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Applied Advanced CalculusENGR233 - Concordia UniversityAlternate Lecture NotesAlexandre Paradis 2019iiiv CONTENTS 5 Conservative Field 5 Line Integrals 5 Surfaces and Surface Integrals 6 Vector Calculus 6 Gradient, Divergence, Curl and Properties 6 Divergence Theorem 6 Green’s and Stokes’s Theorem 6...

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Applied Advanced Calculus ENGR233 - Concordia University Alternate Lecture Notes Alexandre Paradis 2019

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Contents 1 Coordinate Geometry and Vectors 1.1 Basic Concepts: Geometry in 3D 1.2 Vectors in 3-space . . . . . . . . . 1.2.1 Dot Product . . . . . . . . 1.2.2 Cross Product . . . . . . . 1.3 Lines and planes in R3 . . . . . . 1.3.1 distances(no proofs) . . .

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2 Partial Differential and Applications 2.1 Function of Several Variables . . . . 2.2 Definitions and Theorem (no-proof) . 2.3 Partial Derivative . . . . . . . . . . . 2.4 Applications . . . . . . . . . . . . . .

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3 Multiple Integrals 3.1 Double integrals . . . . . . . . . . . . . 3.2 Polar Coordinates and Double Integrals 3.3 Change of Variables . . . . . . . . . . . 3.4 Triple Integrals . . . . . . . . . . . . . 3.4.1 Change of variables . . . . . . .

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4 Curves in 3-Space 71 4.1 Vector Function of one Variable . . . . . . . . . . . . . . . . . 71 4.2 Curves and Parametrization . . . . . . . . . . . . . . . . . . . 76 4.3 Curvature, Torsion and Frenet frame . . . . . . . . . . . . . . 79 5 Vector Fields 85 5.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 iii

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CONTENTS 5.2 Conservative Field . . . . . . . . . . . . . . . . . . . . . . . . 87 5.3 Line Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.4 Surfaces and Surface Integrals . . . . . . . . . . . . . . . . . . 97

6 Vector Calculus 6.1 Gradient, Divergence, Curl and Properties 6.2 Divergence Theorem . . . . . . . . . . . . 6.3 Green’s and Stokes’s Theorem . . . . . . . 6.4 Physical Applications . . . . . . . . . . . . 6.4.1 Fluid Mechanics . . . . . . . . . . . 6.4.2 Electromagnetic Fields . . . . . . . 6.5 Cylindrical and Spherical system . . . . . 6.5.1 Cylindrical . . . . . . . . . . . . . . 6.5.2 Spherical . . . . . . . . . . . . . . .

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105 . 105 . 110 . 114 . 117 . 118 . 119 . 120 . 121 . 121

Introduction Those notes are Alternate Lecture Notes as the subject is not presented in the same order as what I present in the actual course, mainly due to a change in the course approach. This document is a summary of the lecture notes I gave in class to my student in the course ENGR 233, Advanced Calculus, at Concordia University. The notes are greatly inspired by the ones I wrote down as a student, more than 20 years ago, which were based on the book Calculus of Several Variables, by Robert A. Adams. However, I have focus on the application of the material rather than the theory. In my personal point of view, engineering student should understand the mathematics they use, they should know the meaning of the equation they are using, but, more importantly, they should be able to apply the technique in a practical manner. Some suggested problems will eventually be provided in this document. They will be meant to help in the understanding of the concept presented, but might not be enough for a deep understanding. The reader is invited to consult other advanced calculus book if need be. For the moment, for the purpose of the ENGR233 course at Concordia University, all problems related to the material presented in this document will be found in the book Advanced Engineering Mathematics, by Zill, sixth edition, in chapter 7 and 9.

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CONTENTS

Chapter 1 Coordinate Geometry and Vectors This section is the starting point of the course. It focus on the idea of 3space and on the different tools that exist to play in that world, mainly the introduction to vectors, vector products and general lines and planes in three-space. Some of the idea presented in this section will be presented in a different form in later chapter.

