V13.3 Stokes Theorem
3. Proof of Stokes Theorem.
We will prove Stokes theorem for a vector eld of the form P (x, y, z) k . That is, we will
show, with the usual notations,
(3)
P (x, y, z) dz =
C
curl (P k ) n dS .
S
We assume S is given as the graph of z
V14. Some Topological Questions
We consider once again the criterion for a gradient eld. We know that
(1)
F = f
curl F = 0 ,
and inquire about the converse. It is natural to try to prove that
(2)
curl F = 0
F = f
by using Stokes theorem: if curl F = 0, th
18.075 In-Class Practice Test I
Fall 2004
Justify your answers. Cross out what is not meant to be part of your nal
answer. Total number of points: 50
I. (5 pts) Find all solutions of the equation
z 4/3 = 1 + i.
II. (Total 10 pts)
1. (3 pts) Can the funct
18.075 Practice Test I for Inclass Exam # 1
Fall 2004
Justify your answers. Cross out what is not meant to be part of your nal
answer. Total number of points: 50
I. (5 pts) Show that for any complex numbers z1 and z2 ,
|z1 + z2 | |z1 | + |z2 |.
II. (5 pt
Problems: Applications of Spherical Coordinates
Find the average distance of a point in a solid sphere of radius a from:
(a) the center,
(b) a xed diameter, and
(c) a xed plane through the center.
Answer: Recall that the average value of a function f (x,
V15.2-3 Relation to Physics
The three theorems we have studied: the divergence theorem and Stokes theorem in
space, and Greens theorem in the plane (which is really just a special case of Stokes theorem) are widely used in physics and continuum mechanics,
Problems: Triple Integrals
1. Set up, but do not evaluate, an integral to nd the volume of the region below the plane
z = y and above the paraboloid z = x2 + y 2 .
Answer: Draw a picture. The plane z = y slices o an thin oblong from the side of the
parabo
V15.1 Del Operator
1. Symbolic notation: the del operator
To have a compact notation, wide use is made of the symbolic operator del (some call
it nabla):
(1)
=
i+
j+
k
x
y
z
M
Recall that the product of
and the function M (x, y, z) is understood to be
.
NAME:
18.075 Inclass Exam # 2
November 3, 2004
Answer all questions. Justify your answers.
Cross out what is not meant to be part of your
nal answer. Total number of points:67.5. Extra 5
points may be given as bonus.
I. Consider the following integral:
Problems: Flux Through General Surfaces
1. Let F = yi + xk and let S be the graph of z = x2 + y 2 above the unit square in the
xy-plane. Find the upward ux of F through S.
Answer: We can save time by noting that F is a tangential vector eld and the vector
18.075 In-class Practice Test for Exam 2
October 29, 2004
Justify your answers. Cross out what is not meant to be part of your solution.
I. (10pts) By use of contour integration, evaluate the real integral
II. Consider the integral
0
cos x
dx.
1 + x2
cos
NAME:
18.075 Inclass Exam # 1
Wednesday, September 29, 2004
Justify your answers. Cross out what is not meant
to be part of your nal answer. Total number of points: 45.
I. (5 pts) Show that for any complex numbers z1 and
z2 ,
|z1| |z2| |z1 + z2|.
II. (5 p
Limits in Spherical Coordinates
Denition of spherical coordinates
= distance to origin, 0
= angle to z-axis, 0
= usual = angle of projection to xy-plane with x-axis, 0 2
Easy trigonometry gives:
z = cos
x = sin cos
y = sin sin .
The equations for x an
NAME: Sow ON 3
18.075 IllClass Exam 1
Wednesday, September 29, 2004
Justify your answers. Cross out what is not meant
to be part of your nal answer. Total number of points: 45.
I. (5 pts) Show that for any complex numbers Z1 and
Z27
le IZ2H S '21 + 22!-
H