) d S = g(, z) egion S G(,, z) d S = h(, z) egion S G(,, z) d S d f d = f (, ) d f d = f (, ) = = ds = 1 + [ f (, )] 2 + [ f (, )] 2 d A G(,, f (, )) 1 + [ f (, )] 2 + [ f (, )] 2 d A d g d = g (, z) d g d z = g (, z) = ds = 1 + [g (, z )] 2 + [g z (, z )] 2 d A G(, g(, z), z) 1 + [g (, z)] 2 + [g z (, z) ] 2 d A dh d = f (, z) d h d z = f z (, z) ds = 1 + [h (, z) ] 2 + [h z (, z)] 2 d A G(h(, z),, z) 1 + [h (, z)] 2 + [h z (, z)] 2 d A Surface Integral 15
16 Mass of a Surface densit ρ(,, z) mass m = S ρ(,, z) d S Surface Integral 16
17 Orientation of a Surface Oriented Surface there eists a continuous unit normal vector function n defined at each point (,,z) on the surface Orientation the vector field n(,, z) or n(,, z) Upward orientation Downward orientation z = f(,) : positive k component z = f(,) : negative k component Surface g(,, z) = 0 Unit Normal Vector n = 1 g g g = g i + g j + g z k Surface z = f (, ) g(,, z) = z f (, ) = 0 g(,, z) = f (, ) z = 0 Surface Integral 17
18 Level Curve and Surface Functions of two variables z = F (, ) 3 G(,, z) = 0 Level Curve c i = F (, ) 2 F (, ) = 0 when z = 0 0 = F (, ) r '(t) Functions of three variables w = G(,, z) 4 H (,, z,w) = 0 Level Surface c i = G(,, z) 3 G(,, z) = 0 when w = 0 0 = G(,, z) r '(t) Surface Integral 18
19 Gradient & Normal Vector Functions of three variables w = F (,, z) 4 Level Surface c 0 = F (,, z) dc 0 d t = d F dt 0 = F (,, z) d dt + F d dt + F z 3 d z dt = f (t) = g(t) z = h(t) 0 = ( F i + F j + F z k ) ( d dt i + d d t j + d z d t k ) 0 = F (,, z) r '(t) tangent vector F normal to the level surface at P ( 0, 0, z 0 ) Surface Integral 19
20 Surface Integral over a 3-D Vector Field (1) A given point in a 3-d space ( 0, 0, z 0 ) A vector P ( 0, 0, z 0 ), Q( 0, 0, z 0 ), ( 0, 0, z 0 ) domain range 3 functions z ( 0, 0, z 0 ) z n F ( 0, 0, z 0 ) ( 0, 0, z 0 ) P ( 0, 0, Z 0 ) ( 0, 0, z 0 ) Q( 0, 0, z 0 ) ( 0, 0, z 0 ) ( 0, 0, z 0 ) ( 0, 0, z 0 ) onl points that are on the surface F ( 0, 0, z 0 ) = P ( 0, 0, z 0 )i + Q( 0, 0, z 0 ) j + ( 0, 0, z 0 )k consider onl the component of F along n F n Surface Integral 20
21 Surface Integral over a 3-D Vector Field (2) F (,, z) = P(,, z) i + Q(,, z) j + (,, z) k Line Integral over a 3-d Vector Field r (t) = f (t) i + g(t) j + h(t) k c F dr = C P(,, z) d + Q(,, z) d + (,, z) d Surface Integral over a 3-d Vector Field n = g g (F n) ds S = flu total volume of a fluid passing through S per unit time F velocit field of a fluid Surface Integral 21
22 Vector Form of Green's Theorem A force field A smooth curve Work done b F along C C F (, ) = P (, )i + Q(, ) j C : = f (t), = g(t), a t b W = c F dr = c F T d s = C P(, ) d + Q(, ) d C P d + Q d = ( Q P ) d A curl F = F = i j k = z P Q 0 ( Q P ) k ( F ) k = ( Q P ) C P d + Q d = ( F ) k d A Surface Integral 22
23 Stokes' Theorem (1) A force field F (, ) = P (, )i + Q(, ) j Work done b F along C W = c F dr = c F T d s n C T v 2-space = ( F ) k d A 3-space = ( F ) n d A Surface Integral 23
24 Stokes' Theorem (2) A force field F (, ) = P (, )i + Q(, ) j Work done b F along C W = c F dr = c F T d s n T 3-space = ( F ) n d A i j k curl F = F = z P Q v = ( Q ) z i + ( P z ) j + ( Q P ) k n = Surface Integral 24 i j + k 1 + ( ) 2 + ( ) 2 surface g(,, z) = z f (, )
25 Gradient of a 2 Variable Function Function of two variables f (, ) f (, ) = i + j vector ate of change of f in the direction ate of change of f in the direction Slope in the direction = 2 Slope in the direction = 1 z k j = 1 j i = 2i i θ j u Surface Integral 25
26 2-D Divergence z F (, ) F (, +Δ ) F (+Δ, ) F (+Δ, +Δ ) (, ) (, +Δ ) (+Δ, ) (+Δ, +Δ ) Surface Integral 26
27 Chain ule Function of two variable = f (u, v) u = g(, ) v = h(, ) Surface Integral 27
28 eferences [1] [2] [3] M.L. Boas, Mathematical Methods in the Phsical Sciences [4] D.G. Zill, Advanced Engineering Mathematics
Surface Integrals (6A)
urface Integrals (6A) urface Integral tokes' Theorem Copright (c) 2012 Young W. Lim. Permission is granted to cop, distribute and/or modif this document under the terms of the GNU Free Documentation License,
