Introduction to
Advanced Calculus
(1) In Beginning Calculus, we study functions of one variable
f x  : R  R
where the input and output are both real numbers.
(2) In Intermediate Calculus, we study functions of several
variables such as
f  x , y  : R 2  R or g  x , y , z  : R 3  R
where the output is still a real number.
(3) In Advanced Calculus, we shall study functions of
several variables and with outputs in vectors such as
F  x , y  : R 2  R 2 or G  x , y , z  : R 3  R 3
These are called vector-valued functions, or even Vector Fields
when the domain has dimension = dimension of the range, such
as
3
3
Gx , y , z  : R  R
Examples:
F x , y   x  y , 2 xy2
K x, y   y, x  1, x sin y
Gx, y, z   xz, z  y,0
a 2D vector field
a surface in 3D
a 3D vector field whose
z-component is always 0.
Velocity vector field showing wind patterns
A simple case:
when the domain is just R and the range is R3, then the
function is called a space curve - because each output vector
will give a point (namely the endpoint of the vector) in space,
and if we joint all these points together, we will get a curve.
Space Curves
A space curve is a function C from R to R3, for example
C t   cos t , 0.05t  sin t , 2t 
note that we do not use the vector notation because we are
treating the output as a point in space rather than a vector.
30
25
20
15
10
5
-1
0
-0.5
-0.5
0
0.5
0
1
1.5
0.5
1
Space curves in 3D
In order to see these objects in 3D, you need to have a pair
of green-red glasses.
The Red lens should be on your left eye, and you may need
to dim the surrounding lights to get good results. Even so,
it may still take a few seconds for your brain to adjust to
the different images seen by different eyes.
When you are ready, just click and enjoy.
This is a surface in space. You should be able to determine
which part is in the front without the glasses. However the
viewing experience in 3D is a lot more satisfying.
This is a spiral wrapped around a donut. If you move your
head sideways slightly, you may see that the curve rotates
with you.
1
0
-1
4
2
0
-2
-4
4
2
0
-2
-4
x  (4  sin 20t ) cost , y  (4  sin 20t ) sin t , z  cos 20t
Can you tell which part of the curve is in the front without the
glasses?
1
0.5
0
-0.5
-1
-1.5
-2
-3
-2
-1
0
1
2
33
2
1
0
-1
-2
-3
x  (2  cos1.5t ) cost , y  (2  cos1.5t ) sin t , z  sin1.5t
This is a picture of Mars landscape released by NASA
you need to put the red lens on your left eye to see 3D.
Downtown San Diego, again red lens on the left
SDSU
Vector Fields in 3D
These are functions from R3 to R3 . For any given point
P(a,b,c) in space, the arrow initiating from point P
represents the output of the function with input (a,b,c).
Smooth Curves and Tangent vector
A curve C(t) = (x(t), y(t), z(t)) is smooth in its domain with
the functions x(t), y(t), z(t) are all differentiable and
x' t , y' t , z' t   0, 0, 0
throughout the domain.
If the curve C(t) is smooth, then the direction of tangent to
C(t) at a point t0 is given by the vector
x' to , y' to , z' to 
The arc length is given by
b
a
x' t 2  y' t 2  z' t 2 dt
if the curve starts at t = a and ends at t = b.
Line Integrals
Introduction:
Suppose that we have a curve C(t) = ‹ x(t), y(t) › on the xy-plane
and a function f(x, y) defined also on the xy-plane. What is the
area of the curved surface above the curve and below the surface
z = f(x,y)?
The red curve is C(t),
click to see the surface.
60
40
20
0
y
-4
4
-2
2
0
x
0
2
4
-2
-4
Line Integrals
Introduction:
Suppose that we have a curve C(t) = ‹ x(t), y(t) › on the xy-plane
and a function f(x, y) defined also on the xy-plane. What is the
area of the curved surface above the curve and below the surface
z = f(x,y)?
The area is an integral
60

40
20
C
f ds
0
y
-4
4
-2
2
0
x
0
2
4
-2
-4
More precise
definition will be on
the next slide
Line Integrals
In the following example, f (x, y) = 60 - x2 - y2, and
C (t )  
t  sin 2t t  sin 2t
,

