1 Changing variables in multiple integrals

When the method of substitution is used to solve an integral of the form    a b f ( x ) d x   three parts of the integral are changed, the limits, the function and the infinitesimal d x . So if the substitution is of the form x = x ( u ) the u limits, c and d , are found by solving a = x ( c ) and b = x ( d ) and the function is expressed in terms of u as f ( x ( u ) ) .

Figure 28

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Figure 28 shows why the d x needs to be changed. While the δ u is the same length for all u , the δ x change as u changes. The rate at which they change is precisely d d u x ( u ) . This gives the relation

δ x = d x d u δ u

Hence the transformed integral can be written as

a b f ( x ) d x = c d f x u d x d u d u

Here the d x d u is playing the part of the Jacobian that we will define.

Another change of coordinates that you have seen is the transformations from cartesian coordinates ( x , y ) to polar coordinates ( r , θ ) .

Recall that a double integral in polar coordinates is expressed as

f ( x , y ) d x d y = g ( r , θ ) r d r d θ

Figure 29

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We can see from Figure 29 that the area elements change in size as r increases. The circumference of a circle of radius r is 2 π r , so the length of an arc spanned by an angle θ is 2 π r θ 2 π = r θ . Hence the area elements in polar coordinates are approximated by rectangles of width δ r and length r δ θ . Thus under the transformation from cartesian to polar coordinates we have the relation

δ x δ y r δ r δ θ

that is, r δ r δ θ plays the same role as δ x δ y . This is why the r term appears in the integrand. Here r is playing the part of the Jacobian.