More informationLine Integrals (4A) Line Integral Path Independence. Young Won Lim 10/22/12
Line Integrals (4A Line Integral Path Independence Copyright (c 2012 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License,
More informationLine Integrals (4A) Line Integral Path Independence. Young Won Lim 11/2/12
Line Integrals (4A Line Integral Path Independence Copyright (c 2012 Young W. Lim. Permission is granted to copy, distriute and/or modify this document under the terms of the GNU Free Documentation License,
More informationGeneral Vector Space (2A) Young Won Lim 11/4/12
General (2A Copyright (c 2012 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version
More informationMatrix Transformation (2A) Young Won Lim 11/9/12
Matrix (A Copyright (c 01 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1. or any later version published
More informationBackground Complex Analysis (1A) Young Won Lim 9/2/14
Background Complex Analsis (1A) Copright (c) 2014 Young W. Lim. Permission is granted to cop, distribute and/or modif this document under the terms of the GNU Free Documentation License, Version 1.2 or
More informationIntroduction to ODE's (0A) Young Won Lim 3/9/15
Introduction to ODE's (0A) Copyright (c) 2011-2014 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2
More informationHigher Order ODE's (3A) Young Won Lim 7/7/14
Higher Order ODE's (3A) Copyright (c) 2011-2014 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or
More informationHigher Order ODE's (3A) Young Won Lim 12/27/15
Higher Order ODE's (3A) Copyright (c) 2011-2015 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or
More informationDouble Integrals (5A)
Doule Integrls (5A) Doule Integrl Doule Integrls in Polr oordintes Green's Theorem opyright (c) 2012 Young W. Lim. Permission is grnted to copy, distriute nd/or modify this document under the terms of
More informationHigher Order ODE's (3A) Young Won Lim 7/8/14
Higher Order ODE's (3A) Copyright (c) 2011-2014 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or
More informationHigher Order ODE's, (3A)
Higher Order ODE's, (3A) Initial Value Problems, and Boundary Value Problems Copyright (c) 2011-2015 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms
More informationSeparable Equations (1A) Young Won Lim 3/24/15
Separable Equations (1A) Copyright (c) 2011-2015 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or
More informationBackground Trigonmetry (2A) Young Won Lim 5/5/15
Background Trigonmetry (A) Copyright (c) 014 015 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1. or
More informationODE Background: Differential (1A) Young Won Lim 12/29/15
ODE Background: Differential (1A Copyright (c 2011-2015 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version
More informationDefinitions of the Laplace Transform (1A) Young Won Lim 1/31/15
Definitions of the Laplace Transform (A) Copyright (c) 24 Young W. Lim. Permission is granted to copy, distriute and/or modify this document under the terms of the GNU Free Documentation License, Version.2
More informationRoot Locus (2A) Young Won Lim 10/15/14
Root Locus (2A Copyright (c 2014 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version
More informationCT Rectangular Function Pairs (5B)
C Rectangular Function Pairs (5B) Continuous ime Rect Function Pairs Copyright (c) 009-013 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU
More informationExpected Value (10D) Young Won Lim 6/12/17
Expected Value (10D) Copyright (c) 2017 Young W. Lim. Permissios granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later
More informationDifferentiation Rules (2A) Young Won Lim 1/30/16
Differentiation Rules (2A) Copyright (c) 2011-2016 Young W. Lim. Permission is grante to copy, istribute an/or moify this ocument uner the terms of the GNU Free Documentation License, Version 1.2 or any
More informationComplex Series (3A) Young Won Lim 8/17/13
Complex Series (3A) 8/7/3 Copyright (c) 202, 203 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version.2 or
