From Math 2220 Class 31 V2 Schedule Vector Fields From Math 2220 Class 31 Line and Path Integrals Properties Interpretations Dr. Allen Back Change of Coordinates Polar/Sph/Cyl Problems Nov. 7, 2014 Inverses from Algebra Why Cont. Fcns are Integrable
Schedule From Math 2220 Class 31 V2 Schedule Easy Sections: 7.1-7.2. 8.1, 8.3, chapter 4. Vector Fields Line and Path Hard Sections: 7.3-7.6,8.2,8.4. Integrals Properties Interpretations Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable
Schedule From Math 2220 Class 31 V2 Easy Sections: 7.1-7.2. 8.1, 8.3, chapter 4. Schedule Hard Sections: 7.3-7.6,8.2,8.4. Vector Fields Line and Path The hard sections will change to only moderately difficult if you Integrals concentrate on the ideas in the “easy sections” and recognize Properties the more challenging material as just more intricate versions of Interpretations the easy stuff. Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable
Vector Fields - Section 4.3 From Math 2220 Class 31 V2 Schedule Definition: A vector field on a domain D ⊂ R n is a choice of Vector Fields vector � F ( p ) at each p ∈ R n . Line and Path Integrals Properties Interpretations Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable
Vector Fields - Section 4.3 From Math 2220 Class 31 V2 Schedule Definition: Mathematically a vector field is just a function Vector Fields F : D ⊂ R n → R n . � Line and Path Integrals Properties Interpretations Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable
Vector Fields - Section 4.3 From Math 2220 Class 31 V2 Schedule Intuitively: A choice of vector at each point of some domain in Vector Fields the plane or in space. Line and Path Integrals Properties Interpretations Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable
Vector Fields - Section 4.3 From Math Electric Field of a Dipole 2220 Class 31 V2 Schedule Vector Fields Line and Path Integrals Properties Interpretations Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable
Vector Fields - Section 4.3 From Math 2220 Class 31 V2 Schedule When we draw vector fields, generally the spacing between Vector Fields vectors is greater when the vector field is smaller in magnitude. Line and Path Integrals (We’ll eventually be more precise on that.) Properties Interpretations Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable
Vector Fields - Section 4.3 From Math 2220 Class 31 V2 A flow line (or integral curve of a vector field � F ) is a path γ ( t ) so that for any t , the velocity vector γ ′ ( t ) is the same as the Schedule Vector Fields value of the vector field at γ ( t ); i.e. Line and Path Integrals γ ′ ( t ) = � F ( γ ( t )) . Properties Interpretations Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable
Vector Fields - Section 4.3 From Math 2220 Class 31 A flow line (or integral curve of a vector field � F ) is a path γ ( t ) V2 so that for any t , the velocity vector γ ′ ( t ) is the same as the Schedule value of the vector field at γ ( t ); i.e. Vector Fields γ ′ ( t ) = � Line and Path F ( γ ( t )) . Integrals Properties Interpretations We’ll discuss later how flow lines can always be determined Change of Coordinates from vector fields by solving differential equations. Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable
Line and Path Integrals From Math 2220 Class 31 V2 Planar case: Given a path c : [ a , b ] → R 2 representing a curve Schedule C and a scalar function f ( x . y ) or a vector field � F ( x , y ) we can Vector Fields form Line and Path Integrals � Path Integral C f ( x , y ) ds . Properties C � � Line Integral F ( x , y ) · d � s . Interpretations Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable
Line and Path Integrals Calculation of both of these is ordinarily done by “converting From Math 2220 Class 31 them” (really a set of definitions) into ordinary 1-variable V2 integrals as suggested by the formalism Schedule � c ′ ( t ) � dt ds = Vector Fields tracking the length of a Line and Path Integrals small section of the path. Properties c ′ ( t ) dt d � s = Interpretations Change of tracking the linear approximation Coordinates Polar/Sph/Cyl of a secant between nearbye points Problems Inverses from Algebra Why Cont. Fcns are Integrable
Line and Path Integrals From Math 2220 Class 31 V2 Schedule These are calculated by � b Vector Fields � a f ( c ( t )) � c ′ ( t ) � dt Path Integral C f ( x , y ) ds = Line and Path � b Integrals C � a � � F ( c ( t )) · c ′ ( t ) dt . Line Integral F ( x , y ) · d � s = Properties Interpretations Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable
Line and Path Integrals From Math Linear Approximation of secants and lengths along a curve 2220 Class 31 V2 Schedule Vector Fields Line and Path Integrals Properties Interpretations Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable
Line and Path Integrals From Math 2220 Class 31 V2 Schedule Riemann Sums Associated with these Integrals Vector Fields Line and Path Path Integral Σ f ( c ( t ∗ )) � c ′ ( t ∗ ) � ∆ t . Integrals Line Integral Σ � Properties F ( c ( t ∗ )) · c ′ ( t ∗ )∆ t Interpretations Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable
Line and Path Integrals From Math With a Vector Field along the curve 2220 Class 31 V2 Schedule Vector Fields Line and Path Integrals Properties Interpretations Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable
Line and Path Integrals From Math 2220 Class 31 V2 For the helix C given by c ( t ) = ( R cos t , R sin t , bt ), � 0 ≤ t ≤ 4 π , find the value of C ρ gz ds where ρ, g are Schedule constants. Vector Fields Line and Path (Without the g , this would be total mass. Integrals With the g , if z is up it might be total potential energy. Properties The helix shape could, for example, fit the shape of a spring.) Interpretations Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable
Line and Path Integrals From Math 2220 Class 31 V2 Schedule For the helix C given by c ( t ) = ( R cos t , R sin t , bt ), Vector Fields C � � 0 ≤ t ≤ 4 π , find the value of F · d � s where Line and Path � F ( x , y , z ) = ( x 2 , y 2 , z 2 ) . Integrals Properties Interpretations Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable
Properties From Math 2220 Class 31 V2 Another notation for line integrals of Schedule � F ( x , y , z ) = ( P ( x , y , z ) , Q ( x , y , z ) , R ( x , y , z )) : Vector Fields Line and Path � � � Integrals F · d � s = Pdx + Qdy + Rdz . Properties C C Interpretations Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable
Properties From Math 2220 Class 31 V2 � y dx + y 2 dy Schedule Vector Fields C Line and Path for the two paths Integrals Straight line from ( − 1 , 0) to (1 , 0) . Properties Parabola t → ( t , 1 − t 2 ) for − 1 ≤ t ≤ 1 . Interpretations Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable
Properties From Math 2220 Class 31 V2 Schedule Line integrals of vector fields depend on the path as well as the Vector Fields endpoints. Line and Path Integrals Properties Interpretations Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable
Properties From Math 2220 Class 31 Invariance under orientation preserving reparametrization for V2 line integrals. Schedule (Or any reparametrization for path integrals.) Vector Fields Line and Path Line integral of a gradient. If � F = ∇ f then Integrals Properties � F · d � s = f (end) − f (start) . Interpretations Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable
Interpretations From Math 2220 Class 31 V2 Schedule As always, the guiding principle is what does the Riemann sum Vector Fields naturally represent? Line and Path Integrals Properties Interpretations Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable
Interpretations From Math 2220 Class 31 V2 Path Integrals Schedule Arclength Vector Fields Line and Path Mass of a wire given its density. Integrals Bending of a curve form its curvature. Properties Interpretations Area of a fence along a plane curve from its height Change of along the curve. Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable
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