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Dot Product and Cross Product
Takes: two vectors Results: scalar for dot product, vector for cross product Given: (theta) = angle between vectors a and b when their tails are placed at same point a.b = |a| |b| cos(theta) a.b = (a_x * b_x) + (a_y * b_y) + (a_z * b_z) a x b = |a| |b| sin(theta) a x b = TERM 2
Vector Projection
DEFINITION 2 The vector resolute (also known as the vector projection) of two vectors, in the direction of (also " on "), is given by: proj_a b = (a.b) * (unit vector in direction of a) / |a| OR proj_a b = |a.b| / |a| TERM 3
Scalar Projection
DEFINITION 3 The scalar resolute, also known as the scalar projection or scalar component, of a vector in the direction of a vector is given by: scalar proj of b onto a = (a.b) / |a| TERM 4
Unit Vector
DEFINITION 4 In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) whose length is 1 (the unit length). unit vector in direction of a = a / |a| TERM 5
Vector and Parametric Equations of a Line
DEFINITION 5 Given: arbitrary point P_o = (x_o, y_o, z_o) on line L fixed point P = (x, y, z) on line L Let r = vector connecting origin and point P Let r_o = vector connecting origin and point P_o Let v = r - r_o Eq: = t w/t = parameter Eq: x = x_o + ta , y = y_o + tb , z = z_o + tc
Skew Lines
In solid geometry, skew lines are two lines that do not intersect but are not parallel. Note: Given v_1 and v_2 , if v_
- t = v_2 where t is a scalar constant, then v_1 and v_2 are parallel. TERM 7
Vector Equation of a Plane
DEFINITION 7 Given: Arbitrary point P = (x, y, z) with position vector r Fixed point P_o = (x_o, y_o, z_o) with position vector r_o Normal vector to plane n = coefficients of variables Eq: n * (r - r_o) =0 OR a(x - x_o) + b(y - y_o) + z(z - z_o) = TERM 8
Unit Tangent Vector [T(t)] and Unit Normal
Vector [N(t)]
DEFINITION 8 Given: Vector function r(t) and it's derivative r'(t) Unit Tangent Vector = T(t) = r'(t) / |r'(t)| Unit Normal Vector = N(t) = T'(t) / |T'(t)| TERM 9
Arc Length of Vector Function Given
Parametrically
DEFINITION 9 Given: r(t) -> x = f(t) , y = g(t) , z = h(t) Length L = Integrate[ |r(t)| dt , t, a, b] where a and b are the bounds of t TERM 10
Curvature
DEFINITION 10 Given: Vector function r(t) Unit Tangent Vector = T(t) [#8] Curvature k = |T'(t)| / |r'(t)| OR Curvature k = |r'(t) x r''(t)| / ( |r'(t)|^3 )
Directional Derivative Maximum Rate of
Change
Given: Point P = ( x_o, y_o, z_o ) Vector function or field f Gradient of g = grad(f) Unit Vector u Directional derivate of f( P ) in direction of u = D_u f( P ) D_u f( P ) = grad( f( P ) ). u , this is a dot product and is scalar Maximum Rate of Change = |grad( f( P ) )| Direction of Max Rate of Change = grad( f( P ) ) / |grad( f( P ) )| TERM 17
Level Surface
DEFINITION 17 Graph of the points (x, y, z) that satisfy F(x, y, z) = k, where k is a constant TERM 18
Tangent Plane to Level Surface
DEFINITION 18 Given: Level Surface F(x, y, z) = k, where k is constant Point P( x_o, y_o, z_o ) that lies on the level surface Normal vector to surface = n = grad( F( P ) ) Eq of Tangent Plane: (dF( P ) / dx)(x - x_o) + (dF( P ) / dy)(y - y_o) + (dF( P ) / dz)(z - z_o) = 0 Basically, the same equation of plane we are used to (#7), which states Eq of plane is: n * (r - r_o) TERM 19
Minimum and Maximum Values Critical Points
Saddle Points
DEFINITION 19 Given: f(x, y) Critical Point (a,b) if both partial derivatives = 0 Local max at (a,b) of f(a,b) if f(a,b) >= f(x,y) for (x,y) near(a,b) Local min at (a,b) of f(a,b) if f(a,b) TERM 20
2nd Derivative
Test
DEFINITION 20 Let D = f_xx (a,b). f_yy (a,b) - [ f_xy (a,b) ]^2 If D > 0 and f_xx (a,b) > 0, then (a,b) is local min If D > 0 and f_xx (a,b) < 0, then (a,b) is local max If D < 0 then (a,b) is saddle point If D = 0 then test fails
Lagrange Multipliers
Problem: maximize or minimize function(find extrema) of f(x,y) that is subject to some constraint g(x,y) = k Use fact: level curve f(x,y) = c is orthogonal at (x_o, y_o) to the grad( f( x_o, y_o ) ) 1). Find all values of x,y,z, and (lamda) so that: grad( f ) = (lamda) * grad( g ) AND g(x,y,z) = k 2). Evaluate f at all extrema found, the largest is the max, smallest is minimum of f(x,y,z). (lamda) is Lagrand Multiplier TERM 22