More informationComplex Functions (1A) Young Won Lim 2/22/14
Complex Functions (1A) Copyright (c) 2011-2014 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or
More informationIntroduction to ODE's (0P) Young Won Lim 12/27/14
Introuction to ODE's (0P) Copyright (c) 2011-2014 Young W. Lim. Permission is grante to copy, istribute an/or moify this ocument uner the terms of the GNU Free Documentation License, Version 1.2 or any
More informationSignals and Spectra (1A) Young Won Lim 11/26/12
Signals and Spectra (A) Copyright (c) 202 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version.2 or any later
More informationGeneral CORDIC Description (1A)
General CORDIC Description (1A) Copyright (c) 2010, 2011, 2012 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License,
More informationFourier Analysis Overview (0A)
CTFS: Fourier Series CTFT: Fourier Transform DTFS: Fourier Series DTFT: Fourier Transform DFT: Discrete Fourier Transform Copyright (c) 2011-2016 Young W. Lim. Permission is granted to copy, distribute
More informationDFT Frequency (9A) Each Row of the DFT Matrix. Young Won Lim 7/31/10
DFT Frequency (9A) Each ow of the DFT Matrix Copyright (c) 2009, 2010 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GU Free Documentation License,
More informationLinear Equations with Constant Coefficients (2A) Young Won Lim 4/13/15
Linear Equations with Constant Coefficients (2A) Copyright (c) 2011-2014 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation
More informationMatrix Transformation (2A) Young Won Lim 11/10/12
Matrix (A Copyright (c 0 Young W. im. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation icense, Version. or any later version published
More informationRelations (3A) Young Won Lim 3/27/18
Relations (3A) Copyright (c) 2015 2018 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later
More informationDifferentiation Rules (2A) Young Won Lim 2/22/16
Differentiation Rules (2A) Copyright (c) 2011-2016 Young W. Lim. Permission is grante to copy, istribute an/or moify this ocument uner the terms of the GNU Free Documentation License, Version 1.2 or any
More informationDefinite Integrals. Young Won Lim 6/25/15
Definite Integrls Copyright (c 2011-2015 Young W. Lim. Permission is grnted to copy, distribute nd/or modify this document under the terms of the GNU Free Documenttion License, Version 1.2 or ny lter version
More informationIntegrals. Young Won Lim 12/29/15
Integrls Copyright (c) 2011-2015 Young W. Lim. Permission is grnted to copy, distribute nd/or modify this document under the terms of the GNU Free Documenttion License, Version 1.2 or ny lter version published
More informationFourier Analysis Overview (0A)
CTFS: Fourier Series CTFT: Fourier Transform DTFS: Fourier Series DTFT: Fourier Transform DFT: Discrete Fourier Transform Copyright (c) 2011-2016 Young W. Lim. Permission is granted to copy, distribute
More informationComplex Trigonometric and Hyperbolic Functions (7A)
Complex Trigonometric and Hyperbolic Functions (7A) 07/08/015 Copyright (c) 011-015 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation
More informationDetect Sensor (6B) Eddy Current Sensor. Young Won Lim 11/19/09
Detect Sensor (6B) Eddy Current Sensor Copyright (c) 2009 Young W. Lim. Permission is granteo copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version
More informationMagnetic Sensor (3B) Magnetism Hall Effect AMR Effect GMR Effect. Young Won Lim 9/23/09
Magnetic Sensor (3B) Magnetism Hall Effect AMR Effect GMR Effect Copyright (c) 2009 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation
More informationCapacitor Young Won Lim 06/22/2017
Capacitor Copyright (c) 2011 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version
More informationFourier Analysis Overview (0B)
CTFS: Continuous Time Fourier Series CTFT: Continuous Time Fourier Transform DTFS: Fourier Series DTFT: Fourier Transform DFT: Discrete Fourier Transform Copyright (c) 2009-2016 Young W. Lim. Permission
More informationCapacitor in an AC circuit