2
2
8  t  8

C
f ds is defined to be

8
8
60
f ( x(t ), y (t )) x' (t )2  y ' (t )2 dt
40
20
0
y
-4
4
-2
2
0
x
0
2
4
-2
-4
Line Integrals
This type of line integrals can also be defined for function of 3 variables,
except that the physical interpretation will no longer be area.
Line Integral of a real-valued function
Let f (x,y,z) be a real-valued function, and C(t) = (x(t), y(t), z(t))
(for a  t  b) is a smooth curve in the domain of f , then
C
f ds 
b
a
f  xt , yt , z t  x' t 2  y' t 2  z' t 2 dt
More Line Integrals
There are actually two types of line integrals
I. line integrals of a real-valued function,
II. line integrals of a vector field.
We have already seen the 1st type, and the second type has another
physical interpretation.
Physical Interpretation
If a F is a force field (such as gravity), then the line integral of F along
C is the total work done by the force F when a particle moves along the
path C.
If the integral is positive, then the particle will gain energy (and usually
moves faster). Otherwise the particle will slow down.
In the special case that the force is always perpendicular to the path,
the total work done will be 0.
Voyagers are leaving the Solar System
Line integral of a vector field
Let F(x, y, z) = (P(x, y, z), Q(x, y, z), R(x, y, z)) be a vector
field and C(t) = (x(t), y(t), z(t)) (for a  t  b) be a smooth
curve in the domain of F, then
 F  d r   F xt , yt , zt  x' t , y' t , z' t  dt
b
C
a
is called the line integral of F along C.
Example
y
x
y
F ( x, y )   2
, 2

2
2
x y x y
2
1
-2
-1
1
-1
-2
2
x
C (t )   t , 1  t 2  ,  2  t  2
Gradient Vectors fields and the Del Operator
Recall that if a function f (x,y,z) from R3 to R has
continuous 1st order derivative, then we can derive a vector
field
f f f
f  x , y , z  
,
,
x y z
which is called the gradient vector field of f.
In fact, lots of vector fields are gradient vector fields, and
hence we have the following
Gradient Vectors fields and the Del Operator
Definition:
A vector field F is said to be Conservative if it is the
gradient vector field of some differentiable function
f (x,y,z), i.e.
F x , y , z   f x , y , z 
In this case, f (x,y,z) is called a potential function for F.
(  is called the del operator )
Perpetual Machines.
http://www.lhup.edu/%7Edsimanek/museum/unwork.htm
Path Independent Integrals
Definition
A line integral
C F  d r
is said to be independent of path
if its value depends only on the end points of the curve C
and not on the shape of the curve C (as long as it lies in the
domain of F) In other words, if C2 is another path with the
same starting and ending points as C, and the trace of C2
also lies in the domain of F, then
C F  d r  C F  d r
2
Theorem
1. If F is conservative i.e. F = f for some f , then every
line integral of F will be independent of path and
C F  d r  f xb, yb, zb  f xa , ya , za 
where (x(a),y(a),z(a)) is the beginning of the curve and
(x(b),y(b),z(b)) is the end of the path.
2. If every line integral of the form
C F  d r
independent of path, then F is conservative.
is
Corollary
F is conservative if and only if
C F  d r  0
for any closed path lying inside the domain of F.
We are now going to see several examples of conservative
and also non-conservative vector fields.
Examples
Dues to the technical difficulties in showing 3D vector fields,
we shall only show 2D vector fields.
Is F(x,y) = [x - y, x - 2] conservative?
y
4
2
0
-2
-4
-4
-2
0
2
4
x
Is F(x,y) = [3 + 2xy, x2 - 3y2] conservative?
y
4
2
0
-2
-4
-4
-2
0
2
4
x
Is F(x,y) = [2xy + sin y, x3 + 2 cos y] conservative?
y
2
1
0
-1
-2
-2
-1
0
1
2
x
Test for Conservativity
If F(x, y) = P, Q is a vector field from R2 to R2 with
continuous 1st order partial derivatives, then F is
conservative if and only if
Q P

x y
Divergence of a vector field
Definition
Given a vector field F  P ,Q , R
with differentiable
components, we define the divergence of F to be
P Q R
F 


x y z
Physical interpretations:
If F is the velocity field of some fluid, then F (a,b,c) is
the amount of fluid flowing out from the point (a,b,c) in
unit time.
If  F (a,b,c) > 0, then we say that (a,b,c) is a source.
If  F (a,b,c) < 0, then we say that (a,b,c) is a sink.
If  F = 0 through out its domain, then we say that F is
incompressible.
 2 y  2x 
2 x 2  4 xy  2 y 2  2 2 x 2  4 xy  2 y 2  2


