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
γ
in R
3
when a force vector
F
acts is defined to be
w(
γ
)
≡
∫
γ
F
•d
x
(1)
This is to say that term ‘work done in moving along
γ
’ in physics has a specific meaning, the latter
being the value of (1).
(Presumably, this mathematical definition of work meshes well with our
intuitive idea of work done.)
Let me remind you (from the definition in the text) that the shorthand on the right side of
(1) has the following meaning:
Parameterize the path
γ
by choosing an interval, [a, b], on the line,
and a vector valued function
x
(t) for a
≤
t
≤
b which traces out the path
γ
.
With this done, then the
right side of (1) is computed by the ordinary integral
w(
γ
) =
a
b
∫
F
(
x
(t))•
x
´(t) dt
.
(2)
Now, suppose that the path
γ
is the path that the particle would have travelled were its
trajectory
γ
given by Newton’s law; thus
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '11
 GregoryJ.Galloway
 Math, Multivariable Calculus, Force, Mass, 1 m, 2 m, 1 2 W

Click to edit the document details