Capacitor in an AC circuit Copyright (c) 2011 2017 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2
More informationDigital Signal Octave Codes (0A)
Digital Signal Periodic Conditions Copyright (c) 2009-2017 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version
More informationBackground ODEs (2A) Young Won Lim 3/7/15
Background ODEs (2A) Copyright (c) 2014-2015 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any
More informationSignal Functions (0B)
Signal Functions (0B) Signal Functions Copyright (c) 2009-203 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License,
More informationGroup & Phase Velocities (2A)
(2A) 1-D Copyright (c) 2011 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published
More informationPropagating Wave (1B)
Wave (1B) 3-D Wave Copyright (c) 2011 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later
More informationDiscrete Time Rect Function(4B)
Discrete Time Rect Function(4B) Discrete Time Rect Functions Copyright (c) 29-213 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation
More informationPhasor Young Won Lim 05/19/2015
Phasor Copyright (c) 2009-2015 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version
More informationDown-Sampling (4B) Young Won Lim 10/25/12
Down-Sampling (4B) /5/ Copyright (c) 9,, Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version. or any later
More informationThe Growth of Functions (2A) Young Won Lim 4/6/18
Copyright (c) 2015-2018 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published
More informationDigital Signal Octave Codes (0A)
Digital Signal Periodic Conditions Copyright (c) 2009-2017 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version
More informationLecture 04. Curl and Divergence
Lecture 04 Curl and Divergence UCF Curl of Vector Field (1) F c d l F C Curl (or rotor) of a vector field a n curlf F d l lim c s s 0 F s a n C a n : normal direction of s follow right-hand rule UCF Curl
More informationCapacitor and Inductor
Capacitor and Inductor Copyright (c) 2015 2017 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or
More informationCapacitor and Inductor
Capacitor and Inductor Copyright (c) 2015 2017 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or
More informationFaculty of Engineering, Mathematics and Science School of Mathematics
Faculty of Engineering, Mathematics and Science School of Mathematics SF Engineers SF MSISS SF MEMS MAE: Engineering Mathematics IV Trinity Term 18 May,??? Sports Centre??? 9.3 11.3??? Prof. Sergey Frolov
More informationDiscrete Time Rect Function(4B)
Discrete Time Rect Function(4B) Discrete Time Rect Functions Copyright (c) 29-23 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation
More informationDigital Signal Octave Codes (0A)
Digital Signal Periodic Conditions Copyright (c) 2009-207 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version.2
More informationBayes Theorem (10B) Young Won Lim 6/3/17
Bayes Theorem (10B) Copyright (c) 2017 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later
More informationHyperbolic Functions (1A)
Hyperbolic Functions (A) 08/3/04 Copyright (c) 0-04 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version. or
More informationDigital Signal Octave Codes (0A)
Digital Signal Periodic Conditions Copyright (c) 2009-207 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version.2
More informationDown-Sampling (4B) Young Won Lim 11/15/12
Down-Sampling (B) /5/ Copyright (c) 9,, Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version. or any later
More informationBackground LTI Systems (4A) Young Won Lim 4/20/15
Background LTI Systems (4A) Copyright (c) 2014-2015 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2
More informationSecond Order ODE's (2A) Young Won Lim 5/5/15
Second Order ODE's (2A) Copyright (c) 2011-2015 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or
More informationUp-Sampling (5B) Young Won Lim 11/15/12
Up-Sampling (5B) Copyright (c) 9,, Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version. or any later version
More informationCLTI Differential Equations (3A) Young Won Lim 6/4/15
