Module 02: Math Review

Transcription

1 Module 02: Math Review 1

2 Module 02: Math Review: Outline Vector Review (Dot, Cross Products) Review of 1D Calculus Scalar Functions in higher dimensions Vector Functions Differentials Purpose: Provide conceptual framework NOT teach mechanics 2

3 Coordinate System Coordinate system: used to describe the position of a point in space and consists of 1. An origin as the reference point 2. A set of coordinate axes with scales and labels 3. Choice of positive direction for each axis 4. Choice of unit vectors at each point in space Cartesian Coordinate System

4 Vectors

5 Vector A vector is a quantity that has both direction and magnitude. Let a vector be denoted by the symbol A The magnitude of A is denoted by A A

6 Application of Vectors (1) Vectors can exist at any point P in space. (2) Vectors have direction and magnitude. (3) Vector Equality: Any two vectors that have the same direction and magnitude are equal no matter where in space they are located.

7 A Vector Addition B Let and be two vectors. Define a new vector CAB A B,the vector addition of and by the geometric construction shown in either figure

8 Summary: Vector Properties Addition of Vectors 1. Commutativity A+ B= B+ A 2. Associativity ( A+ B ) + C= A + ( B+ C ) 3. Identity Element for Vector Addition such that 4. Inverse Element for Vector Addition such that Scalar Multiplication of Vectors 0 A A+ 0= 0+ A = A A + A = 0 ( ) 1. Associative Law for Scalar Multiplication bc ( A ) = ( bc ) A = ( cb A ) = cb ( A ) 2. Distributive Law for Vector Addition c ( A+ B ) = c A + c B 3. Distributive Law for Scalar Addition ( b + c ) A = b A + c A 4. Identity Element for Scalar Multiplication: number 1 such that 1 A= A

9 Vector Decomposition Choose a coordinate system with an origin and axes. We can decompose a vector into component vectors along each coordinate axis, for example along the x,y, and z-axes of a Cartesian coordinate system. A vector at P can be decomposed into the vector sum, A= A + A + A x y z

10 Unit Vectors and Components The idea of multiplication by real numbers allows us to define a set of unit vectors at each point in space ˆ ˆ ˆ with ih ( i, j, k ) ˆ i = 1, ˆ j = 1, k ˆ = 1 Components: A = ( A,A,A ) x y z A = A ˆ i, A = A ˆ j, A = A kˆ x x y y z z A = A ˆ i + A ˆ j + A k ˆ x y z

11 Vector Decomposition in Two Dimensions i Consider a vector A = ( A,A, 0) x y x- and y components: A x = Acos(θ), A y = Asin(θ) i Magnitude: A = A x A y Direction: A y y A x = Asin(θ) s ( Acos(θ) = tan(θ) θ = tan 1 ( A y / A x )

12 Vector Addition A = Acos(θ ) î i + Asin(θ ) ĵj A A B = Bcos(θ( î + Bsin(θ ĵ B ) ( B ) j Vector Sum: Components C= A+ B C x = A x + B x, C y = A y + B y C x = C cos(θ C ) = Acos(θ A )+ Bcos(θ B ) C y = C sin(θ C ) = Asin(θ A ) + Bsin(θ B ) C = ( A + B ) ˆ i + ( A + B ) ˆ j = C cos( θ ) ˆ i + Csin( θ ˆj x x y y C C )

13 Preview: Vector Description of Motion Position r () t = x () t ˆ i + y () t ˆ j Displacement Δ r () t = Δ x () t ˆ i + Δ y () t ˆj Velocity dx () t ˆ dy () t v () t = i + ˆ j v ˆ ˆ x () t i + v y () t j dt dt Acceleration dv () x ()ˆ t dv y t a () t = i + ˆ j a () ˆ () ˆ x t i + a y t j dt dt

14 Scalar Product A scalar quantity Magnitude: AB AB cos The scalar (dot) product can be positive, zero, or negative Two types of projections: the scalar product is the parallel component of one vector with respect to the second vector times the magnitude of the second vector AB A (cos ) B A B AB A (cos ) B A B

15 Scalar Product Properties A B = B A c A B = c( ( A B ) ( A + B ) C = A C + B C

16 Scalar Product in Cartesian Coordinates With unit vectors ˆˆ i, jand kˆ ˆiˆi ˆjˆj kˆ kˆ 1 ˆiˆj ˆik ˆ ˆjk ˆ 0 ˆˆ ii ˆi ˆi cos(0) 1 ˆˆ ij ˆi ˆj cos( /2) 0 Example: A Aˆi Aˆj A kˆ, B BˆiBˆjB kˆ x y z x y z AB AB AB AB x x y y z z

17 Worked Example: Scalar Product Given two vectors A= ˆi + ˆ j k ˆ Find A B B = 2 ˆ i ˆ j + 3 kˆ Solution: AB = AB + AB + AB AB x x y y z z = (1)( 2) + (1)( 1) + ( 1)(3) = 6

18 Summary: Vector Product Magnitude: equal to the area of the parallelogram defined by the two vectors A B = A B sin θ = A B sin θ = A sin θ B (0 θ π ) ( ) ( ) Direction: determined by the Right-Hand-Rule Rule

19 Properties of Vector Products A B = B A c ( A B ) = A c B = ca B ( A+ B ) C = A C+ B C

20 Vector Product of Unit Vectors Unit vectors in Cartesian coordinates ˆi ˆj ˆi ˆj sin 2 1 ˆiˆi ˆi ˆj sin(0) 0 ˆi ˆjkˆ ˆi ˆi 0 ˆj kˆ ˆi ˆjˆj0 kˆ ˆi ˆj kˆ kˆ 0