 2

,
, 0
2  1
2
2
2
2
2
2
x

y


x  y 1
x  y 1




y
2
1
-2
-1
0
-1
-2
1
2
x
Electric field E
Gauss Law · E = ρ/εo
where ρ is charge density.
Gauss’s law for Magnetism
(2nd of the Maxwell equations)
where B stands for magnetic field.
· B = 0
Curl of a vector field
Definition
Given a vector field F = P, Q, R with differentiable
components, we define the Curl of F to be
 F 
R Q P R Q P

,

,

y z z x x y
Physical interpretations:
If F is the velocity field of some fluid, then × F(a,b,c) is
rotation of the fluid at the point (a,b,c), i.e. the magnitude
of the vector is the rotational speed and the direction is the
direction of rotation according to the right hand rule.
If ×F(a, b, c) = 0,0,0, then we say that F is
irrotational at the point (a, b, c).
If ×F = 0,0,0 throughout the domain of F, then
we say that F is irrotational.
Is the vector field F(x,y,z) = [-y, x, 0] rotational? Conservative?
y
☺
4
2
-4
-2
0
-2
-4
2
4
x
Is the vector field F(x,y,z) = [x + y, x – y, 0] rotational?
Is it Compressible?
y
4
2
-4
-2
0
2
-2
-4
☺
4
x
Test for Conservativity
Theorem
If f (x,y,z) is a function from R3 to R which has continuous 1st
order partial derivatives, then
Curl(f ) = ×f = 0,0,0
If F(x, y, z) = P, Q, R is a vector field from R3 to R3 with
continuous 1st order partial derivatives, then F is
conservative if and only if
CurlF = ×F = 0,0,0
As a consequence, we see that F is conservative if and
only if it is irrotational.
The following vector field is clearly rotational (at many
points), hence not conservative.
y
4
2
-4
-2
0
-2
-4
2
4
x
Clay Mathematics Institute
Millennium Problems
In order to celebrate mathematics in the new millennium, The Clay Mathematics Institute
of Cambridge, Massachusetts (CMI) has named seven Prize Problems. The Scientific
Advisory Board of CMI selected these problems, focusing on important classic questions
that have resisted solution over the years. The Board of Directors of CMI designated a $7
million prize fund for the solution to these problems, with $1 million allocated to each.
One hundred years earlier, on August 8, 1900, David Hilbert delivered his famous lecture
about open mathematical problems at the second International Congress of
Mathematicians in Paris. This influenced our decision to announce the millennium
problems as the central theme of a Paris meeting.
Paris, May 24, 2000
•Birch and Swinnerton-Dyer Conjecture
•Hodge Conjecture
•Navier-Stokes Equations
•P vs NP
•Poincaré Conjecture
•Riemann Hypothesis
•Yang-Mills Theory
Navier-Stokes Equation
Waves follow our boat as we meander
across the lake, and turbulent air currents
follow our flight in a modern jet.
Mathematicians and physicists believe
that an explanation for and the prediction
of both the breeze and the turbulence can be found
through an understanding of solutions to the
Navier-Stokes equations. Although these equations
were written down in the 19th Century, our
understanding of them remains minimal. The
challenge is to make substantial progress toward a
mathematical theory which will unlock the secrets
hidden in the Navier-Stokes equations.
Navier-Stokes Equation
u1
p
 u  (u1 )   ( u1 ) 
 f1 ( x, t )
t
x
u2
p
 u  (u2 )   ( u2 ) 
 f 2 ( x, t )
t
y
u3
p
 u  (u3 )   ( u3 ) 
 f 3 ( x, t )
t
z
u  0
where
 
p x, t
is the pressure
in the fluid
ν is the viscosity
(i.e. the fluid is incompressible.)
u( x, t )  u1 ( x, t ), u2 ( x, t ), u3 ( x, t )
2
2
2
 2  2  2
x
y
z
is an unknown velocity vector
is the Laplacian operator.
f ( x, t )  f1 ( x, t ), f 2 ( x, t ), f 3 ( x, t ),
is a given externally applied force such as gravity.
The challenge is to prove that there are smooth functions
p( x, t ), u( x, t ) that satisfy these equations!
The End
Can you tell which part of the curve is in the front without the
glasses?
Background
220,220,220
Red, 255,0,0
Blue 0, 220, 220
x  (2  cos1.5t ) cost , y  (2  cos1.5t ) sin t , z  sin1.5t