Calculus Lecture 01

8/14/2019 Calculus Lecture 01

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Lecture 1CALCULUS

1. Introduction

1.1 What is vector calculus?

Vector calculus is how to define and measure the variation of

temperature, fluid velocity, force, magnetic flux etc. over all threedimensions of space. In the real 3D engineering world, one wants to

know things like the stress and strain inside a structure, the velocity of

the air flow over a wing, or the induced electromagnetic field around an

aerial. For such questions, it is simply not good enough to deal with

dx

dy and dx ) x( f . We must instead know how to integrate and

differentiate vector quantities with three components (in directions

i, j and k) which depend on three co-ordinates x, y, z.Vector calculus provides the necessary mathematical notation

and techniques for dealing with such issues. First, let’s recall what we

mean by vectors and calculus in isolation.

1.2 Vectors (revision)

Notation: 321321 ,, vvvk v jvivv

length:2

3

2

2

2

1|| vvvv unit vector: v =|| v

vv

Position vector: z y xk z j yi xr ,,

1.3 Scalar field, vector field and Scalar functions

1. A scalar function (of one variable) f (x) or f (t) is a formula thattakes a scalar and returns a scalar. It might be used to describe the

spatial variation of temperature T(x) along a one-dimensional bar

heated at one end, or the time variation of the DC current i (t)

across a certain component in an electrical circuit.

2. A scalar field is a scalar quantity defined over a region of

space. It takes a vector (of positions) and returns a scalar.

)(),,( r f z y x f (or f (x, y) in 2D).

Eg: The variation of temperature T (x, y, z) in this room using

Cartesian co-ordinates. We might also think of the variation of density

or charge density ),,( z y x inside a solid object.

3. A vector field v (x, y, z) is a vector-valued quantity defined

over a region of space. It is defined by a function that takes a vector (of positions) and returns a vector

k z y xv j z y xvi z y xvv ),,(),,(),,( 321



2

j y xvi y xvv ),(),( 21 ( in 2D)

Eg: The spatial variation of fluid velocity v(x, y, z) in a steady flow,

or current ),,( z y x I flowing in a conductor.

Example: An important vector field that we have already encountered is the

gradient vector field . Let f(x,y) be a differentiable function then thefunction that take a point (x0,y0) to gradf(x0,y0) is a vector field since

the gradient of a function at a point is a vector.

For example, if f(x,y) = 0.1xy - 0.2y then gradf(x,y) = 0.1yi +

(0.1x - 0.2) j

The sketch of the gradient is pictured below.

1.4 Vector functionsA vector function (of one variable) v (x) or v (t) takes a scalar

and returns a vector:

.)()()( 321 k t v jt vit vv Such functions might be used to

describe the motion of a particle whose position vector ' is r (t) at time

t; or the external forces F (x) acting at distance x along a 1-

dimensional case.

Differentiation and integration of vector functions are easy! One

simply differences or integrates the components separately.

k t vdt

d

jt vdt

d

it vdt

d

vdt

d

)()()( 321

Example:

A particle moves on a circle of radius 1, such that its position vector is given by

jt it t r cossin)(

Calculate its velocity and acceleration. Show that the velocity and

acceleration are orthogonal.



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E x a m p l e :Sketch the graph of the following vector function.

Rules of differentiation

. Here )(t uu , )(t vv and c is a constant

1. ///vuvu

2. //ucuc

3. ///vuvuvu

4. ///vuvuv xu

Tangent vector to the curve:

k t z jt yit xt r )()()()( ////

1.5 Geometry of Space Curves-Curvature

Let )(t r be a vector description of a curve. Then the distance s( t)

along the curve from the point )(0t r to the point )(t r is , as we have

seen, simply

;)(|)( /

0

duur t s

t

t

Assuming, 0)(/

t r

Now then the vectords

r d

dt ds

t r

dt

r d

dt

r d T

/

)(/

/

is tangent to R and has length one. It is called the uni t tangent

vector.

Consider next the derivative ds

T d T T

ds

T d

ds

T d T T T

ds

d .2...

But we know that 1||. 2 T T T . Thus 0.

ds

T d T , which means

that the vector

ds

T d perpendicular, or normal, to the tangent vector

T. The unit vector alongds

T d denotes by .n

ds

T d

ds

T d

n . The

length of this vector is called the curvature )1

(

and is usually

denoted by the letter . Thusds

T d n and

ds

T d . The unit



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vectords

T d n

1 is called the principal uni t normal vector , and

its direction is sometimes called the principal normal direction.

Example:Consider the circle of radius a and center at the origin:

jt ait at r sincos)(

Example:

If k t jt it t r 22)1()( , find unit tangent vector to the curve.

Torsion of a Space Curve

Let R( t ) be a vector description of a curve. If T is the unit tangent

and n is the principal unit normal, the unit vector b = T × n is called

the binormal vector . Note that the binormal is orthogonal to both T

and n. Let’s see about its derivativeds

bd with respect to arc length s.

First, note that 2||1. bbb , and so 0.

ds

bd b , which means that

being orthogonal to , the derivativeds

bd is in the plane of T and n.

Next, note that b is perpendicular to the tangent vector T , and so

0. T b . Thus 0. ds

bd T . So what have we here? The vector

ds

bd is

perpendicular to both b and T, and so must have the direction of n .

This means

nds

bd and ||

ds

bd . The scalar is called the

torsion.

Example:

Let k ct jt ait at r sincos)( be a space curve, which is

represented by a circular helix. Find unit tangent vector, torsion andcurvature to the curve.

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Calculus Lecture 01