ECE 206: Advanced Calculus 2ece206/Lectures/slides/section1.pdf · ECE 206: Advanced Calculus 2...

ECE 206: Advanced Calculus 2

Department of Electrical and Computer EngineeringUniversity of Waterloo

Fall 2014

Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 1/39

Course details

Instructor

Dr. Oleg Michailovich ([email protected], EIT 4127, ext. 38247)

Office hours

TBD in according to the class and instructor’s preferences

Course websites

ece.uwaterloo.ca/~ece206 (general information, lecture notes& slides, laboratories, supporting material, etc.)

learn.uwaterloo.ca/ (lab and graded assignments, exams andmarks)


Marking scheme

Laboratory assignments (6 assignments × 3%): 18%

Graded assignments (6 assignments × 2%): 12%

Midterm exam: 20%

Final exam: 50%

Solutions for the laboratories may be submitted in pairs while gradedassignments must be completed individually.


Recommended course textbook

Michael D. Greenberg, AdvancedEngineering Mathematics, 2nd ed.

However ....

The class material will be mainly based on the lecture notes by Prof.Andrew Heunis which are available at the course website.


Course outline

1 Multi-dimensional integration

Two-dimensional integrationThree-dimensional integration

2 Scalar and vector fields

Motivating examplesDefinition of scalar and vector fields

3 Curves and paths in space

Motivating examplesPaths and parametric representation of curvesDerivatives along a path and tangent to a curveSimple curves and closed curves

4 Line integral and arc length

Line integral of a vector fieldLine integral of a scalar field and arc length


Course outline (cont.)

5 Conservative vector fields

Gradient of a scalar fieldConservative vector fieldsConservation of energy

6 Green’s theorem in the plane

Green’s theorem for rectanglesGreen’s theorem: the general case

7 Surfaces, surface areas and surface integrals

Parametric representation of surfacesTangents to a surface and smooth surfacesArea of a surfaceSurface integral of a scalar fieldSurface integral of a vector field

8 Vector calculus

The divergence, Laplacian, and curl differential operatorsTheorem of StokesDivergence theorem of Gauss-OstogradskiiThe continuity equation


Course outline (cont.)

9 The basic laws of electricity and magnetism

Static electric fieldsStatic magnetic fieldsTime-varying fields

10 Maxwell’s equations

The Ampere-Maxwell law for time-varying fieldsMaxwell’s equationsElectromagnetic waves without sourcesElectromagnetic waves with sources

11 Cylindrical and spherical coordinates

Polar coordinatesCylindrical coordinatesSpherical coordinates


Questions?


General overview

The course is about vector calculus and the calculus of complexvariables.

The three pillars of vector calculus are: Green’s theorem, Stokes’theorem and the Gauss-Ostrogradskii theorem.

Used in electromagnetism, aerodynamics, fluid mechanics, classi-cal mechanics, quantum mechanics and gravitational physics.

Allows representing the main laws of electricity and magnetismas a set of just four equations, called Maxwell’s equations.

Richard P. Feynman:

“From a long view of the history of mankind - seen,from, say, ten thousand years from now - there can belittle doubt that the most significant event of the 19thcentury will be judged as Maxwell’s discovery of the lawsof electrodynamics”.


Maxwell’s marvellous equations

For given electric field E, magnetic field B, charge density ρ, andcurrent density field J, Maxwell’s equations

∇ ·E =ρ

ε0∇ ·B = 0

∇×E +∂B

∂t= 0

∇×B =µ0J + ε0µ0∂E

∂t

So what are the symbols ∇· and ∇× standing for?


Two Dimensional Integration

We want to integrate a real-valued function

f : D → R

where D ⊂ R2 is a rectangular domain defined as

D =

(x, y) ∈ R2 | a ≤ x ≤ b, c ≤ y ≤ d

= [a, b]× [c, d].


Finite partition

Subdivide the intervals [a, b] and [c, d] into n+ 1 equally spaced pointsxini=0 and yjnj=0, respectively, such that

a = x0 < x1 < . . . < xn = b, c = y0 < y1 < . . . < yn = d,

with spacings

∆x := xi+1 − xi =b− an

, ∆y := yj+1 − yj =d− cn

.

