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31.4 Determining the Area or Volume Element

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 d x d y d z .

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 f ( x , z ) d A over an area A in the x y plane, we can invent a third dimension in the k direction, and imagine we are integrating the flux of the vector f ( x , y ) k ^ through the surface A in that plane.

This would be the integral of ( f ( x , y ) k ^ ) · n ^ d S , and with n ^ = k ^ in the x y plane, this flux integral becomes the integral of f d x d y . In this case as in the volume case, the problem of reducing the surface integral to integrals d x d y does not exist.

Suppose now we have a volume or flux integral and wish to change variables from x , y (and z if a volume integral) to say, u , v (and w 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 d x d y or d x d y d z in terms of d u d v or d u d v d w .
We have already noted how to do this but we repeat it here because of its importance.

Making a small change in u induces changes in x , y and z that can be described by the equation

d P ⟶ u = ( ∂ x ∂ u ) d u i ^ + ( ∂ y ∂ u ) d u j ^ + ( ∂ z ∂ u ) d u k ^

with similar expressions for the changes in position d P ⟶ v and d P ⟶ w caused by changes in v and w .

The volume in x y z space induced by small changes d u , d v and d w is the volume of the parallepiped with sides d P ⟶ u , d P ⟶ v and d P ⟶ w , which is the magnitude of the determinant whose rows or columns are these vectors.

We can factor d u d v d w out of the vectors and obtain that d x d y d z = J d u d v d w where J is the magnitude of the determinant formed by the derivatives of x , y and z with respect to u , v and w .

In the case of area, the comparable expression is the same thing: the ratio of the area element in one set of variables d x d y to the other, d u d v , is the magnitude of the determinant formed by the derivatives of x and y with respect to u and v . 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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