1.1

Basic Concepts: Geometry in 3D

To build a 3-space, and to be able to work in 3-space, one needs a set of 3 axis, perpendicular to each other. This ensure a set of independent axis. In rectangular space, it is custom to use the x, y and z axis. There is no 1

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CHAPTER 1. COORDINATE GEOMETRY AND VECTORS

preferred way to place them, other than they have to be right-hand oriented. If one place his finger on the x axis and rotate them towards the y axis, the thumb should be along the z axis (see drawing).

The point of junction of the three axis is called the origin of the space and it has coordinate: (0,0,0). Note that the coordinate are always written as (x,y,z).

Figure 1.1: Right-handed system

The distance between two points in three space, noted s, is used to determine and set the geometry (metric) in which you are working. In most engineering applications, for everyday use, the metric to use is the Euclidian space. Another notable metric is the Reimann space which can be used in general relativity (in application, consider the principle behind the GPS

1.2. VECTORS IN 3-SPACE

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system). Hence, for the Euclidian space:

s2 “ px2 ´ x1 q2 ` py2 ´ y1 q2 ` pz2 ´ z1 q2 In order to work in the 3-space rectangular coordinate environment, we can divide the space into 8 octant. The most common one used is the first one, made of the region:

0 ď x, 0 ď y, 0 ď z When dealing with theorems, 3-space is often called R3 . Of course, Rn exist. In this case, a plane in such space is called an hyperplane. It will not be discussed more in this course.

1.2

Vectors in 3-space

Definition 1. A vector is a mathematical object represented by a magnitude and a direction. In engineering, it will also have a unit. Definition 2. A scalar is a mathematical object represented only by a magnitude. In engineering, it will also have a unit. A vector in 3 dimension is the linear combinaison of three vectors, one along each of the axis. Thus, one should use the standard unit vectors to represent a component of a vector along a particular axis. Remember: a unit

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CHAPTER 1. COORDINATE GEOMETRY AND VECTORS

vector is a vector of magnitude 1. It serves only to indicate the direction. The three unit vectors: x Ñ ~i, y Ñ ~j, z Ñ ~k

Figure 1.2: Unit vector in rectangular system

Therefore, a complete vector in 3-space can be written as: ~r “ x~i ` y~j ` zk~ or, sometimes,xx, y, z y, but this notation is not the most explicit one. It is useful to avoid the use of the vector symbol. The magnitude of a vector is a quantity that is very useful to calculate. It is written simply as r, or: }~r } “

a

x2 ` y 2 ` z 2

The value of the magnitude of a vector can be seen as the distance, in straight line, between the beginning and the end of the arrow of the vector.

1.2. VECTORS IN 3-SPACE

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In order to describe vectors in 3-space, one should be able to find them. The main method is when one has 2 points and one has to find a vector from one point to the other. Let use define two points and find the vector from point P1 to point P2 . P1 “ x1~i ` y1~j ` z1~k and P2 “ x2~i ` y2~j ` z2~k P1~P2 “ px2 ´ x1 q~i ` py2 ´ y1 q~j ` pz2 ´ z1 q~k

A very useful concept to understand is how to find a unit vector from an arbitrary vector in space. This will be useful to indicate an orientation in space. Normally, such general unit vector is represented using the Greek letter λ.

~r λ~r “ }~r } Example 3. Find a vector, call it ~v , going from (3,2,1) to (5,6,4) and find the direction of the vector. Solution: let us apply the formula seen before: ~ “ 2~i ` 4~j ` 3~k ~v “ p5 ´ 3q~i ` p6 ´ 2q~j ` p4 ´ 1qk

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CHAPTER 1. COORDINATE GEOMETRY AND VECTORS

and 2~i ` 4~j ` 3~k 2~i ` 4~j ` 3~k ? “ λ~v “ ? 4 ` 16 ` 9 29 Now, we have an idea of what is a vector, we have all the same concept in our minds. The question now is how to do products with vector. You may have seen before the concept of dot product, and cross product. Let us review those concepts as they are key concept using vectors and fundamental for geometry purposes.