Let Di,j to be the small rectangular defined by

Di,j =

(x, y) ∈ R2 | xi ≤ x ≤ xi+1, yj ≤ y ≤ yj+1

= [xi, xi+1]×[yj , yj+1]

and let ri,j = (ξi, ηj) ∈ Di,j , which implies

xi ≤ξi ≤ xi+1

yj ≤ηj ≤ yj+1


Riemann sum

Definition

We define the Riemann sum of the function f on the rectangle D as

Sn :=

n∑i=0

n∑j=0

f(ξi, ηj)∆x∆y,

for any n ∈ 1, 2, . . .


Riemann integral

Definition

If the sequence of Riemann sums Sn, n = 1, 2, ... converges to alimit S as n→∞, and the limit S is the same for every choice ofpoints (ξi, ηj) ∈ Di,j , then S is called the (Riemann) integral of thefunction f over the rectangle D.

Various notations for the integral S are∫D

f(x, y)dxdy,

∫D

f(x, y)dA,

∫D

fdxdy,

∫D

fdA.

Remarks:

If the Riemann sums do not converge to any limit, the integral issaid to be undefined.

Fortunately, we need never be concerned with this fact, for theclass of functions which can be integrated over D is simply huge.


Fubini’s theorem

If we fix some x ∈ [a, b], a function f(x, y) depends only on y inthe interval c ≤ y ≤ d.

For any such x, let h1(x) be defined as:

h1(x) :=

∫ d

c

f(x, y)dy ∀a ≤ x ≤ b.

In exactly the same way we also define the function h2(y) as:

h2(y) :=

∫ b

a

f(x, y)dx ∀c ≤ y ≤ d.

Fubini’s theorem∫D

f(x, y)dxdy =

∫ b

a

h1(x)dx =

∫ d

c

h2(y)dy.


Fubini’s theorem (cont.)

A more detailed way to formulate Fubini’s theorem is:∫D

f(x, y)dxdy =

∫ b

a

∫ d

c

f(x, y)dy

dx =

∫ d

c

∫ b

a

f(x, y)dx

dy.

Fubini’s theorem reduces evaluation of an integral over a rectang-le to the successive evaluations of two integrals over intervals.

These are called iterated integrals.

Both choices will work but in practice it is often the case thatone choice involves less work than the other.

Example: Integrate f(x, y) = x2 + y2 over D := [−1, 1]× [0, 1].


Integrals over non-rectangular domains

What if D ⊂ R2 is not a rectangular domain in the x− y plane.

To define such D suppose that φ1 : [a, b]→ R and φ2 : [a, b]→ Rare given continuous functions over some fixed interval a ≤ x ≤ bsuch that

φ1(x) ≤ φ2(x), ∀a ≤ x ≤ y.

Definition

The region D ⊂ R2 is called y-simple with lower function φ1(x), upperfunction φ2(x) and common interval of definition a ≤ x ≤ b, when

D =

(x, y) ∈ R2 | a ≤ x ≤ b, φ1(x) ≤ y ≤ φ2(x).


Non-rectangular domains (cont.)

Let the constants c and d be defined as:

c < φ1(x) ≤ φ2(x) < d, ∀a ≤ x ≤ y.

Then, D is contained within E (i.e., D ⊆ E), where

E =

(x, y) ∈ R2 | a ≤ x ≤ b, c ≤ y ≤ d

= [a, b]× [c, d].


Non-rectangular domains (cont.)

Now define the function f∗ : E → R2 as given by

f∗(x, y) =

f(x, y), if (x, y) ∈ D0, if (x, y) ∈ E\D

It is then evident that∫D

f(x, y)dxdy =

∫E

f∗(x, y)dxdy

Now, by Fubini’s theorem∫E

f∗(x, y) dxdy =

∫ b

a

∫ d

c

f∗(x, y)dy

dx =

=

∫ b

a

∫ φ2(x)

φ1(x)

f∗(x, y)dy

dx =

∫ b

a

∫ φ2(x)

φ1(x)

f(x, y)dy

dx.


