From Math 2220 Class 30 Change of Coordinates Polar/Sph/Cyl Dr. - - PowerPoint PPT Presentation

From Math 2220 Class 30 V2c Schedule Left and Right Inverses Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable

From Math 2220 Class 30

• Dr. Allen Back
• Nov. 5, 2014

From Math 2220 Class 30 V2c Schedule Left and Right Inverses Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable

Schedule

12 lectures, 4 recitations left including today.

From Math 2220 Class 30 V2c Schedule Left and Right Inverses Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable

Schedule

a Most of what remains is vector integration and the integral theorems. b We’ll start 7.1, 7.2,4.2 on Friday. c If you are not in physics, please bear with us on this material. There are good reasons why it is always in multivariable calculus courses, though it will not be that evident at an elementary level.

From Math 2220 Class 30 V2c Schedule Left and Right Inverses Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable

Left and Right Inverses

Right inverses and existence of solutions:s Set theoretically, for f : A → B, a function g : B → A satisfying f ◦ g = IdB (i.e. f (g(q)) = q for all q ∈ B) is EQUIVALENT to the equation f (p) = q having a solution p ∈ A for any q ∈ B.

From Math 2220 Class 30 V2c Schedule Left and Right Inverses Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable

Left and Right Inverses

Left inverses and uniqueness of solutions: Set theoretically, for f : A → B, a function g : B → A satisfying g ◦ f = IdA (i.e. g(f (p)) = p for all p ∈ A) is EQUIVALENT to the equation f (p) = q whenever (i.e. for which q) it has a solution, having that solution be unique among elements of A.

From Math 2220 Class 30 V2c Schedule Left and Right Inverses Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable

Left and Right Inverses

Of course in linear algebra, we learn that for linear transformations f : Rn → Rn given by f (x) = Ax, either of the above are equivalent to det(A) = 0.

From Math 2220 Class 30 V2c Schedule Left and Right Inverses Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable

Left and Right Inverses

Recall also for f : Rn → Rn and f (x) = Ax: a Linear transformations take parallelograms to

• parallelograms. (Similarly in higher dim.)

b If e1, e2, . . . , en are the standard basis vectors, then the i’th column of the n × n matrix A is given by Aei. c So, e.g. it is easy to map the unit square with sides e1, e2 in R2 (and lower left corner at the

• rigin) to any parallelogram with sides v1, v2 (and

lower left corner at the origin. (A =

• v1

v2

• .)

d So A−1 can take us from an arbitrary parallelogram with sides v1, v2 to the unit square.

From Math 2220 Class 30 V2c Schedule Left and Right Inverses Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable

Left and Right Inverses

Recall also for f : Rn → Rn and f (x) = Ax: e By composition of two of the above, we can map any parallelogram with sides w1, w2 in R2 (and lower left corner at the origin) to any parallelogram with sides v1, v2 (and lower left corner at the origin. f Throwing in a translation lets us deal with parallelograms for which the origin is not a vertex.

From Math 2220 Class 30 V2c Schedule Left and Right Inverses 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 30 V2c Schedule Left and Right Inverses 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 30 V2c Schedule Left and Right Inverses 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 30 V2c Schedule Left and Right Inverses 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 30 V2c Schedule Left and Right Inverses 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 30 V2c Schedule Left and Right Inverses 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 30 V2c Schedule Left and Right Inverses 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 30 V2c Schedule Left and Right Inverses 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 30 V2c Schedule Left and Right Inverses 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 30 V2c Schedule Left and Right Inverses 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 30 V2c Schedule Left and Right Inverses 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 30 V2c Schedule Left and Right Inverses 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 30 V2c Schedule Left and Right Inverses 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 30 V2c Schedule Left and Right Inverses 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 30 V2c Schedule Left and Right Inverses 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 30 V2c Schedule Left and Right Inverses 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 30 V2c Schedule Left and Right Inverses 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 30 V2c Schedule Left and Right Inverses 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 30 V2c Schedule Left and Right Inverses 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 30 V2c Schedule Left and Right Inverses 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 30 V2c Schedule Left and Right Inverses 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 30 V2c Schedule Left and Right Inverses 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 30 V2c Schedule Left and Right Inverses 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 30 V2c Schedule Left and Right Inverses 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 30 V2c Schedule Left and Right Inverses 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 30 V2c Schedule Left and Right Inverses 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 30 V2c Schedule Left and Right Inverses Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable

Why Cont. Fcns are Integrable

From Math 2220 Class 30 V2c Schedule Left and Right Inverses 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 30 V2c Schedule Left and Right Inverses 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.