Math 21a Supplement 2 on Work and Energy
As mentioned at the end of Section 5.1 in the text, the work done in moving a particle along
a path in R3 when a force vector F acts is defined to be
This is to say that term work done in moving along i
Math 21a Supplement 1 on Work and Energy
Newtons law asserts that the position vector r(t) of a particle of mass m under the
influence of a force F obeys the equation
m r = F .
a) W ork
Suppose that the components of the force vector F do not depend o
Math 21a Handout on Triple Integrals
The purpose of this handout is to provide a few more examples of triple integrals. In
particular, I provide one example in the usual x-y-z coordinates, one in cylindrical coordinates and
one in spherical coordinates.
Math 21a Handout on Surface Area
The text argued that after parameterizing a surface (or part of one) by a function X(u, v),
with u and v coordinates on a region R in R2, then the integral of a function f on the surface is the
same as the integral
f dS f
Math 21a Supplement on Relativity
Matrices play an interesting role in Einsteins theory of special relativity. To set the stage,
imagine two observers with the following characteristics: The first observer is at rest at the origin
on the x-axis. This obse
Math 21a Supplement on
Suppose that an object (planet, asteroid, whatever) travels through space under the
gravitational influence of a star. Newtons laws allow one to describe trajectory of the
planet. This supplement describes the situa
Math 21a Handout on Lagrange Multipliers - Fall 2000
The principal purpose of this handout is to supply some additional examples of the Lagrange multiplier
method for solving constrained equations for three unknowns. In this regard, remember that the basi
What is Integration Good For?
Or how I learned to stop worrying and love the integral,
Or why the heck are we learning all of this?
April 19, 2000
This handout is to explain what an integral means. This handout does not explain how
Math 21a Supplement on Integration and Electro-Magnetism
The Math 21a Supplement on Electricity and Magnetism presented the equations that
describe all known electric and magnetic phenomena. In particular, recall that Maxwells equations
(ix) right side should be a b + a c.
Proof of Theorem 1.2
(i) a > b and b > c means a b P
(ii) a > b and c d means a b P
(iii) a > b and c > 0 means a b P
Theorem 1.6 should be Theorem 1.3
If B A, and B is nonempty,
Math 21a Supplement on Electricity and Magnetism
Vector fields on R3 play a central role in Maxwells theory of electricity and magnetism. In
particular, the electric field in space is described in Maxwells theory by a vector valued function of
space and t
Math 21a Handout on Differential Equations
This handout introduces the subject of differential equations. But it barely scratches the
surface of this vast, growing and extremely useful area of mathematics.
1. Differential equations in the sciences
Math 21a Supplement on Charge Density
Surface integrals arise in the physics of the manner in which electric charge is distributed
over the surface of an electrically charged object. To a first approximation, the story goes as
follows: A typical solid obj
Math 21a Supplement on Center of Mass
Integrals over volumes occur naturally when studying the motions of extended objects.
The fact is that the point particle approximation in Physics is often far from accurate, and in these
cases, the spatial extent of
Math 21a Supplement on Torque and Angular Momentum
According to Newton, the position vector, r(t), of a particle changes with time, t, under the
influence of a force vector F according to the rule
m r = F .
Here, m is the mass of the particle.