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
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
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
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 4.4: 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
A pplications:
( i) F ind a n i rrotational field F having a specified divergence p.
Let be defined by (13) and let F : = ' 7.
( ii) Find a scalar potential for field F in ( i)
(iii) Find a solenoidal field F in a domain D having a spcefied curl . ) Note
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
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
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
Math 252
HW-4
Due 9:00am Friday Feb 27
1. (a) Write e , e , ez at point P = xe1 + ye2 + ze3 in terms of (x, y, z) and
e1 , e2 , e3 .
yi + xj
. Write F in terms of the cylindrical coordinates and
x2 + y2
compute the divergence and curl of F using both the
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MATH 252 HW7 Due 9:00am Friday Mar 27
1. Problem 2 (Section 5.5).
2. (a) Derive Greens theorem from Stokes theorem.
(b) Derive Greens theorem from the 2D divergence theorem.
3. Supopose S is a smooth oriented surface. Suppose its boundary 83 is
positively
MATH 252 Spring 2015
Homework 0
1. Show that if ~u + ~v and ~u ~v are orthogonal, then the vectors ~u and ~v must have the
same length.
2. Let A(2, 1, 3), B(2, 1, 2), C(2, 4, 1), D(0, 1, 2) be four points in R3 .
(a) Show that these four points are not ly
Math 252
HW-3
Due 9:00am Friday Jan 30
1. Suppose r(s) is a curve on a sphere with radius 2. Let (s) be the curvature and (s) = 1/(s) be the radius of curvature. Let (T, N, B) be the Frenet
frame. Prove that
1 d
r = N +
B,
ds
where is the torsion.
Hint: