V13.1-2 Stokes Theorem
1. Introduction; statement of the theorem.
The normal form of Greens theorem generalizes in 3-space to the divergence theorem.
What is the generalization to space of the tangential form of Greens theorem? It says
F dr =
curl F d
FINITE ELEMENT ANALYSIS OF SOLIDS AND FLUIDS I
Assigned: Session 23
Prof. K. J. Bathe
Problem 1 (20 points):
a) static correction
R=R M i ri
V10.2 The Divergence Theorem
2. Proof of the divergence theorem.
We give an argument assuming rst that the vector eld F has only a k -component:
F = P (x, y, z) k . The theorem then says
P k n dS =
The closed surface S projects into a reg
V4.3 Physical meaning of curl
3. An interpretation for curl F.
We will start by looking at the two dimensional curl in the xy-plane. Our interpretation
will be that the curl at a point represents twice the angular velocity of a small paddle wheel
V8. Vector Fields in Space
Just as in Section V1 we considered vector elds in the plane, so now we consider vector
elds in three-space. These are elds given by a vector function of the type
F(x, y, z) = M (x, y, z) i + N (x, y, z) j + P (x, y, z) k .
V9.1 Surface Integrals
Surface integrals are a natural generalization of line integrals: instead of integrating over
a curve, we integrate over a surface in 3-space. Such integrals are important in any of the
subjects that deal with continuous media (soli
V9.3-4 Surface Integrals
3. Flux through general surfaces.
For a general surface, we will use xyz-coordinates. It turns out that here it is simpler
to calculate the innitesimal vector dS = n dS directly, rather than calculate n and dS
separately and multi
V9.2 Surface Integrals
2. Flux through a cylinder and sphere.
We now show how to calculate the ux integral, beginning with two surfaces where n
and dS are easy to calculate the cylinder and the sphere.
Example 1. Find the ux of F = z i +x j +y k outward t
V10.1 The Divergence Theorem
1. Introduction; statement of the theorem.
The divergence theorem is about closed surfaces, so lets start there. By a closed surface
S we will mean a surface consisting of one connected piece which doesnt intersect itself, and
We use triple integration to calculate the gravitational attraction that a solid body V of
mass M exerts on a unit point mass placed at the origin.
If the solid V is also a point mass, then according to Newtons law of gravitation,
V12. Gradient Fields in Space
1. The criterion for gradient elds. The curl in space.
We seek now to generalize to space our earlier criterion (Section V2) for gradient elds
in the plane.
Criterion for a Gradient Field.
(1) F = f
2.035: Midterm Exam - Part 2 (Take home)
My education was dismal. I went to a school for mentally
Do not spend more than 4 hours.
Please give reasons justifying each (nontrivial) step in your c
2.035: Midterm Exam - Part 1 (In-class)
You may use the notes you took in class and any other handwritten notes in your own
handwriting. No other sources should be used.
Problem 1: Explain each of the following concepts precisely (ye
V11. Line Integrals in Space
1. Curves in space.
In order to generalize to three-space our earlier work with line integrals in the plane, we
begin by recalling the relevant facts about parametrized space curves.
In 3-space, a vector function of one variab
Limits in Iterated Integrals
3. Triple integrals in rectangular and cylindrical coordinates.
You do these the same way, basically. To supply limits for
the region D, we integrate rst with respect to z. Therefore we
dz dy dx over
1. Hold x and y xed,
Partial Dierential Equations
An important application of the higher partial derivatives is that they are used in partial
dierential equations to express some laws of physics which are basic to most science and
engineering subjects. In this section, we wil
2.035: Selected Topics in Mathematics with Applications
Final Exam Spring 2007
Every problem in the calculus of variations has a solution,
provided the word solution is suitably understood.
David Hilbert (1862-1943)
Work any 5 problems.
Pick-up exam: 12:3
18.01 Solutions to Exam 1
Problem 1(15 points) Use the denition of the derivative as a limit of dierence quotients to
compute the derivative of y = x + x for all points x > 0. Show all work.
Solution to Problem 1 Denote by f (x) the function x + x . B