From Math 2220 Class 31 Line and Path Integrals Properties - - PowerPoint PPT Presentation

From Math 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

From Math 2220 Class 31

• Dr. Allen Back
• Nov. 7, 2014

From Math 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

Schedule

Easy Sections: 7.1-7.2. 8.1, 8.3, chapter 4. Hard Sections: 7.3-7.6,8.2,8.4.

From Math 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

Schedule

Easy Sections: 7.1-7.2. 8.1, 8.3, chapter 4. Hard Sections: 7.3-7.6,8.2,8.4. The hard sections will change to only moderately difficult if you concentrate on the ideas in the “easy sections” and recognize the more challenging material as just more intricate versions of the easy stuff.

From Math 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

Definition: A vector field on a domain D ⊂ Rn is a choice of vector F(p) at each p ∈ Rn.

From Math 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

Definition: Mathematically a vector field is just a function

• F : D ⊂ Rn → Rn.

From Math 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

Intuitively: A choice of vector at each point of some domain in the plane or in space.

From Math 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

Electric Field of a Dipole

From Math 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

When we draw vector fields, generally the spacing between vectors is greater when the vector field is smaller in magnitude. (We’ll eventually be more precise on that.)

From Math 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

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 value of the vector field at γ(t); i.e. γ′(t) = F(γ(t)).

From Math 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

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 value of the vector field at γ(t); i.e. γ′(t) = F(γ(t)). We’ll discuss later how flow lines can always be determined from vector fields by solving differential equations.

From Math 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

Planar case: Given a path c : [a, b] → R2 representing a curve C and a scalar function f (x.y) or a vector field F(x, y) we can form Path Integral

• C f (x, y) ds.

Line Integral

• C

F(x, y) · d s.

From Math 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

Calculation of both of these is ordinarily done by “converting them” (really a set of definitions) into ordinary 1-variable integrals as suggested by the formalism ds = c′(t) dt tracking the length of a small section of the path. d s = c′(t) dt tracking the linear approximation

• f a secant between nearbye points

From Math 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

These are calculated by Path Integral

• C f (x, y) ds =

b

a f (c(t)) c′(t) dt

Line Integral

• C

F(x, y) · d s = b

a

F(c(t)) · c′(t) dt.

From Math 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

Linear Approximation of secants and lengths along a curve

From Math 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

Riemann Sums Associated with these Integrals Path Integral Σf (c(t∗))c′(t∗)∆t. Line Integral Σ F(c(t∗)) · c′(t∗)∆t

From Math 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

With a Vector Field along the curve

From Math 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

For the helix C given by c(t) = (R cos t, R sin t, bt), 0 ≤ t ≤ 4π, find the value of

• C ρgz ds where ρ, g are

constants. (Without the g, this would be total mass. With the g, if z is up it might be total potential energy. The helix shape could, for example, fit the shape of a spring.)

From Math 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

For the helix C given by c(t) = (R cos t, R sin t, bt), 0 ≤ t ≤ 4π, find the value of

• C

F · d s where

• F(x, y, z) = (x2, y2, z2).

From Math 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

Properties

Another notation for line integrals of

• F(x, y, z) = (P(x, y, z), Q(x, y, z), R(x, y, z)) :
• C
• F · d

s =

• C

Pdx + Qdy + Rdz.

From Math 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

Properties

• C

y dx + y2 dy for the two paths Straight line from (−1, 0) to (1, 0). Parabola t → (t, 1 − t2) for −1 ≤ t ≤ 1.

From Math 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

Properties

Line integrals of vector fields depend on the path as well as the endpoints.

From Math 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

Properties

Invariance under orientation preserving reparametrization for line integrals. (Or any reparametrization for path integrals.) Line integral of a gradient. If F = ∇f then

• F · d

s = f (end) − f (start).

From Math 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

Interpretations

As always, the guiding principle is what does the Riemann sum naturally represent?

From Math 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

Interpretations

Path Integrals Arclength Mass of a wire given its density. Bending of a curve form its curvature. Area of a fence along a plane curve from its height along the curve.

From Math 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

Interpretations

Line Integrals Work Electric Potential Differential Equations applications

From Math 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

Change of Coordinates

∆A ∼ r∆r∆θ by geometry

From Math 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

Change of Coordinates

The Polar Coordinate Transformation

From Math 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

Change of Coordinates

∆A ∼

• xu

xv yu yv

• ∆u∆v

From Math 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

Change of Coordinates

For a C 1 1:1 onto map F(u, v) = (x(u, v), y(u, v) taking D∗ ⊂ R2 in the uv-plane to D ⊂ R2 in the xy plane, dA = dx dy =

• xu

xv yu yv

• dudv

lets us move between xy and uv integrals. (Similarly for more than 2 variables.)

