Mth 234 Multivariable Calculus
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Cartesian coordinates in space (12.1) • Overview of vector calculus. • Cartesian coordinates in space. • Right-handed, left-handed Cartesian coordinates. • Distance formula between two points in space. • Equation of a sphere.
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Mth 234 Multivariable Calculus
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Overview of Multivariable calculus Mth 132 Calculus I: f : R → R, f (x), differential calculus. Mth 133 Calculus II: f : R → R, f (x), integral calculus. Mth 234 Multivariable Calculus: f : R2 → R, 3
f : R → R, 3
r:R→R ,
f (x, y)
)
f (x, y, z)
r(t) = hx(t), y(t), z(t)i
scalar-valued. ª
vector-valued.
We study how to differentiate and integrate such functions.
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Mth 234 Multivariable Calculus
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Cartesian coordinates in space (12.1) • Overview of vector calculus. • Cartesian coordinates in space. • Right-handed, left-handed Cartesian coordinates. • Distance formula between two points in space. • Equation of a sphere.
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Mth 234 Multivariable Calculus
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Review: Cartesian coordinates on the plane (R2 ) Every point on a plane is labeled by an ordered pair (x, y). z
y
z0
(x0,y0,z 0)
y
0
(x0,y0) z0 y
0
x0
x
x
0
x
x0
y
y0
Cartesian coordinates in space (R3 ) Every point in space is labeled by an ordered triple (x, y, z). &
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Example: Find the set S = {x > 0, y > 0, z = 0} ⊂ R3 . Solution: z
y>0
z=0 y x>0 x
S
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Example: Find the set S ⊂ R3 given by S = {0 6 x 6 1, − 1 6 y 6 2, z = 1}. Solution:
z S −1
2
y
1 x C &
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Mth 234 Multivariable Calculus
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Cartesian coordinates in space (12.1) • Overview of vector calculus. • Cartesian coordinates in space. • Right-handed, left-handed Cartesian coordinates. • Distance formula between two points in space. • Equation of a sphere.
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Mth 234 Multivariable Calculus
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There are two types of Cartesian coordinate systems except by rotations: Right-handed (RH) and Left-handed (LH)
z
z (x0,y0,z 0)
(x0,y0,z 0)
z0
z0 x0
x
y0
Right Handed
y
y0 y
x0
x
Left Handed
No rotation transforms one into the other
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Mth 234 Multivariable Calculus
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This coordinate system is right handed y
z
z y x
x
z
y x
This coordinate system is left handed z
z
z
y
x x
y x
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Mth 234 Multivariable Calculus
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Remark: The same classification occurs in Cartesian coordinates on the plane. y
y
Right Handed
x
x Left Handed
• In R3 we will define the cross product of vectors. • This product has different results in RH or LH Cartesian coordinates. • There is no cross product in R2 . In class we use RH Cartesian coordinate systems &
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Mth 234 Multivariable Calculus
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Cartesian coordinates in space (12.1) • Overview of vector calculus. • Cartesian coordinates in space. • Right-handed, left-handed Cartesian coordinates. • Distance formula between two points in space. • Equation of a sphere.
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Mth 234 Multivariable Calculus
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Distance formula between two points in space ¯ ¯ ¯ Theorem 1 The distance P1 P2 ¯ between the points P1 = (x1 , y1 , z1 ) and P2 = (x2 , y2 , z2 ) is given by ¯ ¯ p ¯P1 P2 ¯ = (x2 − x1 )2 + (y2 − y1 )2 + (z2 − z1 )2 . The distance between points in space is crucial to define the idea of limit to functions in space
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Proof: Pythagoras Theorem. P2
z
(z2− z1 )
P1 (x2− x1 )
a y
x
(y2− y1 )
¯ ¯ ¯P1 P2 ¯2 = a2 + (z2 − z1 )2 , a2 = (x2 − x1 )2 + (y2 − y1 )2 . ¤ &
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Example: Find the distance between P1 = (1, 2, 3) and P2 = (3, 2, 1). Solution: ¯ ¯ p ¯P1 P2 ¯ = (3 − 1)2 + (2 − 2)2 + (1 − 3)2 √ = 4+4 √ √ ¯ ¯ ¯ ¯ P1 P2 = 2 2 . = 8 ⇒ C
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Example: Use the distance formula to determine whether three points in space are collinear. Solution: y
y P2
d32
d21
d32 P2
P3 P1
d 31
P1
d 21 d 31
x
d21 + d32 > d31 Not collinear,
P3
x
d21 + d32 = d31 collinear. C
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Mth 234 Multivariable Calculus
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Cartesian coordinates in space (12.1) • Overview of vector calculus. • Cartesian coordinates in space. • Right-handed, left-handed Cartesian coordinates. • Distance formula between two points in space. • Equation of a sphere.
