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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Introduction Vector and Coordinate Geometry in 3-Space
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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Introduction Vector and Coordinate Geometry in 3-Space
What is the course about
Differentiation and Taylor’s expansionModule 1–3
Integration and Stokes’ theoremModule 4–6
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Introduction Vector and Coordinate Geometry in 3-Space
What is the course about
Differentiation and Taylor’s expansion
Vector functions of several variables
Limit values, continuity, derivatives, etc.
Parametrisation, different coordinate systems, applications
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Introduction Vector and Coordinate Geometry in 3-Space
What is the course about
Integration and Stokes’ theoremLine-, surface-, volume integrals
Green’s theorem, Divergence theorem, Stokes’ theorem
Applications: physics, engineering, industries, etc.
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Introduction Vector and Coordinate Geometry in 3-Space
Prerequisites
Calculus in One Variable
Linear Algebra
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Introduction Vector and Coordinate Geometry in 3-Space
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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Introduction Vector and Coordinate Geometry in 3-Space
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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Introduction Vector and Coordinate Geometry in 3-Space
Today’s Lecture
Analytic geometry in Three Dimensions
Cylindrical and Spherical Coordinates
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Introduction Vector and Coordinate Geometry in 3-Space
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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Introduction Vector and Coordinate Geometry in 3-Space
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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Introduction Vector and Coordinate Geometry in 3-Space
Analytic Geometry in Three Dimensions
Cartesian Coordinates
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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Introduction Vector and Coordinate Geometry in 3-Space
Analytic Geometry in Three Dimensions
Cartesian Coordinates
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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Introduction Vector and Coordinate Geometry in 3-Space
Analytic Geometry in Three Dimensions
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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Introduction Vector and Coordinate Geometry in 3-Space
Analytic Geometry in Three Dimensions
Euclidean n-Space
Non-Euclidean space?!1
Figur: Non-Euclidean
1https://www.euclideanspace.com/maths/geometry/space/nonEuclid/index.htm
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Introduction Vector and Coordinate Geometry in 3-Space
Analytic Geometry in Three Dimensions
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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Introduction Vector and Coordinate Geometry in 3-Space
Analytic Geometry in Three Dimensions
Topological concepts
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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Introduction Vector and Coordinate Geometry in 3-Space
Analytic Geometry in Three Dimensions
Topological concepts
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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Introduction Vector and Coordinate Geometry in 3-Space
Analytic Geometry in Three Dimensions
Topological concepts
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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Introduction Vector and Coordinate Geometry in 3-Space
Analytic Geometry in Three Dimensions
Topological concepts
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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Introduction Vector and Coordinate Geometry in 3-Space
Analytic Geometry in Three Dimensions
Cylindrical and SphericalCoordinates
Cylindrical coordinates
Spherical coordinates
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Introduction Vector and Coordinate Geometry in 3-Space
Analytic Geometry in Three Dimensions
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
Introduction Vector and Coordinate Geometry in 3-Space
Analytic Geometry in Three Dimensions
Cylindrical coordinates, (r , θ, z)
Here
r =
√x2 + y2,
0 ≤ θ ≤ 2π,z = z
Figur: Cylindrical coordinates 23 / 28
Introduction Vector and Coordinate Geometry in 3-Space
Analytic Geometry in Three Dimensions
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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Introduction Vector and Coordinate Geometry in 3-Space
Analytic Geometry in Three Dimensions
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
Introduction Vector and Coordinate Geometry in 3-Space
Analytic Geometry in Three Dimensions
Spherical coordinates, (R, φ, θ)
Here,
R =
√x2 + y2 + z2,
0 ≤ φ ≤ π,0 ≤ θ ≤ 2π
Figur: Spherical coordinates26 / 28
Introduction Vector and Coordinate Geometry in 3-Space
Analytic Geometry in Three Dimensions
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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Introduction Vector and Coordinate Geometry in 3-Space
Analytic Geometry in Three Dimensions
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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Introduction Vector and Coordinate Geometry in 3-Space
Analytic Geometry in Three Dimensions
Review
What we have learned today
Cartesian coordinates
Topological concepts (open, closed, boundary, etc.)
Cylindrical / spherical coordinates
Solve suggested exercises (see CANVAS - syllabus)!
28 / 28
Introduction Vector and Coordinate Geometry in 3-Space
Analytic Geometry in Three Dimensions
Review
What we have learned today
Cartesian coordinates
Topological concepts (open, closed, boundary, etc.)
Cylindrical / spherical coordinates
Solve suggested exercises (see CANVAS - syllabus)!
28 / 28
Introduction Vector and Coordinate Geometry in 3-Space
Analytic Geometry in Three Dimensions
Review
What we have learned today
Cartesian coordinates
Topological concepts (open, closed, boundary, etc.)
Cylindrical / spherical coordinates
Solve suggested exercises (see CANVAS - syllabus)!
28 / 28
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