SF1686 Calculus in Several Variables - Lecture 1

Introduction Vector and Coordinate Geometry in 3-Space

SF1686 Calculus in Several VariablesLecture 1

Sunghan Kim

Department of Mathematics, KTH

Fall 2020, Period 1

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SF1686 Calculus in Several Variables

Welcome to the course!

Lecturer: Sunghan Kim,

Examiners: Henrik Shahgholian/ Hans ThunbergTAs: Gerard Farré Puiggalí / Anton Ottoson

Website All information is on Canvas:https://kth.instructure.com/courses/20311

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What is the course about?

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What is the course about

Differentiation and Taylor’s expansionModule 1–3

Integration and Stokes’ theoremModule 4–6

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Differentiation and Taylor’s expansion

Vector functions of several variables

Limit values, continuity, derivatives, etc.

Parametrisation, different coordinate systems, applications

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Integration and Stokes’ theoremLine-, surface-, volume integrals

Green’s theorem, Divergence theorem, Stokes’ theorem

Applications: physics, engineering, industries, etc.

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Prerequisites

Calculus in One Variable

Linear Algebra

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Suggestions

Before the lecture: Read the lecture note.Feel free to supplement with other videos.

During the lecture: Review and active participation.

After the lecture: Work with exercises in the book.Utilise the exercise class & TAs.

Active use of:http://demonstrations.wolfram.com

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http://demonstrations.wolfram.com


Rules

Since we are on Zoom..

Each class: 2 parts

Each part: 45 mins (lecture) - 15 mins (Q&A break)

Questions are only allowed during Q&A break

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Today’s Lecture

Analytic geometry in Three Dimensions

Cylindrical and Spherical Coordinates

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Analytic Geometry in Three Dimensions

Cartesian Coordinates

Three dimensional space (or 3-space, R3, etc.):

The Cartesian coordinate system is the standard system toanalyse (or represent) the given geometry:

x-, y -, z-axis: orthogonal system;

P = (x , y , z)

The origin isO = (0,0,0)

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Distance, d , between two points

P = (x , y , z), P′ = (x ′, y ′, z′) :

d =

√(x − x ′)2 + (y − y ′)2 + (z − z′)2.

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Draw the following sets in R3:

1) Octant: x ≥ 0, y ≥ 0, z ≥ 0

2) Plane: y = 2x − 1

3) Cylinder: x2 + y2 = 1

4) Sphere: x2 + y2 + z2 = 1

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The Euclidean n-Space, n ≥ 2

The n-dimensional space (or Rn, n-space, etc.)

P = (x1, x2, · · · , xn)

Origin:O = (0,0, · · · ,0)

Distance: √(x1 − y1)2 + (x2 − y2)2 + · · ·+ (xn − yn)2

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Euclidean n-Space

Non-Euclidean space?!1

Figur: Non-Euclidean

1https://www.euclideanspace.com/maths/geometry/space/nonEuclid/index.htm

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https://www.euclideanspace.com/maths/geometry/space/nonEuclid/index.htm

https://www.euclideanspace.com/maths/geometry/space/nonEuclid/index.htm



Topological concepts

Let a ∈ Rn.

A neighbourhood of a is a set of the form

Br (a) = {x ∈ Rn : |x − a| < r }.

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Let M ⊂ Rn and a ∈ Rn.We call a an interior point of M, if

Br (a) ⊂ M , for some r > 0.

We call a an exterior point of M, if

Br (a) ⊂ Mc , for some r > 0.

We call a a boundary point of M, if

Br (a) ∩M , ∅, Br (a) ∩Mc , ∅, for all r > 0.

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M is said to be open, if every point of M is an interior point.

M is said to be closed if Mc is an open set (so all theboundary points belong to M).

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Determine whether the following sets in R2 are open or closed.Specify their boundaries.

x2 + y2 < 1

y = x2

y > x2, y < x + 1

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A set can be neither open nor closed.y > x2, y ≤ x + 1Proof) Points on y = x + 1, y > x2 are boundary points, sothe set is not open.Points on y = x2, y ≤ x + 1 are boundary points, which arenot included, so the set is not closed either.

It can also be open and closed simultaneously.M = R2 or ∅.

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Cylindrical and SphericalCoordinates

Cylindrical coordinates

Spherical coordinates

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Cylindrical coordinates, (r , θ, z)

One way to represent a point (x , y , z) in the space is

x = r cosθ,y = r sinθ,z = z

Figur: Cylindrical coordinates 22 / 28



Cylindrical coordinates, (r , θ, z)

Here

r =

√x2 + y2,

0 ≤ θ ≤ 2π,z = z

Figur: Cylindrical coordinates 23 / 28



Cylindrical coordinates, examples

Draw the following areas and represent them in the cylindricalcoordinates:

1) Cylinder: x2 + y2≤ 1, x , y , z ≥ 0

2) Cone: x2 + y2 = z2, z ≥ 0

3) Sphere: x2 + y2 + z2 = 1, −y ≤ x ≤ y , z ≥ 0

4) Paraboloid:x2

4+

y2

9= z

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Spherical coordinates, (R, φ, θ)

Another way is to consider

x = R sinφ cosθ,

y = R sinφ sinθ,

z = R cosφ

Figur: Spherical coordinates 25 / 28



Spherical coordinates, (R, φ, θ)

Here,

R =

√x2 + y2 + z2,

0 ≤ φ ≤ π,0 ≤ θ ≤ 2π

Figur: Spherical coordinates26 / 28



Spherical coordinates, examples

Draw the following areas and represent them in the sphericalcoordinates:

1) Sphere: x2 + y2 + z2 = 1, −y ≤ x ≤ y , z ≥ 0

2) Ellipsoid:x2

4+

y2

9+ z2 = 1

3) Cone: x2 + y2 = z2, z ≥ 0

4) (Difficult) Cylinder: x2 + y2 = 1.

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Review

What we have learned today

Cartesian coordinates

Topological concepts (open, closed, boundary, etc.)

Cylindrical / spherical coordinates

Solve suggested exercises (see CANVAS - syllabus)!

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SF1686 Calculus in Several Variables - Lecture 1

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