CLTI Differential Equations (3A) Copyright (c) 2011-2015 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version
More informationDigital Signal Octave Codes (0A)
Digital Signal Periodic Conditions Copyright (c) 2009-207 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version.2
More informationDispersion (3A) 1-D Dispersion. Young W. Lim 10/15/13
1-D Dispersion Copyright (c) 013. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1. or any later version published
More informationAudio Signal Generation. Young Won Lim 1/29/18
Generation Copyright (c) 2016-2018 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version
More informationDirectional Derivatives in the Plane
Directional Derivatives in the Plane P. Sam Johnson April 10, 2017 P. Sam Johnson (NIT Karnataka) Directional Derivatives in the Plane April 10, 2017 1 / 30 Directional Derivatives in the Plane Let z =
More informationMLC Practice Final Exam
Name: Section: Recitation/Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages 1 through 13. Show all your work on the standard
More informationDigital Signal Octave Codes (0A)
Digital Signal Periodic Conditions Copyright (c) 2009-207 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version.2
More informationPage Points Score Total: 210. No more than 200 points may be earned on the exam.
Name: PID: Section: Recitation Instructor: DO NOT WRITE BELOW THIS LINE. GO ON TO THE NEXT PAGE. Page Points Score 3 18 4 18 5 18 6 18 7 18 8 18 9 18 10 21 11 21 12 21 13 21 Total: 210 No more than 200
More informationBandpass Sampling (2B) Young Won Lim 3/27/12
andpass Sapling (2) Copyright (c) 2009-2012 Young W. Li. Perission is granted to copy, distribute and/or odify this docuent under the ters of the GNU Free Docuentation License, Version 1.2 or any later
More informationGroup Velocity and Phase Velocity (1A) Young Won Lim 5/26/12
Group Velocity and Phase Velocity (1A) Copyright (c) 211 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version
More informationBilateral Laplace Transform (6A) Young Won Lim 2/16/15
Bilterl Lplce Trnsform (6A) 2/6/5 Copyright (c) 25 Young W. Lim. Permission is grnted to copy, distribute nd/or modify this document under the terms of the GNU Free Documenttion License, Version.2 or ny
More informationResolution (7A) Young Won Lim 4/21/18
(7A) Coyright (c) 215 218 Young W. Lim. Permission is granted to coy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version ublished
More informationCLTI System Response (4A) Young Won Lim 4/11/15
CLTI System Response (4A) Copyright (c) 2011-2015 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2
More informationGroup Delay and Phase Delay (1A) Young Won Lim 7/19/12
Group Delay and Phase Delay (A) 7/9/2 Copyright (c) 2 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version.2
More informationUndersampling (2B) Young Won Lim 4/4/12
Undersapling (2) Copyright (c) 2009-2012 Young W. Li. Perission is granted to copy, distribute and/or odify this docuent under the ters of the GNU Free Docuentation License, Version 1.2 or any later version
More informationSequential Circuit Timing. Young Won Lim 11/6/15
Copyright (c) 2011 2015 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published
More informationBilateral Laplace Transform (6A) Young Won Lim 2/23/15
Bilterl Lplce Trnsform (6A) Copyright (c) 25 Young W. Lim. Permission is grnted to copy, distribute nd/or modify this document under the terms of the GNU Free Documenttion License, Version.2 or ny lter
More informationCT Correlation (2A) Young Won Lim 9/9/14
CT Correlation (2A) Copyright (c) 200-204 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version.2 or any later
More informationHilbert Inner Product Space (2B) Young Won Lim 2/7/12
Hilbert nner Product Space (2B) Copyright (c) 2009-2011 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version
More informationSome common examples of vector fields: wind shear off an object, gravitational fields, electric and magnetic fields, etc
