Multiple integrals · Definition

Change of variables in multiple integrals

multivariable calculusmultiple integrals

Before you start you need to know

It is the substitution rule of single-variable integrals taken to the plane: an awkward region becomes a rectangle or a disk in well-chosen variables.

Definition

A change of variables turns the integral of a function \(f\) over a region \(R\) of the \(xy\)-plane into one over a region \(S\) of the \(uv\)-plane, correcting the area with the Jacobian \(J\) of the transformation.

\[\begin{aligned}\iint_R f\,dx\,dy&=\iint_S f(x(u,v),y(u,v))\,|J|\,du\,dv\\J&=\frac{\partial(x,y)}{\partial(u,v)}=\begin{vmatrix}x_u&x_v\\y_u&y_v\end{vmatrix}\end{aligned}\]

A transformation \(T(u,v)=(x(u,v),y(u,v))\) maps a small rectangle with sides \(du\) and \(dv\) to a curved parallelogram whose area is \(|J|\,du\,dv\), where \(J\) is the determinant of the matrix of partial derivatives. Hence \(dA=|J|\,du\,dv\): the Jacobian measures how much \(T\) stretches or shrinks areas locally. Polar coordinates are the case \(J=r\), and with three variables the \(3\times3\) determinant appears, giving \(r\) in cylindrical and \(\rho^2\sin\varphi\) in spherical coordinates.

To apply the method, \(T\) must be \(C^1\), one-to-one (except on the boundary) and have \(J\neq0\) in the interior of \(S\). If you are given \(u=u(x,y)\), \(v=v(x,y)\), compute \(\frac{\partial(u,v)}{\partial(x,y)}\) and use \(\frac{\partial(x,y)}{\partial(u,v)}=1\big/\frac{\partial(u,v)}{\partial(x,y)}\), avoiding solving for \(x,y\) when it is not needed.

Guiding example: for \(x=2u,\ y=3v\) the unit disk \(S\) maps onto the ellipse \(\tfrac{x^2}{4}+\tfrac{y^2}{9}\le1\) with \(|J|=6\), so its area is \(6\pi\). The strategy is to choose \(u,v\) so that the boundary of \(R\) lies on curves \(u=\text{const}\) and \(v=\text{const}\), or so that the integrand simplifies, as in \(e^{(y-x)/(y+x)}\).

Typical mistakes: forgetting the factor \(|J|\), using \(\frac{\partial(u,v)}{\partial(x,y)}\) instead of its reciprocal, not taking the absolute value, and describing the new limits of \(S\) incorrectly. It helps to sketch \(S\) by computing where the vertices or boundary curves of \(R\) go.

Think about it

What does it mean that \(J=0\) at a point?

That there the transformation squashes area to zero (it collapses the rectangle into a curve or a point) and is not locally invertible.

Why is the factor \(r\) in polar coordinates and not \(1\)?

Because a sector of angle \(d\theta\) at distance \(r\) has arc length \(r\,d\theta\); the rectangle \(dr\times r\,d\theta\) has area \(r\,dr\,d\theta\), which is \(|J|\,dr\,d\theta\).

Exercises 10

  1. For the change \(x=2u,\ y=3v\), what is \(|J|=\left|\frac{\partial(x,y)}{\partial(u,v)}\right|\)?
  2. For \(x=u+v,\ y=u-v\), the value of \(|J|\) is:
  3. For polar coordinates \(x=r\cos\theta,\ y=r\sin\theta\), the Jacobian \(\frac{\partial(x,y)}{\partial(r,\theta)}\) is:
  4. If \(x=u^2-v^2,\ y=2uv\), the Jacobian \(\frac{\partial(x,y)}{\partial(u,v)}\) is:
  5. With \(x=2u,\ y=3v\), the area of the ellipse \(\tfrac{x^2}{4}+\tfrac{y^2}{9}\le1\) is:
  6. If \(u=x+y,\ v=x-y\), then \(dx\,dy=\)
  7. If at a point \(J=-3\), what does the transformation do near it?
  8. Let \(R\) be the region given by \(x=u+v,\ y=u-v\) with \(0\le u,v\le1\). Compute \(\iint_R (x^2-y^2)\,dA\).
  9. Let \(R\) be the triangle with vertices \((0,0),(1,0),(0,1)\). With \(u=y-x,\ v=x+y\), compute \(\iint_R e^{(y-x)/(y+x)}\,dA\).
  10. Find the area of the region \(1\le xy\le2,\ 1\le y/x\le2\) using \(u=xy,\ v=y/x\) (positive quadrant).

References

  • James Stewart, Daniel K. Clegg, Saleem Watson, Calculus: Early Transcendentals, 9th edition, Cengage, 2021. §15.9
  • Joel Hass, Christopher Heil, Przemyslaw Bogacki, Maurice D. Weir, George B. Thomas Jr., Thomas' Calculus, 15th edition, Pearson, 2023. §15.7
  • Jerrold E. Marsden, Anthony Tromba, Vector Calculus, 6th edition, W. H. Freeman, 2012. Cap. 6