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Home | 18.013A | Chapter 31 |
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Thus far we have addressed questions that arise in doing a flux integral.
In the case of a volume integral, the issues discussed up to now do not exist.
The integrand is the integrand itself, which is a scalar field, and the volume element is .
However the issue of how to express this volume element when you change variables does arise, and we consider that question here. The question and its answer are quite similar in any dimension.
We note at this point that area integrals are easy special cases of flux integrals. If we want to integrate over an area in the plane, we can invent a third dimension in the direction, and imagine we are integrating the flux of the vector through the surface in that plane.
This would be the integral of , and with in the plane, this flux integral becomes the integral of . In this case as in the volume case, the problem of reducing the surface integral to integrals does not exist.
Suppose now we have a volume or flux integral and wish to change variables from (and if a volume integral) to say, (and if same).
We do this when we believe that doing so makes the integrand easier to handle, or sometimes if it simplifies the limits of integration.
We wish to determine how to express
or
in terms of
or
.
We have already noted how to do this but we repeat it here because of its importance.
Making a small change in induces changes in and that can be described by the equation
with similar expressions for the changes in position and caused by changes in and .
The volume in space induced by small changes and is the volume of the parallepiped with sides and , which is the magnitude of the determinant whose rows or columns are these vectors.
We can factor out of the vectors and obtain that where is the magnitude of the determinant formed by the derivatives of and with respect to and .
In the case of area, the comparable expression is the same thing: the ratio of the area element in one set of variables to the other, , is the magnitude of the determinant formed by the derivatives of and with respect to and . In each case the magnitude, also called the absolute value, of this determinant is called the Jacobian of this change of variables.
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