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CHAPTER 9 VECTOR CALCULUS-PART 1
WEN-BIN JIAN (簡紋濱)
DEPARTMENT OF ELECTROPHYSICS
NATIONAL CHIAO TUNG UNIVERSITY
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OUTLINE
1. VECTOR FUNCTIONS
2. MOTION ON A CURVE
3. CURVATURE AND ACCELERATION
4. PARTIAL DERIVATIVES
5. DIRECTIONAL DERIVATIVE
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1. VECTOR FUNCTIONS
Extends from 1D to 3D Vector Space:
x-axis
0 1 2 3 4 5 6 7-8-7-6 -5-4 -3 -2-1
Length:
Directional vector, unit vector:
Three-dimensional vector space:
x
y
z
Ox0
y0
z0
The three unit vectors in the x, y, and z axises: or
The positional vector:
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1. VECTOR FUNCTIONS
Scalar Functions, Functions of Scalar or Functions of Vector:: scalar function (1D vector function), scalar (1D) to scalar (1D) mapping
: three-dimensional scalar function, 3D to scalar mapping
– level surface
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1. VECTOR FUNCTIONS
Vector Functions of Vectors: , 2D -> 2D
For example:
For example:
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1. VECTOR FUNCTIONS
Vector Function – Curve, 3D -> 1D Using ParameterVector function – Curve: the three independent variables x, y, z need to satisfy two conditions – two constraints
Example: , satisfy the conditions & .
one parameter description, let
Example: , satisfy the conditions & .
one parameter description, let
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1. VECTOR FUNCTIONS
Example: Find the vector function that describes the intersection of the plane and the paraboloid .
one parameter description, let
Find the Intersection Curve
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1. VECTOR FUNCTIONS
Limits of a Vector Function
DEFINITION: Limits of a Vector Function of a Curve
If →
, →
, →
exist,
→ → → →
THEOREM: Properties of Limits
If →
and →
1. Any scalar , →
2.→
3.→
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1. VECTOR FUNCTIONS
Continuity & Derivative of a Vector function
DEFINITION: Continuity of a Vector Function of a Curve
A vector function is said to be continuous at if(i) is defined, (ii)
→exists, and (iii)
→.
DEFINITION: Derivative of a Vector Function
∆ →
which can be obtained by differentiation of components.If
Geometric Interpretation of - Tangent Vectors
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t=3
1. VECTOR FUNCTIONS
Calculate The Tangent Line of a CurveExample: Find the tangent line at for the vector function of a curve .
derive the tangent vector:
the tangent vector at
the positional vector at
the tangent line at
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1. VECTOR FUNCTIONS
Calculate The Tangent Vector
Example: If and ,
please calculate ⃗
.
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1. VECTOR FUNCTIONS
Differentiation & Integration of The CurveTHEOREM: Rules of Differentiation
If and are differentiable vector functions and is a differentiable scalar function, then
RULE: Integrals of The Vector Function of a Curve
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1. VECTOR FUNCTIONS
The Length of a CurveRULE: Calculation of The Length of a Section of a Curve
If , the length of the curve is
,
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1. VECTOR FUNCTIONS
The Arc Length Parameter
Example: Please calculate the arc length , where
.
Please change the vector function from the variable to the length variable .
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OUTLINE
1. VECTOR FUNCTIONS
2. MOTION ON A CURVE
3. CURVATURE AND ACCELERATION
4. PARTIAL DERIVATIVES
5. DIRECTIONAL DERIVATIVE
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2. MOTION ON A CURVE
Motion of a ParticleLet’s start from the position of a particle as a function of time
the velocity, speed, and acceleration of the particle is
Example: The particle under uniform circular motion is described by the positional function , please calculate its acceleration.
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2. MOTION ON A CURVE
Trajectory of a ParticleExample: Assume the initial velocity of the projectile is
, where is the angle between the initial velocity and the horizontal. The initial position is the origin.
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3. CURVATURE AND ACCELERATION
Unit Tangent: tangent vector for the vector function of the curve is .
The unit tangent is , indicating the direction of the tangent line.
Curvature : The change of the unit tangent per unit length.
Calculation of Curvature
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3. CURVATURE AND ACCELERATION
Example: The positional vector of a circle is .
Curvature of a Circle
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3. CURVATURE AND ACCELERATION
assume the tangent unit vector and the normal unit vector start from the positional vector
The Normal Vector: , note that this is not a unit vector.
Tangential and Normal Acceleration ( and (or ))
𝑟 𝑡 : positional vector, function of curve
𝑟′ 𝑡 =⃗
: tangent vector
𝑇 𝑡 = 𝑟 𝑡 / 𝑟 𝑡 : unit tangent
𝑛 𝑡 = 𝑑𝑇 𝑡 /𝑑𝑡: normal vector
𝑁 𝑡 = 𝑇 𝑡 / 𝑇 𝑡 : unit normal
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3. CURVATURE AND ACCELERATION
The Normal Vector: The Unit Normal Vector:
The Binormal Unit Vector (defined in 3D only)
and form the osculating plane.
and form the normal plane.
and form the rectifying plane.
Normal and Binormal Vectors
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3. CURVATURE AND ACCELERATION
Example: Please calculate the unit tangent, unit normal, and unit binormal vectors of the curve . Please calculate the curvature of the curve.
Tangemt, Normal and Binormal Vectors
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3. CURVATURE AND ACCELERATION
use inner and cross product
Another Method for The Calculation of The Curvature
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3. CURVATURE AND ACCELERATION
Example: If , please calculate the curvature.
/
Curvature of Twisted Cubic
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OUTLINE
1. VECTOR FUNCTIONS
2. MOTION ON A CURVE
3. CURVATURE AND ACCELERATION
4. PARTIAL DERIVATIVES
5. DIRECTIONAL DERIVATIVE
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4. PARTIAL DERIVATIVES
Example: Find out the level curves of the function .
Level curves:
If , let , .
one parameter description
If , let , .
If , let .
Level Curves for Functions of Two Variables
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4. PARTIAL DERIVATIVES
Level Surface for Functions of Three Variables
We start from three variables x, y, z in 3D space. The three variables are subjected to the constraint of , which gives the level surfaces.
Example: Level Surfaces for Functions of Three Variables
Describe the level surface of the function .
Functions of Three Variables
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4. PARTIAL DERIVATIVES
Total Derivatives of by is ,
.
Partial Derivatives of by , assume that x and y are
independent, , then ,
.
Notation:
If is differentiable, .
Commutative: ,
Chain Rule:
Partial Derivatives of Functions
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4. PARTIAL DERIVATIVES
Example: if , , , find and .
Example: if , , , , find .
Partial Derivatives of Functions
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5. DIRECTIONAL DERIVATIVE
As an example, , its level curve is presented in the graph.
Here we introduce the partial derivatives, and , what’s the meaning of ? It points to the direction of the gradient.
Gradient Calculation (Scalar Functions to Vector Functions)
𝛻𝐹 = 𝛻𝐹 𝚤̂ + 𝛻𝐹 𝚥̂ = 𝐹 𝚤̂ + 𝐹 𝚥̂
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5. DIRECTIONAL DERIVATIVE
Derivation of Gradient Calculation: what’s the change of the scalar function in a small displacement ?
A small variation in direction:
, ∆
∆
A small variation in direction:
, ∆
∆
The gradient calculation:
Gradient Calculation
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𝐹 𝑥, 𝑦, 𝑧 = 𝑥 + 𝑦 + 𝑧
5. DIRECTIONAL DERIVATIVE
The Gradient Calculation in 3D SpaceFor example, the scalar function
𝛻𝐹 =𝜕𝐹
𝜕𝑥𝚤̂ +
𝜕𝐹
𝜕𝑦𝚥̂ +
𝜕𝐹
𝜕𝑧𝑘
Gradient Calculation
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5. DIRECTIONAL DERIVATIVE
Maximum Value of the Directional DerivativeArgument I:If is a unit vector ( ) but not along the , then you may expect the change of is dependent on the angle between the
two vectors as , the concept
indicates that the maximum change of is along the gradient direction which is the direction of the steepest ascent or descent.Argument II:Starting from the two variable function , the level curve is
. A small variation is zero, .
. It indicates that the direction of the gradient of the function is perpendicular to the level curves.We conclude that the direction of is perpendicular to the level curves .
Concepts of Gradient Calculation
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5. DIRECTIONAL DERIVATIVE
Example: Calculate the gradient of the function .
𝑓 𝑥, 𝑦 = 5𝑦 − 𝑥 𝑦
Gradient Calculation
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5. DIRECTIONAL DERIVATIVE
Example: Calculate the gradient of the function at .
At ,
, ,
Gradient Calculation
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5. DIRECTIONAL DERIVATIVE
Example: Find the directional derivative of at in the direction of .
The unit vector of the direction
in the direction :
Gradient Calculation
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5. DIRECTIONAL DERIVATIVE
Example: The temperature in a rectangular box is approximated by , , ,
. Find the direction of rapid cool off at .
The rapid warm up direction is . The
rapid cool off direction is .
Gradient Calculation