Fubini’s theorem for y-simple regions


Suppose that D is any y-simple region with lower function φ1(x),upper function φ2(x) and common interval of definition a ≤ x ≤ b,and f : D → R is a given function. Then∫

D

f(x, y)dxdy =

∫ b

a

∫ φ2(x)

φ1(x)

f(x, y)dy

dx.

Example: For φ1(x) = 0, φ2(x) =√

1 + cosx, and x ∈ [0, 2π], findthe integral of f(x, y) = 2y.

Example: Evaluate the integral∫Dx2y dxdy, where D is a triangular

area bounded by the lines x = 0, y = 0, and x+ y = 1.


Fubini’s theorem for x-simple regions

To define an x-simple region suppose that ψ1 : [c, d]→ R2 andψ2 : [c, d]→ R2 are given continuous functions over some fixedinterval c ≤ y ≤ d such that

ψ1(y) ≤ ψ2(y), ∀c ≤ y ≤ d.

Definition

The region D ⊂ R2 is called x-simple with left function ψ1(y), rightfunction ψ2(x) and common interval of definition c ≤ y ≤ d, when

D =

(x, y) ∈ R2 | ψ1(y) ≤ x ≤ ψ2(y), c ≤ y ≤ d.

The x-simple region D ⊂ R2 above is bounded on the left by thegraph of ψ1(y) and bounded on the right by the graph of ψ2(y).


Fubini’s theorem for x-simple regions


Suppose that D is any x-simple region as defined above. Then∫D

f(x, y)dxdy =

∫ d

c

∫ ψ2(y)

ψ1(y)

f(x, y)dx

dy.


Regular regions

Regular regions

Regular regions are regions that are both x-simple and y-simple at thesame time. Put another way, a region D is regular when it is given by

D =

(x, y) ∈ R2 | a ≤ x ≤ b, φ1(x) ≤ y ≤ φ2(x)

=

=

(x, y) ∈ R2 | c ≤ y ≤ d, ψ1(y) ≤ x ≤ ψ2(y).

Fubini’s theorem for simple regions

Suppose that D ⊂ R2 is a regular region, that is both y-simple (withφ1(x), φ2(x), and x ∈ [a, b]), as well as x-simple (with ψ1(y), ψ2(y),and y ∈ [c, d]).∫

D

fdA =

∫ b

a

∫ φ2(x)

φ1(x)

f(x, y)dy

dx =

∫ d

c

∫ ψ2(y)

ψ1(y)

f(x, y)dx

dy.


Examples

Example: Show that thetriangular shaped region D ⊂ R2

on the right is a regular region.

Example: Show that the disc ofradius r centred at the point (α, β)in R2 is a regular region.


Some comments

When a region D ⊂ R2 is regular, one of the two iterated integ-rals may be difficult to compute whereas the other integral maybe easy to compute.

If D ⊂ R2 is some region which is either x-simple or y-simple,then taking f to be the function with constant value

f(x, y) = 1, ∀(x, y) ∈ D

we have ∫D

dxdy =

∫D

dA = area of D.


Three Dimensional Integration

Suppose we have a real-valued function

f : Ω→ R,

where Ω ⊂ R3 is a rectangular parallelepiped.

The set Ω can be defined as: Ω = [a, b]× [c, d]× [e, g].


Three Dimensional Integration (cont.)

Let’s subdivide the intervals [a, b], [c, d], and [e, g] into n+ 1equally-spaced points, resulting in xni=0, ynj=0, and znk=0,respectively. Namely,

a = x0 < x1 < . . . < xn = b

c = y0 < y1 < . . . < yn = d

e = z0 < z1 < . . . < zn = g

with spacings ∆x, ∆y, and ∆z given by

∆x =b− an

, ∆y =d− cn

, ∆z =e− gn

.

