Re: Differentiation of Line Integral

That is the same formula as given by Hans Lamecker. It is missing the
absolute value over the Jacobian 1/(a-b).

No, it is not missing the Jacobian. See below.

I said it's missing the abs() function from around the Jacobian. That is
the definition I have used to. See below.


Huh? What is a determinant doing in this problem?

Well, that is how the change of variables generalizes to higher
dimensions, giving the volume of the volume-element (not sure of the
right translation) in a given point. Included out of a habit, no effect
in dimension 1.

You are confusing change of variables in an iterated
integral with change of variable in a line integral.

Yes, you are right in the sense that the generalization I gave is not
enough to be used with higher dimensional line integrals (and thus the
det was really confusing there). Here is the generalization I meant to
simplify in there that unifies the task of changing variables in an
integral, whether a line integral or some other:

For a transform
g(p) = r

where
p e R^m
r e R^n
m <= n

dR = |sqrt(det(g'(p)^T g'(p)))| dP

where
^T is for transpose, and g'(p) is always thought as a matrix.

Example 1: m = 1
----------------

Let C be a set a points, parametrizable by a curve g(t) over [a..b].

The line-element is
dr = |sqrt(det(g'(t)^T g'(t)))| dt

because g' is a one parameter function, g'(t) is a column vector:
dr = |sqrt(det(||g'(t)||^2))| * dt
= ||g'(t)|| dt

so if
f : R^n -> R
then

int[C] f(r)dr = int [a..b] f(g(t)) ||g'(t)|| dt

giving the usual definition of the scalar line integral.

Example 2: m = n
----------------

dR = |sqrt(det(g'(p)^T g'(p)))| dP
= |sqrt(det(g'(p)^T) det(g'(p)))| dP
= |sqrt(det(g'(p))^2)| dP
= |det(g'(p))| dP

These transforms include your example of moving from xy plane to uv
plane. A specific example is moving to polar coordinates. In 3D, an
example is moving to spherical coordinates. Same way in higher
dimensions...

Specifically when n = m = 1, we have the familiar change of variables in
one dimension.

Example 3: m = n - 1
--------------------

This is a variable change over a surface integral.

--
Kalle Rutanen
http://kaba.hilvi.org
.

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