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A sphere is a set of points at fixed distance from a center Definition 1 A sphere centered at P0 © S = P = (x, y, z) :
= (x0 , y0 , z0 ) of radius R is ¯ ¯ ª ¯P0 P ¯ = R .
z
R
y
x
That is, (x, y, z) ∈ S iff (if and only if) (x − x0 )2 + (y − y0 )2 + (z − z0 )2 = R2 .
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Mth 234 Multivariable Calculus
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A Ball is a set of points contained in a sphere Definition 2 A ball centered at P0 = (x0 , y0 , z0 ) of radius R is ¯ ¯ © ª ¯ ¯ B = P = (x, y, z) : P0 P < R . That is, (x, y, z) ∈ B iff (x − x0 )2 + (y − y0 )2 + (z − z0 )2 < R2 .
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Example: Plot a sphere centered at P0 = (0, 0, 0) of radius R > 0. Solution: z
R
y
x
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Example: Plot the sphere x2 + y 2 + z 2 + 4y = 0 Solution: Technique: Complete the square. 0 = x2 + y 2 + 4y + z 2 h ³4´ ³ 4 ´2 i ³ 4 ´2 + z2 = x2 y 2 + 2 y+ − 2 2 2 ³ ´ 4 2 2 =x + y+ + z 2 − 4. 2 x2 + y 2 + 4y + z 2 = 0
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⇔
x2 + (y + 2)2 + z 2 = 22 .
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Example: Plot the sphere x2 + y 2 + z 2 + 4y = 0 Since x2 + y 2 + 4y + z 2 = 0
⇔
x2 + (y + 2)2 + z 2 = 22 ,
we conclude that P0 = (0, −2, 0) and R = 2, therefore, z
−2
y
x
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Exercise: • Given constants a, b, c, and d ∈ R, show that x2 + y 2 + z 2 − 2a x − 2b y − 2c z = d is the equation of a sphere iff d > −(a2 + b2 + c2 ).
(1)
• Furthermore, show that if Eq. (1) is satisfied, then the expressions for the center P0 and the radius R of the sphere are given by p P0 = (a, b, c), R = d + (a2 + b2 + c2 ).
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Vectors on a plane and in space (12.2) • Vectors in R2 and R3 . • Vector components in Cartesian coordinates. • Magnitude of a vector and unit vectors. • Addition and scalar multiplication.
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A vector in R2 or R3 is an oriented line segment Definition 3 A vector in Rn , with n = 2, 3, is an ordered pair of −−→ n points in R , denoted as P1 P2 , where P1 , P2 ∈ Rn . The point P1 is called the initial point and P2 is called the terminal point.
P1 P2
P2
P1
• A vector is drawn by an arrow pointing to the terminal point. −−→ • A vector is denoted not only by P1 P2 but also by an arrow over a letter, like ~v , or by a boldface letter, like v . &
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Mth 234 Multivariable Calculus
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The order of the points determines the direction.
P1 P2 P1
P2
P2 P1
P2
P1
−−→ −−→ The vectors P1 P2 and P2 P1 have opposite directions.
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Mth 234 Multivariable Calculus
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Vectors on a plane and in space (12.2) • Vectors in R2 and R3 . • Vector components in Cartesian coordinates. • Magnitude of a vector and unit vectors. • Addition and scalar multiplication.
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Components of a vector in Cartesian coordinates Theorem 2 Given the points P1 = (x1 , y1 ), P2 = (x2 , y2 ) ∈ R2 , the −−→ vector P1 P2 determines a unique ordered pair denoted as follows, −−→ P1 P2 = h(x2 − x1 ), (y2 − y1 )i. −−→ Proof: Draw the vector P1 P2 in Cartesian coordinates. y P2 y2 P1P2 y1
( y2− y ) 1
P1 ( x 2− x1 )
x1
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x2
x
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A similar result holds for vectors in space. Theorem 3 Given the points P1 = (x1 , y1 , z1 ), P2 = (x2 , y2 , z2 ) ∈ R3 , −−→ the vector P1 P2 determines a unique ordered triple denoted as follows, −−→ P1 P2 = h(x2 − x1 ), (y2 − y1 ), (z2 − z1 )i. −−→ Proof: Draw the vector P1 P2 in Cartesian coordinates. z
P1 P2 P1
P2 ( z2− z1 )
( x2− x1 ) y
x &
( y2 − y1 )
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Mth 234 Multivariable Calculus
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Example: Find the components of a vector with initial point P1 = (1, −2, 3) and terminal point P2 = (3, 1, 2). Solution: −−→ P1 P2 = h(3 − 1), (1 − (−2)), (2 − 3)i
⇒
−−→ P1 P2 = h2, 3, −1i . C
Example: Find the components of a vector with initial point P3 = (3, 1, 4) and terminal point P4 = (5, 4, 3). Solution: −−→ P3 P4 = h(5 − 3), (4 − 1), (3 − 4)i
⇒
−−→ P3 P4 = h2, 3, −1i . C
−−→ −−→ P1 P2 and P3 P4 have the same components although they are different vectors. &
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The vector components do not determine a unique vector. y u v vx
P 0P 0
vy
vy vx
x
− → The vectors u, v and 0P have the same components but they are all different, since they have different initial and terminal points. −−→ Definition 4 Given a vector P1 P2 = hvx , vy i, the standard − → position vector is the vector 0P , where 0 = (0, 0) is the origin of the Cartesian coordinates and P = (vx , vy ). &
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Vectors are used to describe motion of particles.
z
v(t) a(t) r(t) r(0)
y
x The position r(t), velocity v (t), and acceleration a(t) at the time t of a moving particle are described by vectors in space. &
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Vectors on a plane and in space (12.2) • Vectors in R2 and R3 . • Vector components in Cartesian coordinates. • Magnitude of a vector and unit vectors. • Addition and scalar multiplication.
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The length of a vector is the distance between its initial and terminal points −−→ Definition 5 The magnitude or length of a vector P1 P2 is the distance from the initial point to the terminal point. −−→ • If the vector P1 P2 has components −−→ P1 P2 = h(x2 − x1 ), (y2 − y1 ), (z2 − z1 )i, ¯−−→¯ then its magnitude, denoted as ¯P1 P2 ¯, is given by ¯−−→¯ p ¯P1 P2 ¯ = (x2 − x1 )2 + (y2 − y1 )2 + (z2 − z1 )2 . • If the vector v has components v = hvx , vy , vz i, then its magnitude, denoted as |v | is given by q |v | = vx2 + vy2 + vz2 . &
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Mth 234 Multivariable Calculus
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Example: Find the length of a vector with initial point P1 = (1, 2, 3) and terminal point P2 = (4, 3, 2). −−→ Solution: First find the component of the vector P1 P2 , that is, −−→ −−→ P1 P2 = h(4 − 1), (3 − 2), (2 − 3)i ⇒ P1 P2 = h3, 1, −1i. Therefore, its length is ¯−−→¯ p ¯P1 P2 ¯ = 32 + 12 + (−1)2
⇒
¯−−→¯ √ ¯P1 P2 ¯ = 11 . C
Example: If the vector v represents the velocity of a moving particle, then its length |v | represents the speed of the particle.
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Unit vectors have length one. Definition 6 A vector v is called a unit vector iff |v | = 1. D 1 2 3 E Example: Show that v = √ , √ , √ is a unit vector. 14 14 14 Solution:
r
4 9 1 + + |v | = 14 14 14 r 14 = ⇒ |v | = 1 . 14 C
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The following vectors are examples of unit vectors. i = h1, 0, 0i,
j = h0, 1, 0i,
k = h0, 0, 1i.
z
k
i
j
y
x
These vectors will be very useful to write any other vector.
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Vectors on a plane and in space (12.2) • Vectors in R2 and R3 . • Vector components in Cartesian coordinates. • Magnitude of a vector and unit vectors. • Addition and scalar multiplication.
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Vectors can be added and multiplied by scalars. Definition 7 Given the vectors v = hvx , vy , vz i, w = hwx , wy , wz i in R3 , and a number a ∈ R, then the the addition of (v + w) and the scalar multiplication (av ) are given by v + w = h(vx + wx ), (vy + wy ), (vz + wz )i, av = havx , avy , avz i. The vector −v = (−1)v is called the opposite of vector v . Remark: The difference of two vectors is the addition of one vector and the opposite of the other vector, that is, v − w = v + (−1)w. In components we obtain the formula v − w = h(vx − wx ), (vy − wy ), (vz − wz )i. &
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The addition of two vectors is equivalent to the parallelogram law y
V+W vy
V
V
(v+w) y
vx
W wy wx
x
(v+w)x
The vector (v + w) is the diagonal of the parallelogram formed by vectors v and w when they are in their standard position. &
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The addition and difference of two vectors. v+w
v−w
aV
v
a>1 −V
V aV
a = −1
0