1.2.1

Dot Product

The dot product is a product between two vectors, with a scalar as a result. This scalar is found from the multiplication of the magnitude of each vector with the cosine of the angle between the two vectors. So, using two vectors ~u and ~v : ~u ¨ ~v “ }~u}}~v }cosθ “ ux vx ` uy vy ` uz vz

Figure 1.3: dot product projection idea A few properties are nice to know about the dot product, provided that u, v, and w are vectors and t is a scalar: 1. ~u ¨ ~v “ ~v ¨ ~u

1.2. VECTORS IN 3-SPACE

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2. ~u ¨ p~v ` w ~ q “ ~u ¨ ~v ` ~u ¨ w ~ 3. t~u ¨ ~v “ tp~v ¨ ~uq 4. ~u ¨ ~u “ }~u} 5. ~u ¨ ~v “ 0 Ñ ~u K ~v The use of dot product is to do projection of a vector into a particular direction. In terms of engineering applications, it is used to find a quantity that is only function in a direction. For example, the physical quantity called the Work. The work is defined as the force along a path, in the direction of motion. Think of the most efficient way to open a door. You push in the direction of the displacement. Perpendicular to the path, it is useless. If you push toward the hinges of the door, it will not move.

W “

ż

~ ¨ d~s F

Let us now see two geometric concept related to the dot product. 1. Saclar Projection: The scalar projection s of any vector ~u in the direction of another vector ~v is the dot product of ~u with the unit vector of ~v , provided that v is not zero. s“

~u ¨ ~v “ }~u}cosθ }~v }

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CHAPTER 1. COORDINATE GEOMETRY AND VECTORS It is the magnitude of a vector in a prescribed direction. It is also presented as s “ ~u ¨ λ~v 2. Vector Projection: The vector projection u~v of ~u in the direction of ~v is the multiplication of the scalar projection of ~u on ~v by the unit vector of ~v .

u~v “

p~u ¨ ~v q~v }~v }2

It is a vector in the direction of ~v , having the magnitude of the scalar projection.

1.2.2

Cross Product

For any vector ~u and ~v in R3 , the cross product of those vectors, noted ~u ˆ ~v is the unique vector w ~ satisfying the three conditions: 1. w ~ K ~v and w ~ K ~u 2. }~u ˆ ~v } “ }~u}}~v }sinpθq 3. ~u , ~v and ~u ˆ ~v form a right hand triad. In other words, the vector resulting form the cross product is perpendicular to the plane made by the two original vectors. In terms of components, one can calculate the cross products of two vectors, in three-space, as follow: w ~ “ ~u ˆ ~v “ puy vz ´ uz vy q~i ` puz vx ´ ux vz q~j ` pux vy ´ uy vx qk~

1.2. VECTORS IN 3-SPACE

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However, this formula is not fun to memorize and can lead to some errors if not applied properly. Two other method can be used to find the result of a cross product in rectangular coordinate: 1. for small vectors: ijkij— to the right, negative, to the left, positive.

Figure 1.4: cross product trick

2. matrix form:

ˇ ˇ ˇ ˇ ~k ˇ ˇ ~i ~j ˇ ˇ ˇ ˇ ~u ˆ ~v “ ˇˇ ux uy uz ˇˇ ˇ ˇ ˇ ˇ ˇ vx vy vz ˇ

The value of the determinant will give the result of the cross product. Here are a few properties for the cross product, again, u,v, and w are vectors, t is a scalar: 1. ~u ˆ ~u “ 0 2. ~u ˆ ~v “ ´~v ˆ ~u 3. p~u ` ~v q ˆ w ~ “ ~u ˆ w ~ ` ~v ˆ w ~ 4. t~u ˆ ~v “ tp~v ˆ ~uq