21 Components of Vector Product A = A ˆ i + A ˆ j + A k ˆ, B = B ˆ i + B ˆ j + B kˆ x y z x y z A B = ( A B A B ) ˆ i + ( A B A B ) ˆ j + ( A B A B ) k ˆ = y z z y z x x z x y y x ˆ i ˆ j k ˆ A x A y Az B B B x y z

22 Worked Example: Vector Product Find a unit vector perpendicular to A= ˆ i+ ˆ j kˆ and B= 2 ˆi ˆ j + 3kˆ.

23 One Variable Calculus

24 Review: 1D Calculus Think about scalar a functions in 1D: f (x) x Think of this as height of mountain vs position 24

25 Derivatives How does function change with position? o f (x) df dx f '(a) = df dx x = a = slope x = a x Rate of change of f at x = a? 25

26 By the way Taylor Series Approximate function? Use derivatives! es f(x)=si in(2πx 1.0 x)what is f(x) near x=0.35? X

27 By the way Taylor Series Approximate function? Use derivatives! es (x)=sin n(2πx) f What is f(x) near x=0.35? T 0 (x) = f (0.35) X 1.00 Red curve is our approximation to f(x) near x=0.35 using one term in the Taylor series 27

28 By the way Taylor Series Approximate function? Use derivatives! es f(x)=si in(2πx x) What is f(x) near x=0.35? T 1 (x) = f (0.35) + f '(0.35)( x 0.35) X 1.00 Red curve is our approximation to f(x) near x=0.35 using two terms in the Taylor series 28

29 By the way Taylor Series Approximate function? Use derivatives! es f(x)=si in(2πx x) X 1.00 What is f(x) near x=0.35? T 2 (x) = f (0.35) + f '(0.35)( x 0.35) f ''(0.35) ( x 0.35 ) 2 2 Red curve is our approximation to f(x) near x=0.35 using three terms in the Taylor series 29

30 By the way Taylor Series Approximate function? Use derivatives! f(x)= =sin(2 2πx) What T ( x ) is = f(x) f(0.35) near x=0.35? 10 T 10 ( x ) + f '(0.35) ( x 0.35 ) X f ( ) x 2 2 ''(0.35) eleven more terms! Red curve is our approximation to f(x) near x=0.35 using 11 terms in the Taylor series N ( ) n x a d f In general T ( ) N x = n n! dx n = 0 x= a n 30

31 Taylor Series Most Commonly Used Only to 1st Order f(x) )=sin(2 2πx) Most Common: 1 st Order T (x) = f (a) + 1 f '(a)( x a ) X For hints as to when to use Taylor, look for approximate or when x is small or small angle or close to 31

32 Integration Sum function while walking along axis b f (x) f (x)dx =? a x = a x = b x Geometry: Find Area Also: Sum Contributions 32

33 Move to More Dimensions We ll start in 2D 33

34 Scalar Functions in 2D Function is height of mountain: z = F ( x, y ) Z Y X 34

35 Partial Derivatives How does function change with position? o In which direction are we moving? Z F y 0 F x > 0 Y X 35

36 Gradient What is fastest way up the mountain? Z Y X 36

37 Gradient Gade Gradient tells es you oudirection to move: ˆ F ˆ F ˆ ˆ ˆ F = i + j i + j +k x y x y z F > 0 F 0 x x F 0 F > 0 y y 37

38 f ( x,, y)ds) = C Line Integral Sum function while walking under surface along given curve Just like 1D integral, except now not just along x 38

39 2D Integration Sum function while walking under surface F ( x x, y )da Surface Just Geometry: Finding Volume Under Surface 39

40 N-D Integration in General Now think contribution from each piece Find area of surface? da Surface Volume of object? Object dv Mass Density Mass of object? dm = ρ dv Object Object IDEA: Break object into small pieces, visit each, asking What is contribution? 40

41 You Now Know It All Small Extension to Vector Functions 41

42 Can t Easily Draw Multidimensional Vector Functions In 2D various representations: ese s Vector Field Diagram Grass Seeds / Iron Filings 42

43 Integrating g Vector Functions Two types of questions generally asked: 1) Integral of vector function yielding vector dm Ex.: Mass Distribution g = G r 2 object r ˆ IDEA: Use Components - Just like scalar FrdA () = ˆ ˆ ˆ i F () r da + j F () r da + k F () r da x y z 43

44 Integrating g Vector Functions Two types of questions generally asked: 2) Integral of vector function yielding scalar Line Integral Ex.: Work W = F d s Curve IDEA: While walking along the curve how much of the function lies along our path 44

45 Integrating g Vector Functions One last example: Flux Q: How much does field E penetrate the surface? Flux Φ = E d A E Surface 45

46 Arc Length on Circle One Important Geometry Fact: Relation between arc length on circle and included angle θ R L = Rθ 46

47 Rectangular a Coordinates dv = dx dy dz da = dx dy da = dx dz da = dy dz Differentials Draw picture and think! 47

48 Cylindrical Coordinates dv = ρdϕ dρ dz da = ρdϕ dz da = ρdϕ dρ da = d dρ dz Differentials Draw picture and think! 48

49 Spherical Coordinates dv = r sinθdϕ rdθ dr da = r sinθdϕ rdθ Differentials r sinθ Draw picture and think! 49

50 Electricity and Magnetism: Math Vectors: Review Dot Product: How parallel? Cross Product: How perpendicular? Derivatives: Rate of change (slope) of function Gradient tells you how to go up fast Integrals: Visit each piece and ask contribution 50

51 MIT OpenCourseWare SC Physics II: Electricity and Magnetism Fall 2010 For information about citing these materials or our Terms of Use, visit: