CHAPTER 9 VECTOR CALCULUS-PART 1

WEN-BIN JIAN (簡紋濱)

DEPARTMENT OF ELECTROPHYSICS

NATIONAL CHIAO TUNG UNIVERSITY

OUTLINE

1. VECTOR FUNCTIONS

2. MOTION ON A CURVE

3. CURVATURE AND ACCELERATION

4. PARTIAL DERIVATIVES

5. DIRECTIONAL DERIVATIVE

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:

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

1. VECTOR FUNCTIONS

Vector Functions of Vectors: , 2D -> 2D

For example:

For example:

 

 

 

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

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

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.→

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

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

1. VECTOR FUNCTIONS

Calculate The Tangent Vector

Example: If and ,

please calculate ⃗

.

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 

 

 

 

 

 

 

 

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

 

,

 

 

 

 

 

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 .

 

     

OUTLINE

1. VECTOR FUNCTIONS

2. MOTION ON A CURVE

3. CURVATURE AND ACCELERATION

4. PARTIAL DERIVATIVES

5. DIRECTIONAL DERIVATIVE

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.

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.

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

3. CURVATURE AND ACCELERATION

Example: The positional vector of a circle is .

Curvature of a Circle

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

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

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

3. CURVATURE AND ACCELERATION

use inner and cross product

Another Method for The Calculation of The Curvature

3. CURVATURE AND ACCELERATION

Example: If , please calculate the curvature.

 

 

/

Curvature of Twisted Cubic

OUTLINE

1. VECTOR FUNCTIONS

2. MOTION ON A CURVE

3. CURVATURE AND ACCELERATION

4. PARTIAL DERIVATIVES

5. DIRECTIONAL DERIVATIVE

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

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

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

4. PARTIAL DERIVATIVES

Example: if , , , find and .

Example: if , , , , find .

Partial Derivatives of Functions

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

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

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

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

5. DIRECTIONAL DERIVATIVE

Example: Calculate the gradient of the function .

𝑓 𝑥, 𝑦 = 5𝑦 − 𝑥 𝑦

Gradient Calculation

5. DIRECTIONAL DERIVATIVE

Example: Calculate the gradient of the function at .

At ,

, ,

Gradient Calculation

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

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