Ch_03 - Vector I - Summary

# Ch_03 - Vector I - Summary - Chapter 3 Vector Differential...

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Chapter 3 Vector Differential Calculus

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Table of Contents 1. Vector Algebra in 2-Space and 3-Space 2. Inner Product (Dot Product) 3. Vector Product (Cross Product) 4. Vector and Scalar Functions and Fields. Derivatives 5. Curves, Tangents, Arc Length, Curvature, Torsion 6. Curves in Mechanics. Velocity and Acceleration 7. Gradient of a Scalar Field. Directional Derivative 8. Divergence of a Vector Field 9. Curl of a Vector Field Appendix A-1 Types of Derivative A-2 Orthogonal Curvilinear Coordinates
Vector Differential Calculus 1. Vector Algebra in 2-Space and 3-Space 1.1. Scalar and Vector Scalar mole fraction, concentration, Thermodynamic state function, , Tp Vector velocity, shear stress, heat/mass flux, Transport properties Arrow or directed line segments Initial point P to terminal point Q Length (magnitude) or norm : || aa Unit vector: a vector of length 1. Zero vector (no magnitude): 0 Notation boldface with direction and magnitude (exception 0 -vector ) 123 [ , , ] aaa a in 3D 1.2. Equality of Vectors Notation boldface with direction and magnitude (exception 0 -vector ) ab (same length and direction) 1.3. Components of Vectors Cartesian coordinate system [ , , ] a 222  a 12 1 ax x  , 22 1 ay y , 32 1 az z Position vector [, , ] xyz r 111 (, ,) Px y z Qx y z a

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Vector Differential Calculus 1. Vector Algebra in 2-Space and 3-Space 1.4. Vector Addition, Scalar Multiplication Scalar multiplication () 0 ck c k   aa a a0 Vector Addition  ab ba ( )   uv w u vw cc c ab a b (commutative law) (associative law) (distributive law) 123 [ , , ] a c a c a a 112 233 [ , , ] ababab   Geometric representation 1.5. Unit Vectors; , , i j k , , ijk are the unit vector in the positive directions of the axis of a Cartesian coord. 1 2 3 [, , ] aaa a a a  ai j k [1, 0, 0] i , [0, 1, 0] j , [0, 0, 1] k Position vector r with coordinate system [ , , ] xyz x y z ri j k Basis , , Components , , Representation position vector r Space Time vector in fluid motion ( , , , ) ( , , , ) ( , , , ) ( , , , ) xyzt  uu ij k u v uv w v w w i j k r  :, , Px y z
Vector Differential Calculus 1. Vector Algebra in 2-Space and 3-Space 1.6. Vector Space 3 R Linear combination of a and b 12 cc ca b Linear combination of given vectors (1) (2) ( ) ,, , n aa a 1( 1 ) 2( 2 ) () nn c  a and if and only if the only solution of 1 ) 2 ) c a0 is 0 n c   , then the given vectors (1) (1) ( ) , n a are called “ linearly independent ”. Let C divide by BA in : ( 1 ) If midpoint, 1/2  11 1 2 ) (1 ) 0 OB OA c c c   cb b ab a b bc a b 1 0  Any three vectors with common origin be on a straight line lmn  c0 0  with 22 2 0 .

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## This note was uploaded on 04/27/2011 for the course CHEM 101 taught by Professor Suh during the Spring '11 term at University of Toronto.

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Ch_03 - Vector I - Summary - Chapter 3 Vector Differential...

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