Let Ωi,j,k be the (small) parallelepiped given by

Ωi,j,k = [xi, xi+1]× [yj , yj+1]× [zk, zk+1],

and also let ri,j,k = (ξi, ηj , ζk) ∈ Ωi,j,k be an arbitrary but fixedpoint.


Riemann integral in R3

Definition

The Riemann sum of f on the parallelepiped Ω is defined as

Sn :=

n∑i=0

n∑j=0

n∑k=0

f(ξi, ηj , ζk)∆x∆y∆z,

for any n = 1, 2, . . .

Definition

If the sequence of Riemann sums Snn=1,2,... converges to a limit Sas n→∞, and the limit S is the same for every choice of points ri,j,k,then S is called the integral of the function f over the parallelepiped Ω.

Some standard notations for the Riemann integral are:∫Ω

f(x, y, z)dxdydz,

∫Ω

f(x, y, z)dV,

∫Ω

fdxdydz,

∫Ω

fdV.


Fubini’s theorem in R3

Define the rectangle D1 in the x− y plane by D1 = [a, b]× [c, d],and define the function

h1(x, y) =

∫ g

e

f(x, y, z)dz, ∀(x, y) ∈ D1.

Similarly, we can define the rectangle D2 in the x− z plane asD2 = [a, b]× [e, g] and define the function

h2(x, z) =

∫ d

c

f(x, y, z)dy, ∀(x, z) ∈ D2.

Finally, we define the rectangle D3 in the y − z plane asD3 = [c, d]× [e, g] and define the function

h3(y, z) =

∫ b

a

f(x, y, z)dx, ∀(y, z) ∈ D3.


Fubini’s theorem in R3

Theorem: Fubini for rectangular parallelepiped in R3

Suppose that f : Ω→ R where Ω is a rectangular parallelepiped andthe functions h1(x, y), h2(x, z) and h3(y, z) as well as their respectiverectangular domains D1, D2 and D3 are defined as above. Then,∫

Ω

fdV =

∫D1

h1(x, y)dxdy =

∫D2

h2(x, z)dxdz =

∫D3

h3(y, z)dydz.

Each of the above three integrals is a two dimensional integral,which can be reduced to iterated integrals over intervals.

In particular, in the case of h1(x, y), we have∫D1

h1dxdy =

∫ b

a

∫ d

c

h1(x, y)dy

dx =

∫ d

c

∫ b

a

h1(x, y)dx

dy.


Integration over non-rectangular domains

First, we fix any rectangular parallelepiped Ξ ∈ R3 which is largeenough to contain Ω, that is Ω ⊂ Ξ.

Define f∗ : Ξ→ R by

f∗(x, y, z) =

f(x, y, z), if (x, y, z) ∈ Ω

0, if (x, y, z) ∈ Ξ\Ω

We then define the integral of f over the region Ω by∫Ω

fdV =

∫Ξ

f∗dV.

Since Ξ is a rectangular parallelepiped, the integral on the rightcan be evaluated using the 3D Fubini Theorem.


Integration over non-rectangular domains (cont.)

First, we formulate a particularly useful type of region Ω ⊂ R3

over which we can evaluate integrals.

To this end from now on we write

R2xy := the x− y plane in R3.

Given a common domain D ⊂ R2xy, let the functions γ1 : D → R

and γ2 : D → R be defined such that

γ1(x, y) ≤ γ2(x, y), ∀(x, y) ∈ D.

Now let Ω ⊂ R3 be the set of all points (x, y, z) ∈ R3 which arebetween the surfaces S1 and S2 given by

S1 : = (x, y, γ1(x, y)) | (x, y) ∈ D ,S2 : = (x, y, γ2(x, y)) | (x, y) ∈ D .


Fubini for z-simple domains in R3

Definition

The region Ω ⊂ R3 is called z-simple with lower function γ1(x, y),upper function γ2(x, y) and common domain of definition D ⊂ R2

xy, ifit can be defined as

Ω =

(x, y, z) ∈ R3 | (x, y) ∈ D and γ1(x, y) ≤ z ≤ γ2(x, y).

