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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
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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.