Vector Analysis Vector Fields Suppose a region in the plane or space is occupied by a moving fluid such as air or water. Imagine this fluid is made up of a very large number of particles that at any instant
More informationGeneral Vector Space (3A) Young Won Lim 11/19/12
General (3A) /9/2 Copyright (c) 22 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version.2 or any later version
More informationFSM Examples. Young Won Lim 11/6/15
/6/5 Copyright (c) 2 25 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version.2 or any later version published
More informationSOLUTIONS TO THE FINAL EXAM. December 14, 2010, 9:00am-12:00 (3 hours)
SOLUTIONS TO THE 18.02 FINAL EXAM BJORN POONEN December 14, 2010, 9:00am-12:00 (3 hours) 1) For each of (a)-(e) below: If the statement is true, write TRUE. If the statement is false, write FALSE. (Please
More informationBayes Theorem (4A) Young Won Lim 3/5/18
Bayes Theorem (4A) Copyright (c) 2017-2018 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any
More informationIntroduction to ODE's (0A) Young Won Lim 3/12/15
Introduction to ODE's (0A) Copyright (c) 2011-2015 Young W. Lim. Permission is grnted to copy, distribute nd/or modify this document under the terms of the GNU Free Documenttion License, Version 1.2 or
More informationNo calculators, cell phones or any other electronic devices can be used on this exam. Clear your desk of everything excepts pens, pencils and erasers.
Name: Section: Recitation Instructor: READ THE FOLLOWING INSTRUCTIONS. Do not open your exam until told to do so. No calculators, cell phones or any other electronic devices can be used on this exam. Clear
More informationMultiple Choice Review Problems
Multiple Choice Review Problems 1. (NC) Which graph best represents the position of a particle, st ( ), as a function of time, if the particle's velocity is negative and the particle's acceleration is
More informationHamiltonian Cycle (3A) Young Won Lim 5/5/18
Hamiltonian Cycle (3A) Copyright (c) 015 018 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1. or any
More informationMATH 332: Vector Analysis Summer 2005 Homework
MATH 332, (Vector Analysis), Summer 2005: Homework 1 Instructor: Ivan Avramidi MATH 332: Vector Analysis Summer 2005 Homework Set 1. (Scalar Product, Equation of a Plane, Vector Product) Sections: 1.9,
More informationAudio Signal Generation. Young Won Lim 2/2/18
Generation Copyright (c) 2016-2018 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version
More informationImplication (6A) Young Won Lim 3/17/18
Implication (6A) 3/17/18 Copyright (c) 2015 2018 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or
More informationCreated by T. Madas VECTOR OPERATORS. Created by T. Madas
VECTOR OPERATORS GRADIENT gradϕ ϕ Question 1 A surface S is given by the Cartesian equation x 2 2 + y = 25. a) Draw a sketch of S, and describe it geometrically. b) Determine an equation of the tangent
More informationConservative fields and potential functions. (Sect. 16.3) The line integral of a vector field along a curve.
onservative fields and potential functions. (Sect. 16.3) eview: Line integral of a vector field. onservative fields. The line integral of conservative fields. Finding the potential of a conservative field.
More informationProbability (10A) Young Won Lim 6/12/17
Probability (10A) Copyright (c) 2017 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later
More informationWithout fully opening the exam, check that you have pages 1 through 12.
Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages 1 through 12. Show all your work on the standard
More informationDefinitions of the Laplace Transform (1A) Young Won Lim 2/9/15
Definition of the aplace Tranform (A) 2/9/5 Copyright (c) 24 Young W. im. Permiion i granted to copy, ditriute and/or modify thi document under the term of the GNU Free Documentation icene, Verion.2 or
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. D) D: (-, 0) (0, )
Midterm Practice Test MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the domain and graph the function. ) G(t) = t - 3 ) 3 - -3 - - 3 - - -3
More information