MATH2069 LECTURE 8
CRITICAL POINTS ON SURFACES
Second Derivative Test
If f and all its first and second partial derivatives are continuous in the
neighbourhood of (a, b) and fx (a, b) = fy (a, b) = 0 then
2
> 0 at (a, b).
(i) f has a local maximum at (a,
THE UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS AND STATISTICS
MATH 2069
MATHEMATICS 2A Solutions to the Complex Analysis Exam 2012.
4ia. Express the circle |z| = 3 in polar form.
b. Find the image of the circle |z| = 3 under the mapping
w = z2
8
MATH2069 (FINAL) LECTURE 23
DIVERGENCE THEOREM
Let F(x, y, z) = F1 i + F2 j + F3 k be a vector field in R3 such that F1 , F2 and F3
have continuous partial derivatives. Assume also that V is a closed and bounded
solid with smooth boundary surface S. Then
MATH2069 LECTURE 19
SURFACE INTEGRALS
If a surface S is defined parametrically by
, D
r(, ) = x(, )i + y(, )j + z(, )k,
dS = kr r k dd
g(x(, ), y(, ), z(, ) kr r k dd
g(x, y, z) dS =
D
S
Suppose now that is a region in the x y plane and z = f (x, y) is a
MATH2069 LECTURE 13
TRIPLE INTEGRALS
A triple integral of f (x, y, z) over a solid S in R3 in Cartesian coordinates takes
the form
ZZZ
Z b Z g2 (x) Z h2 (x,y)
f (x, y, z)dzdydx
f (x, y, z) dV =
a
S
g1 (x)
h1 (x,y)
ZZZ
1 dV = Volume(S).
S
Recall that for a
MATH2069 LECTURE 10
DOUBLE INTEGRALS
For a region in the x y plane and a surface z = f (x, y) in R3 the double integral
f (x, y)dy dx.
evaluates the volume of the solid above and below z = f (x, y).
Comparison of
b
a
f (x)dx and
f (x, y)dy dx.
When eva
MATH2069 LECTURE 14
TRIPLE INTEGRALS IN CYLINDRICAL COORDINATES
A triple integral of f (r, , z) over a solid S in R3 in Cylindrical coordinates takes
the form
ZZZ
Z b Z g2 () Z h2 (r,)
f (r, , z)rdzdrd
f (r, , z) dV =
a
S
g1 ()
h1 (r,)
We also have
ZZZ
1
MATH2069 LECTURE 21
GRAD, DIV AND CURL
= grad() =
i+
j+
k.
x
y
z
F = div(F) =
F1 F2 F3
+
+
.
x
y
z
F = curl(F) =
i
j
x
y
F1
F2
k
z
F3
grad : scalar to vector
div : vector to scalar
curl : vector to vector
Where the vector differential operator is
THE UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS AND STATISTICS
JULY, 2010
MATH2069
MATHEMATICS 2A
(1) TIME ALLOWED 3 hours
(2) TOTAL NUMBER OF QUESTIONS 6
(3) ANSWER ALL QUESTIONS
(4) THE QUESTIONS ARE OF EQUAL VALUE
(5) ANSWER EACH QUESTION IN A
MATH2069 LECTURE 4
PARTIAL DIFFERENTIATION
Suppose z = f (x, y). Define
f
(x,y)
= lim f (x+x,y)f
x
x0
x
f
y
= lim
y0
f (x,y+y)f (x,y)
y
Notation
f
= fx = zx ,
x
f
= fy = zy
y
z = f (x, y) is continuous at (a, b) if
lim
f (x, y) = f (a, b)
(x,y)(a,b)
In th
MATH2069 LECTURE 18
GREENS THEOREM IN THE PLANE
Let F(x, y) = F1 i + F2 j be a vector eld such that F1 and F2 have continuous
partial derivatives and be a simply connected region in R2 with a smooth
boundary curve C. Then
F2 F1
F dr =
dA
y
C
x
We close M
MATH2069 LECTURE 16
CHANGE OF VARIABLES
x
u
x = x(u, v) and y = y(u, v) = J =
y
u
x
v (x, y)
= dxdy = |J|dudv
=
y (u, v)
v
We have seen in the previous lectures that when we slide from one coordinate system to
another we need to insert a scale factor
MATH2069 LECTURE 2
CURVES IN SPACE
The parametric vector equation of a line in R3 passing through the point P and parallel
to the vector v is given by
x
r = y = P + vt; t R
z
For a path in space
r(t) = x(t)i + y(t)j + z(t)k
at time t we have
Velocity Ve
Solutions to MATH2069 2013 exam
Question 1
i)
a) The intersection point(s) are given by the solutions of
t3 + 1 = u2 + 3u 1
t = u
t2 = 2u
which has the unique solution
t=u=2
corresponding to the point (9, 2, 4).
b) We have
r (t) = 3t2 i j + 2t k
and
s (u)
MATH2069 LECTURE 20
FLUX INTEGRALS
f
x
F1
F2
dS =
Fn
f dxdy
y
S
F3
1
dS =
Fn
S
(Cartesian form)
F1
F2 (r r ) dd
D
F3
(Parametric Form)
dS which are a special type of
Fn
In this lecture we will examine flux integrals
S
surface integral used
First a leftover example from the last lecture.