Theorem: Fubini for z-simple regions in R3

Suppose that Ω is any z-simple region with lower function γ1(x, y),upper function γ2(x, y) and common domain of definition D ⊂ R2

xy. Iff : Ω→ R is a given function then∫

Ω

fdV =

∫D

∫ γ2(x,y)

γ1(x,y)

f(x, y, z)dz

︸︷︷︸

h1(x,y)

dxdy


Fubini for z-simple domains in R3 (cont.)

The above theorem reduces calculation of the three dimensionalintegral over Ω to calculation of the two dimensional integral overD ⊂ R2

xy.


Further simplifications

If D ⊂ R2xy is y-simple with lower function φ1(x), upper function

φ2(x), and common interval of definition [a, b], then D is given by

D =

(x, y) ∈ R2 | x ∈ [a, b] and φ1(x) ≤ y ≤ φ2(x)

and therefore∫D

h1(x, y)dxdy =

∫ b

a

∫ φ2(x)

φ1(x)

h1(x, y)dy

dx.

In this case, the integral of f over Ω is given by∫Ω

fdV =

∫ b

a

∫ φ2(x)

φ1(x)

[∫ γ2(x,y)

γ1(x,y)

f(x, y, z)dz

]︸︷︷︸

h1(x,y)

dy

dx.

Thus, the integration is reduced to computations of iterated in-tegrals over intervals.


Further simplifications (cont.)

If D ⊂ R2xy is x-simple with left function ψ1(y), right function

ψ2(y), and common interval of definition [c, d], then D is given by

D =

(x, y) ∈ R2 | y ∈ [c, d] and ψ1(y) ≤ x ≤ ψ2(y)

and therefore∫D

h1(x, y)dxdy =

∫ d

c

∫ ψ2(y)

ψ1(y)

h1(x, y)dx

dy.

In this case, the integral of f over Ω is given by∫Ω

fdV =

∫ d

c

∫ ψ2(y)

ψ1(y)

[∫ γ2(x,y)

γ1(x,y)

f(x, y, z)dz

]︸︷︷︸

h1(x,y)

dx

dy.

Note that if D is simple, it can be considered to be either y- orx-simple.


Examples

Example: Integrate the functionf(x, y, z) := y over the region Ωshown on the right.

Example: Find the volume of theregion bounded by the paraboloidz = x2 + y2 and the plane z = 2y.


Fubini for x- and y-simple domains in R3

Suppose that ρ1 and ρ2 are continuous functions defined on acommon region D ⊂ R2

xz such that ρ1(x, z) ≤ ρ2(x, z), for all(x, z) ∈ D.Then, Ω ⊂ R3 is called a y-simple region with lower functionρ1(x, z), upper function ρ2(x, z) and common domain of defi-nition D ⊂ R2

xz when

Ω = (x, y, z) | (x, z) ∈ D & ρ1(x, z) ≤ y ≤ ρ2(x, z)

Analogously, suppose η1 and η2 are continuous functions definedon a common region D ⊂ R2

yz such that η1(y, z) ≤ η2(y, z), for all(y, z) ∈ D.Then, Ω ⊂ R3 is called a x-simple region with lower functionη1(y, z), upper function η2(y, z) and common domain of defi-nition D ⊂ R2

yz when

Ω = (x, y, z) | (y, z) ∈ D & η1(y, z) ≤ x ≤ η2(y, z)


Fubini for general regions in R3

Suppose that Ω ⊂ R3 is a given region and f : Ω→ R is a givenfunction.

If Ω is a z-simple region then:∫Ω

fdV =

∫D

∫ γ2(x,y)

γ1(x,y)

f(x, y, z)dz

dxdy

If Ω is a y-simple region then:∫Ω

fdV =

∫D

∫ ρ2(x,z)

ρ1(x,z)

f(x, y, z)dy

dxdz

If Ω is a x-simple region then:∫Ω

fdV =

∫D

∫ η2(y,z)

η1(y,z)

f(x, y, z)dx

dydz


ECE 206: Advanced Calculus 2ece206/Lectures/slides/section1.pdf · ECE 206: Advanced Calculus 2...

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