Tutorial 9-Math 252
dy
1. If x2 y + yz = 0, xyz + 1 = 0, nd dx and dz at (x, y, z ) =
dz
(1, 1, 1).
2. If x2 + yu + xv + w = 0, x + y + uvw + 1 = 0, then regarding
x and y as functions of u, v, and w, nd
x
u
and
y
u
at (x, y, u, v, w) = (1, 1, 1, 1, 1).
3
A summary of Friday's lecture was given [Transparancy 1], showing the definitions
of curvature, unit tangent vector, principal normal, tangential acceleration, and normal
(centripetal) acceleration.
A detailed example was worked out [Transparancies 2 and
Last lecture we proved the first vector identity in section 1.14 of the textbook, using
tensor notation. In this lecture, we proved the other three vector identities of section
1.14 [Transparancies 1 and 2]. (In Assignment #02, you are asked to prove thes
We continued with our discussion of polar coordinates in the xy-plane [Transparancy
1]. The acceleration vector was obtained by differentiating the velocity vector, and
being careful about differentiating the radial and angular unit vectors. Four differen
We started off [Transparancy 1] by considering the velocity vector when the
magnitude of the position vector stays constant. It was shown that the velocity vector
is perpendicular to the position vector under these circumstances.
We then looked in some de
We started looking at acceleration and curvature along a spacecurve (section 2.3 of
the textbook). Acceleration has two components [Transparancy 1], tangential and
centripetal. The latter is defined in terms of a radius of curvature. The curvature
constan
3.3 Divergence
Now we will begin to look at vector derivatives. The rst of these is the divergence of a
vector eld.
1. Denition. Given a vector eld F , its divergence is
divF =
F =
F1 F2 F3
+
+
x
y
z
presuming those partial derivatives exist.
2. Examples:
3.4 to 3.6: Curl and the Laplacian
1. Denition of Curl
2. Standard Examples
3. Product Rule.
curl(F ) = curlF + grad F .
1
4. Paddle Wheel Analogy
Figure 1: From http:/www.instrumart.com/pages/227/in-line-turbine-paddle-wheel-owmeters.
Reminder: Near a p
Sections 5.2Notes
1. Recall the divergence theorem in the form
F dV =
D
F ndS
S
where D is closed and S is the boundary of D.
2. Recall also the vector identity
( ) =
+
whose proof is not hard.
3. Let F = and apply the divergence theorem:
4. Greens rst
Section 4.7 - 4.9: Surface and Volume Integrals
1. Surface Integral of a scalar function
2. Surface integral of a vector function: surface given parametrically (use of R/u
R/v )
3. Using rectangular coordinates
1
4. Example
5. Physical Example
2
6. Volu
Sections 4.5 Notes
1. Vector Potentials
If F is a vector eld (i.e. vector-valued function) such that F =
G, then G is
called a vector potential for F .
Theorem: Suppose F is continuously dierentiable in D R3 , where we will assume
D is star-shaped in our
Section 4.4: Fields with Curl Zero
1. Fields with curlF = 0 are called irrotational. (Fluids context).
This means
2. Theorem If F is continuously dierentiable in a region D R3 , then F is conservative
(i.e. F = grad) i curlF = 0.
3. Example
4. Example
1
5
Sections 4.2 and 4.3 Notes
1. Basic Concepts from Topology
(a) -neighbourhood
(b) interior point of a set
(c) boundary point of a set
(d) exterior point of a set
(e) open set: every point is interior
(f) connected (arcwise): any two points can be joined b
Section 4.1: Line Integrals
1. Recall Arc Length:
b
|dR| =
C
dx
dt
|R (t)|dt =
C
a
2
+
dy
dt
2
+
dz
dt
2 1/2
2. Arc Length in Curvilinear Coords.
3. Example in Cylindrical: arc length of a portion of the helix given by
() = e + e + cos( + )k.
1
dt.
4. Fou
MATH 252-3
Spring 2005
Vector Calculus
Final Exam
Tuesday, 19 April 2005
Attempt all of the following questions; there are 11 problems, for a total of 120 points. The
total time available is three hours (180 minutes).
1. (8 points)
(a) Use tensor notation
SIMON FRASER UNIVERSITY
DEPARTMENT OF MATHEMATICS
Midterm solutions, both versions
MATH 252 D100 Spring 2013
Instructor: Archibald
Feb. 20, 2013, 8:30 9:20 a.m.
Name:
(please print)
family name
given name
student number
SFU-email
SFU ID:
Signature:
Instru
Math 252
HW-2
Due 9:00am Friday Jan 23
1
1. (a) Prove that the curvature of a circle with radius r is .
r
(b) Find the unite tangent vector T, the normal vector N, and the binormal
vector B of the curve parametrized by
r(t) = (cos3 t, sin3 , 2 sin2 t).
2.
Math 252
HW-5
Due 9:00am Friday Mar 6
1. Consider the transformation
x = u21 u22 ,
y = 2u1 u2 ,
z = u3 .
(a) Show that (u1 , u2 , u3 ) form right-handed orthogonal curvilinear coordinates.
(b) Find f where f = u21 + u22 + u23 . Write f in terms of (x, y,
Homework Assignment #5
MATH 252: Vector Calculus
Due date: Monday February 20, 2017
(happy reading break!)
Answer the following questions from the textbook (Davis and Snider, 7th edition):
Section 2.4, page 102103: #7 (transferred from Homework #4), #11,
Homework Assignment #2
MATH 252: Vector Calculus
Due date: Friday January 20, 2017
Answer the following questions from the textbook (Davis and Snider, 7th edition):
Section 2.2, page 8586: #5, 6, 8, 14.
Section 2.3, page 9598: #4, 7, 8, 16.
Then answer