Double Integrals Over General
Regions
DEFINITION 22 Find where the graphs intersect. Type 1: Solve for x, x is last term of integration and it's range is two constants. Type 2: Solve for y, y is last term of integration and it's range is two constants. If one type gives an unsolvable integral, use the other type. TERM 23
Triple Integrals
DEFINITION 23 Find intersections. Project graph onto planes when necessary. TERM 24
Cylindrical Coordinates
DEFINITION 24 Takes the form: (r, (theta), z) where: r = distance from the z-axis theta = angle from the positive x-axis, counter-clockwise z = the same z from rectangular coordinates Changing Cylindrical Coordinates to Rectangular Coordinates: x = r * cos(theta) y = r * sin(theta) z = z r^2 = x^2 + y^2 -> Think circles *Note: When integrating, add extra r to term TERM 25
Spherical Coordinates
DEFINITION 25 Takes form: ( p, (theta), (phi) ) where: p = distance from origin theta = angle from the positive x-axis, counter-clockwise phi = angle from positive z-axis, counter-clockwise Changing Spherical Coordinates to Rectangular Coordinates: x = p * sin(phi) * cos(theta) y = p * sin(phi) * sin(theta) z = p * cos(phi) Note: p = cos(phi) if sphere, pcos(theta) = p*sin(phi) if cone *Note: When integrating, add extra p^2 * sin(phi)
Line Integrals of Vector Fields
Line integral of continuous vector field F over C given by r(t): Integral of F. dr over c = Integrate[ F( r(t) ). r'(t), t, a, b] *Note: dr = Unit Tangent(#8) This is how we find Work as it applies to Physics. *Note: vector field F is conservative if dP/dy = dQ/dx TERM 32
Fundamental Theorem of Line Integrals
DEFINITION 32 The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field (any irrotational vector field can be expressed as a gradient) can be evaluated by evaluating the original scalar field at the endpoints of the curve. Integrate[ grad( f ) , r, a, b] = f( r(b) ) - f( r(a) ) *Note: vector field F is conservative if dP/dy = dQ/dx TERM 33
Fundamental Theorem of Line Integrals 2
DEFINITION 33 To switch from F = grad( f ) to just f: We know F = < df/dx , df/dy, df/dz > f(x,y,z) = integral of df/dx + g(y,z) df/dy = dg/dy, so g(y,z) = integral of df/dy with respect to y Now f(x,y,z) = integral of df/dx + g(y,z) + h(z) df/dz = integral of dh/dz, so h(z) = integral of df/dz with respect to z Looks much simpler written on paper! TERM 34
Green's Theorem
DEFINITION 34 In physics and mathematics, Green's theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C. Integral of P. dx + Q. dy = Double Integral [dQ/dx - dP/dx] TERM 35
Curl
DEFINITION 35 Given vector field F curlF = grad() x F
Divergence
Given vector field F divF = grad(). F This translates to: divF = dP/dx + dQ/dy + dR/dz *Fact: If P,Q,R all have continuous 2nd order partial derivatives, then div(curlF) = 0 TERM 37
Parametric Surfaces
DEFINITION 37 Given: point P on surface and on smooth curve C position vector r(u_o, v_o) of P Both dr/du and dr/dv are tangent to the surface at point P Therefore, the tangent plane to this surface is the plane containing both vector dr/du and dr/dv Area = Double Integral of |dr/du x dr/dv| TERM 38
Surface Integrals
DEFINITION 38 Double Integral [ f( r(u,v) ). |dr/du x dr/dv| ] over D Area of S = Double Integral[ 1. dS ] over surface S Normal Unit Vector n: (dr/du x dr/dv) / |dr/du x dr/dv| n represents the orientation of the surface Given graph of surface by z = g(x,y) and G(x,y) = z - g(x,y)=0 n = grad( G ) / |grad( G )| TERM 39
Flux
DEFINITION 39 Def: The surface integral of surface S over vector field F Double Integral F. dS = Double Integral F. n ds, both over S *Remember: n = (dr/du x dr/dv) / |dr/du x dr/dv| OR n = grad( G ) / |grad( G )| *Also: ds = |1 + dz/dx + dz/dy| TERM 40
Stoke's Theorem
DEFINITION 40 Given nice surface S and it's positively oriented boundary C and a vector field F: Integral of F. dr over C = Double Integral of curlF. ds over S *Amazing: if S_1 and S_2 have same boundary C, then the Double Integral curlF. ds over S_1 = Double Integral curlF. ds over S_