Example 1 : The solid S is bounded below by the plane z = b and above by the
sphere x2 + y 2 + z 2 = a2 , where a b 0.
a) Sketch the cap S and its projection onto the x y plane.
b) Find the volume of S and c
MATH2069 LECTURE 1
REVISION OF VECTOR ALGEBRA
Scalars are quantities which have only a magnitude (such as temperature, mass,
time and speed).
Vectors have a magnitude and a direction.
The n dimensional vector space Rn is given by
x
1
x
2
n
R = . x1 , x2
MATH2069 LECTURE 3
SURFACES IN SPACE
Standard Surfaces
a
ax + by + cz = d is a plane with normal vector n = b .
c
x2 + y 2 + z 2 = r2 is a sphere centre the origin radius r.
z = (x2 + y 2 ) is a paraboloid of revolution. ( R, = 0)
x = (y 2 + z 2 ) is
Solutions to Math 2069 Exam 2012, Vector Calculus
section
Question 1:
Part (i)
An object moves along a circular helix with position given by
r(t) = a cos t i + a sin t j + bt k.
Find
a) the velocity vector of the object;
Solution. The velocity is given by
MATH2069 LECTURE 17
LINE INTEGRALS
Line integrals are used to calculate the work done in moving a particle P from A
to B along a path C through a force field F.
Z
Z
F dr =
(F1 dx + F2 dy + F3 dz)
C
C
In general, this integral depends not only on F but als
MATH2069 LECTURE 9
LAGRANGE MULTIPLIERS
The method of Lagrange multipliers
In R2 , to find the local minima and maxima of f (x, y) subject to the constraint
g(x, y) = 0 we find the values of x, y and that simultaneously satisfy the
equations
f
g
= ,
x
x
f
MATH2069 LECTURE 7
TANGENT PLANES AND ERROR ESTIMATES
Given a level surface (x, y, z) = C in R3 we have, at any point P on the surface,
grad() the surface (x, y, z) = C. Similarly for level curves in R2 .
If z = f (x, y) then z
z
z
x +
y.
x
y
Given a poi
MATH2069 LECTURE 6
VECTOR AND SCALAR FIELDS, THE GRAD OPERATOR
= grad() =
i+
j+
k.
x
y
z
grad : scalar to vector
Given a scalar eld , the directional derivative of in the direction of the vector
where b
is the unit vector in the direction of b.
b is gi
MATH2069 LECTURE 5
CHAIN RULE
If z = f (x, y) and x = x(t) and y = y(t) then
z
z x z y
=
+
t
x t
y t
If z = f (x, y) and x = x(u, v) and y = y(u, v) then
z
z x z y
=
+
u
x u y u
and
z
z x z y
=
+
v
x v y v
A common situation we encounter is that z is a fu
Chapter 1
The Complex Plane
1.1 Revision of First Year
1.2 Topology of the Complex Plane
1.3 Mapping by Complex Functions
MATH2069 Complex Notes,
2016
p. 1
1.1 Revision of First Year
We use the symbol i as the square root of 1, so the other
square root i
Chapter 5
The Complex Logarithm
5.1 The Log function
5.2 Complex exponents
MATH2069 Complex Notes,
2016
p. 1
5.1 The Complex Logarithm
As the exponential is not one-one on C, it has no true inverse
function.
But we have already managed to solve equations
Chapter 11
Real Improper Integrals
MATH2069 Complex Notes,
2016
p. 1
Recap of definition
The only improper integrals we will consider are ones over
infinite intervals.
Let f : R R be a continuous function and a R. Recall:
Z
Z r
1)
f (x) dx converges if
UNIVERSITY OF NEW SOUTH WALES
MATH2069
MATHEMATICS 2A
VECTOR CALCULUS QUIZ 2 SAMPLE 1
This sheet must be filled in and stapled to the front of your answers
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THE UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS AND STATISTICS
JULY, 2009
MATH2069
MATHEMATICS 2A
(1) TIME ALLOWED 3 hours
(2) TOTAL NUMBER OF QUESTIONS 6
(3) ANSWER ALL QUESTIONS
(4) THE QUESTIONS ARE OF EQUAL VALUE
(5) ANSWER EACH QUESTION IN A