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
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
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
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
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.
Ans
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 theor
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
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
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
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
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 int
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 co
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
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 fo