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CHAPTER 1. COORDINATE GEOMETRY AND VECTORS 5. ~u ¨ p~u ˆ ~v q “ ~v ¨ p~u ˆ ~v q “ 0

Example 4. Do the cross product of p2~i ´ ~j ` 2~kq and p´~i ´ 2~j ` ~kq. Solution ˇ ˇ ˇ ~i ˇ ˇ ˇ 2 ˇ ˇ ˇ ˇ ´1

ˇ ˇ ~k ˇ ˇ ˇ ´1 2 ˇˇ “ 3~i ´ 4~j ´ 5~k ˇ ˇ ´2 1 ˇ ~j

In terms of applications, we will use the cross product to calculate some physical quantity involving a particular direction. For example, the velocity when working in rotation, the angular momentum, or the moment of a force (torque). Other topics will be discussed in class.

A final note

Let ~u, ~v , and w ~ be three vector in 3-space. The volume of the parallelepiped made by the three vector is:

~u ¨ p~v ˆ w ~ q “ ~v ¨ pw ~ ˆ ~uq “ w ~ ¨ p~u ˆ ~v q

1.3. LINES AND PLANES IN R3

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Figure 1.5: volume representation

1.3

Lines and planes in R3

Let us start with the concept of planes. We will represent a position in 3-space using a vector, called a position vector. r~0 “ x0~i ` y0~j ` z0~k Now, consider this point part of a plane, which is a 2 dimensions subspace in 3-space. Let us called a new vector, ~n “ A~i ` B~j ` C~k, the normal vector. If that vector is not zero, there exist exactly one plane (or flat surface) passing through the point identified by r~0 and perpendicular to the normal vector. One can use the properties of the dot product to create the equation of the plane using the position vector and the normal vector.

~n ¨ p~r ´ r~0 q “ 0 with ~r “ x~i ` y~j ` z~k.

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CHAPTER 1. COORDINATE GEOMETRY AND VECTORS

Figure 1.6: plane in 3-space In an expanded manner, remembering that A, B, C, x0 , y0 and z0 are numerical values:

Apx ´ x0 q ` Bpy ´ y0 q ` Cpz ´ z0 q “ 0 Example 5. Find an equation of the plane that passes through the three points P=(1,1,0), Q = (0,2,1) and R = (3,2,-1). solution: the first thing to do is to find the normal vector. To do so, let us define two vector on the plane, P~Q and P~R: P~Q “ ´~i ` ~j ` ~k and P~R “ 2~i ` ~j ´ ~k Now, the normal vector is found using the cross product: ˇ ˇ ˇ ~i j~ ~k ˇ ˇ ~n “ P~Q ˆ P~R “ ˇˇ ´1 1 1 “ ´2~i ` ~j ´ 3~k ˇ ˇ ˇ 2 1 ´1 ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

1.3. LINES AND PLANES IN R3

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Applying the previous formula: ~n ¨ p~r ´ r~0 q “ p´2~i ` ~j ´ 3~k q ¨ ppx ´ 1q~i ` py ´ 1q~j ` zk~ q “ 0 ´2x ` y ´ 3z ` 1 “ 0 Let us now focus a little more on lines in R3 . The first thing to mention is that we will use parametric equations. This imply a parameter, we will call it t (a real number), and the variables will be function of that parameter: x(t), y(t) and z(t). Let r~0 be the position vector of a point P0 and let ~v “ a~i ` b~j ` ck~ a non zero vector. The unique line passing by P0 and parallel to the vector ~v can be found by identifying the position vector of P (~r) as ~r “ r~0 ` t~v .

Figure 1.7: line in 3-space

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CHAPTER 1. COORDINATE GEOMETRY AND VECTORS In scalar form (parametric form):

x “ x0 ` at y “ y0 ` bt z “ z0 ` ct The standard form, for a symmetric equation (a, b, c non zero):

t“

z ´ z0 x ´ x0 y ´ y0 “ “ b a c

Example 6. Find the equation of the line passing by the points (1,2,3) and (5,4,3). Solution let P0 “ p1, 2, 3q and P “ p5, 4, 3q. We have: r~0 “ ~i ` 2~j ` 3~k and~r “ 5~i ` 4~j ` 3~k so: ~v “ ~r ´ r~0 “ 4~i ` 2~j~r “ p1 ` 4tq~i ` p2 ` 2tq~j ` 3~k or x “ 1 ` 4t y “ 2 ` 2t

1.3. LINES AND PLANES IN R3

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1.3.1

distances(no proofs)

This small subsection is just a list of nice formulas for minimal distance, perpendicular to lines.

Point to plane: plane: Ax ` By ` Cz “ D s“

|Ax0 ` By0 ` Cz0 ´ D| ? A2 ` B 2 ` C 2

Point to a line:

s“

}p~ r0 ´ r~1 q ˆ ~v } }~v }

Two lines:

s“

|p~ r2 ´ r~1 q ¨ pv~1 ˆ v~2 q| }v~1 ˆ v~2 }

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CHAPTER 1. COORDINATE GEOMETRY AND VECTORS

Chapter 2 Partial Differential and Applications The content of this chapter is of fundamental importance in any applied science. In engineering, it is of vital importance, mainly in thermodynamics, fluid mechanics, turbomachinery and waves applications. A quick survey of function of several variables will be made, then the introduction to partial derivative will be made.Some applications will be presented also.

2.1

Function of Several Variables

Definition 7. A function f of n real variables is a rule that assigns a unique real number f px1 , x2 , ..., xn q to each point in some subset D(f ) of Rn . D(f) is called the domain of f and the set of real numbers f px1 , x2 , ..., xn q obtained 17

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CHAPTER 2. PARTIAL DIFFERENTIAL AND APPLICATIONS

from points in the domain is called the range of f. In terms of many applications, it is convenient to work with z “ f px, y q or f px, y, z, tq “ 0. In three dimension, the function f(x,y) form a surface above the xy plane. In Rn , we speak more of a hypersurface.

To help visualizing the surface in 3-space, a good technique is to draw some level curve. Basically, it means drawing on a 2-D plane some ”cut” of the surface by doing f px, yq “ C, C being a constant. It is the technique used to draw topographic maps. Each line is a different altitude. Figure 1 and 2 below show the function f px, yq “ x2 ´ y 2 as it is in 3-space and the contour plot (or level curves). The graph were generated by the software Wolfram Alpha, the online version.

Figure 2.1: function in 3-space

2.2. DEFINITIONS AND THEOREM (NO-PROOF)

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Figure 2.2: Level/contour curves The same can be applied to 4-space by building 3 dimension hypersurface. However, this is slightly more complex and is more easily done with the help of a computer.

2.2

Definitions and Theorem (no-proof )

This section can be skipped in an introductory course. It is there only to provide some reference if one is more interested in the mathematics behind the applications. A set of definitions will be presented in order to logically accept the next section. Definition 8. If P0 “ px0 , y0 q is a point in the plane and if r ą 0, then the set of points P “ px, yq in the plane whose distance from P0 are less than r is called an open disk of radius r about P0 , or a neighbourhood of P0 .

Nr pP0 q “ tpx, yq : px ´ x0 q2 ` py ´ y0 q2 ă r 2 u

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CHAPTER 2. PARTIAL DIFFERENTIAL AND APPLICATIONS

Definition 9. Let S be a set of points in the plane:

• P0 is a boundary point of S if every neighbourhood of P0 contains at least one point in S and at least one point not in S. The set of all boundary of S is called the Boundary of S.

• S is closed (a closed set) if every boundary point of S belong to S (part of the set).

• S is called an open set if no boundary point of S are in the set S.

• The interior of S is the set of all points of S not par...