From Math 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

Change of Coordinates

Polar: dA = r dr dθ. Cylindrical: dV = r dr dθ dz. Spherical: dV = ρ2 sin φ dρ dθ dφ.

From Math 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

Change of Coordinates

The Basic Spherical Coordinate Picture

From Math 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

Polar/Spherical/Cylindrical Problems

Problem:

• D(x2 + y2)

3 2 dx dy for D the disk x2 + y2 ≤ 1.

From Math 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

Polar/Spherical/Cylindrical Problems

Problem: ∞

−∞ e−x2 dx.

From Math 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

Polar/Spherical/Cylindrical Problems

Problem: Volume of a right circular cone of height H and largest radius R.

From Math 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

Polar/Spherical/Cylindrical Problems

Problem: Volume of the portion of the Earth above latitude 45◦. Ambiguous as written; could mean all points with colatitude φ in spherical coordinates less than π

4 .

Or could refer to all points whose z value is above the z value at this latitude. Either integral could be set up . . . .

From Math 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

Polar/Spherical/Cylindrical Problems

Problem: Volume, using spherical coordinates, of a ball of radius R with a hole of radius a (centered on a diameter) drilled out of it.

From Math 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

Inverses from Algebra

Right inverses and existence of solutions

From Math 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

Inverses from Algebra

Left inverses and uniqueness of solutions

From Math 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

Inverses from Algebra

Consider the map F : D∗ ⊂ R2 → D ⊂ R2 defined by F(x, y) = (x2 − y2, x + y) Label the components of image points F(x, y) as (u, v); i.e. we think of the above transformation as u = x2 − y2 v = x + y

From Math 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

Inverses from Algebra

To study one-to-oneness and ontoness of F, consider the algebra: u v = x2 − y2 x + y = x − y v = x + y

From Math 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

Inverses from Algebra

Adding and subtracting the above two equations: x = 1 2

• v + u

v

• y

= 1 2

• v − u

v

From Math 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

Inverses from Algebra

You may take as given the fact that these formulae check showing that G(u, v) = (1 2

• v + u

v

• , 1

2

• v − u

v

• )

gives the inverse of F where everything is defined; i.e. F(G(u, v)) = (u, v) G(F(x, y) = (x, y)

From Math 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

Inverses from Algebra

What is the natural domain of the function G? In other words, describe the largest subset (call it U) of the uv plane on which G is defined.

From Math 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

Inverses from Algebra

Find the largest set V in the xy plane so that for all (x, y) ∈ V , F(x, y) belongs to the domain U of the function G which you found above.

From Math 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

Inverses from Algebra

Find a point (x, y) so that F(x, y) = (4, 2). More generally, briefly explain why the equation F(G(u, v)) = (u, v) shows that F : V → U is onto.

From Math 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

Inverses from Algebra

Note F(1, 1) = (0, 2). Can there be a different point (x, y) besides (1, 1) with F(x, y) = (0, 2)? More generally, briefly explain why the equation G(F(x, y)) = (x, y) shows that F : V → U is one-to-one.

From Math 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

Why Cont. Fcns are Integrable

Two Definitions Continuity at each Point of a set S: ∀x ∈ S and ∀ǫ > 0 ∃δ > 0 so that |x − y| < δ ⇒ |f (x) − f (y)| < ǫ. Uniform Continuity on a Set S: ∀ǫ > 0 ∃δ > 0 so that ∀x, y ∈ S, |x − y| < δ ⇒ |f (x) − f (y)| < ǫ.

From Math 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

Why Cont. Fcns are Integrable

An Important Theorem Bolzano Weierstrass: Every bounded sequence in Rn has a convergent subsequence. This theorem is also at the heart of the proof that continuous functions with closed and bounded domains are automatically bounded and attain their extrema.

From Math 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

Why Cont. Fcns are Integrable

The idea of why Bolzano Weirrstrass holds:

From Math 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

Why Cont. Fcns are Integrable

Based on Bolzano Weierstrass, one can show that every continuous function on a closed and bounded set is uniformly continuous there.

From Math 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

Why Cont. Fcns are Integrable

Suppose one is considering the integrability of a continuous function f over a rectangle of area A.

From Math 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

Why Cont. Fcns are Integrable

From Math 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

Why Cont. Fcns are Integrable

Suppose one is considering the integrability of a continuous function f over a rectangle of area A. Consider any regular partition each of whose constituent rectangles is smaller in diameter than the δ given in the definition of uniform continuity.

From Math 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

Why Cont. Fcns are Integrable

Suppose one is considering the integrability of a continuous function f over a rectangle of area A. Consider any regular partition each of whose constituent rectangles is smaller in diameter than the δ given in the definition of uniform continuity. Then any two Riemann sums for this partition will differ by at most ǫA. This allows one to show that all Riemann sums settle down to a single limit as the partitions becomes sufficiently fine.