ECE 206: Advanced Calculus 2 Department of Electrical and Computer - - PowerPoint PPT Presentation

ECE 206: Advanced Calculus 2

Department of Electrical and Computer Engineering University of Waterloo Fall 2014

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

Course details

Instructor

• Dr. Oleg Michailovich (olegm@uwaterloo.ca, 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 and marks)

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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 graded assignments must be completed individually.

Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 3/39

Recommended course textbook

Michael D. Greenberg, Advanced Engineering 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.

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Course outline

1 Multi-dimensional integration

Two-dimensional integration Three-dimensional integration

2 Scalar and vector fields

Motivating examples Definition of scalar and vector fields

3 Curves and paths in space

Motivating examples Paths and parametric representation of curves Derivatives along a path and tangent to a curve Simple curves and closed curves

4 Line integral and arc length

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

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Course outline (cont.)

5 Conservative vector fields

Gradient of a scalar field Conservative vector fields Conservation of energy

6 Green’s theorem in the plane

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

7 Surfaces, surface areas and surface integrals

Parametric representation of surfaces Tangents to a surface and smooth surfaces Area of a surface Surface integral of a scalar field Surface integral of a vector field

8 Vector calculus

The divergence, Laplacian, and curl differential operators Theorem of Stokes Divergence theorem of Gauss-Ostogradskii The continuity equation

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Course outline (cont.)

9 The basic laws of electricity and magnetism

Static electric fields Static magnetic fields Time-varying fields

10 Maxwell’s equations

The Ampere-Maxwell law for time-varying fields Maxwell’s equations Electromagnetic waves without sources Electromagnetic waves with sources

11 Cylindrical and spherical coordinates

Polar coordinates Cylindrical coordinates Spherical coordinates

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Questions?

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General overview

The course is about vector calculus and the calculus of complex variables. 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 magnetism as 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 be little doubt that the most significant event of the 19th century will be judged as Maxwell’s discovery of the laws

• f electrodynamics”.

Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 9/39

Maxwell’s marvellous equations

For given electric field E, magnetic field B, charge density ρ, and current density field J, Maxwell’s equations ∇ · E = ρ ǫ0 ∇ · B = 0 ∇ × E + ∂B ∂t = 0 ∇ × B =µ0J + ǫ0µ0 ∂E ∂t So what are the symbols ∇· and ∇× standing for?

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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].

Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 11/39

Finite partition

Subdivide the intervals [a, b] and [c, d] into n + 1 equally spaced points {xi}n

i=0 and {yj}n j=0, respectively, such that

a = x0 < x1 < . . . < xn = b, c = y0 < y1 < . . . < yn = d, with spacings ∆x := xi+1 − xi = b − a n , ∆y := yj+1 − yj = d − c n . 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

Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 12/39

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, . . .}

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Riemann integral

Definition If the sequence of Riemann sums {Sn, n = 1, 2, ...} converges to a limit S as n → ∞, and the limit S is the same for every choice of points (ξi, ηj) ∈ Di,j, then S is called the (Riemann) integral of the function 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 is said to be undefined. Fortunately, we need never be concerned with this fact, for the class of functions which can be integrated over D is simply huge.

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Fubini’s theorem

If we fix some x ∈ [a, b], a function f(x, y) depends only on y in the 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.

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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 that

• ne choice involves less work than the other.

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

Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 16/39

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] → R are given continuous functions over some fixed interval a ≤ x ≤ b such that φ1(x) ≤ φ2(x), ∀a ≤ x ≤ y. Definition The region D ⊂ R2 is called y-simple with lower function φ1(x), upper function φ2(x) and common interval of definition a ≤ x ≤ b, when D =

• (x, y) ∈ R2 | a ≤ x ≤ b,

φ1(x) ≤ y ≤ φ2(x)

• .

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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].

Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 18/39

Non-rectangular domains (cont.)

Now define the function f ∗ : E → R2 as given by f ∗(x, y) =

• f(x, y),

if (x, y) ∈ D 0, 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.

Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 19/39

Fubini’s theorem for y-simple regions

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 + cos x, and x ∈ [0, 2π], find the integral of f(x, y) = 2y. Example: Evaluate the integral

• D x2y dxdy, where D is a triangular

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

Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 20/39

Fubini’s theorem for x-simple regions

To define an x-simple region suppose that ψ1 : [c, d] → R2 and ψ2 : [c, d] → R2 are given continuous functions over some fixed interval 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), right function 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 the graph of ψ1(y) and bounded on the right by the graph of ψ2(y).

Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 21/39

Fubini’s theorem for x-simple regions

Fubini’s theorem for y-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.

Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 22/39

Regular regions

Regular regions Regular regions are regions that are both x-simple and y-simple at the same 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 (with φ1(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.

Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 23/39

Examples

Example: Show that the triangular shaped region D ⊂ R2

• n the right is a regular region.

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

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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 may be 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.

Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 25/39

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].

Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 26/39

Three Dimensional Integration (cont.)

Let’s subdivide the intervals [a, b], [c, d], and [e, g] into n + 1 equally-spaced points, resulting in {x}n

i=0, {y}n j=0, and {z}n k=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 − a n , ∆y = d − c n , ∆z = e − g n . 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 fixed point.

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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 {Sn}n=1,2,... converges to a limit S as 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.

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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 as D2 = [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 as D3 = [c, d] × [e, g] and define the function h3(y, z) = b

a

f(x, y, z)dx, ∀(y, z) ∈ D3.

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Fubini’s theorem in R3

Theorem: Fubini for rectangular parallelepiped in R3 Suppose that f : Ω → R where Ω is a rectangular parallelepiped and the functions h1(x, y), h2(x, z) and h3(y, z) as well as their respective rectangular 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.

Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 30/39

Integration over non-rectangular domains

First, we fix any rectangular parallelepiped Ξ ∈ R3 which is large enough 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 right can be evaluated using the 3D Fubini Theorem.

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Integration over non-rectangular domains (cont.)

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

• ver which we can evaluate integrals.

To this end from now on we write R2

xy := the x − y plane in R3.

Given a common domain D ⊂ R2

xy, 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 are between 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} .

Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 32/39

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, if

it 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. If

f : 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

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

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

xy.

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Further simplifications

If D ⊂ R2

xy 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.

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Further simplifications (cont.)

If D ⊂ R2

xy 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- or x-simple.

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Examples

Example: Integrate the function f(x, y, z) := y over the region Ω shown on the right. Example: Find the volume of the region bounded by the paraboloid z = x2 + y2 and the plane z = 2y.

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Fubini for x- and y-simple domains in R3

Suppose that ρ1 and ρ2 are continuous functions defined on a common 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 function ρ1(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 defined

• n 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 function η1(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)}

Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 38/39

Fubini for general regions in R3

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

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