∞Calculus IV Unit 16 – Change of Variables in Multiple Integrals

Key Concepts

• Change of variables is a technique used to simplify the evaluation of multiple integrals by transforming the integral to a new coordinate system
• The Jacobian matrix represents the linear transformation between the original and new coordinate systems
• The Jacobian determinant measures the change in volume or area elements during the coordinate transformation
• The Change of Variables Theorem relates the original integral to the transformed integral using the Jacobian determinant
• Common coordinate transformations include polar, cylindrical, and spherical coordinates
  • Polar coordinates $(r, \theta)$ are useful for integrating over circular or radial domains
  • Cylindrical coordinates $(r, \theta, z)$ are suitable for problems with cylindrical symmetry
  • Spherical coordinates $(r, \theta, \phi)$ are used for integrating over spherical regions
• Iterated integrals are evaluated in the new coordinate system by adjusting the limits of integration and the order of integration
• The Jacobian determinant is included as a multiplicative factor in the transformed integral to account for the change in volume or area elements

Motivation and Applications

• Change of variables simplifies the evaluation of multiple integrals by transforming the integral to a coordinate system that better aligns with the geometry of the problem
• Transforming to a suitable coordinate system can reduce the complexity of the integrand and make the integration process more manageable
• Change of variables is particularly useful when the region of integration has a specific geometric shape (circular, cylindrical, or spherical)
• Applications of change of variables include:
  • Calculating volumes and surface areas of solids with symmetries
  • Evaluating integrals in physics and engineering problems involving circular or spherical domains
  • Computing probability distributions and expected values in statistics
• Change of variables can also be used to derive important identities and relationships in vector calculus, such as Green's Theorem and Stokes' Theorem

Jacobian Matrix and Determinant

• The Jacobian matrix is a matrix of partial derivatives that represents the linear transformation between the original and new coordinate systems

• For a transformation from $(x, y)$ to $(u, v)$, the Jacobian matrix is given by:

  $J = \begin{bmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \end{bmatrix}$

• The Jacobian determinant, denoted as $|J|$ or $\det(J)$, is the determinant of the Jacobian matrix

• The Jacobian determinant measures the change in volume or area elements during the coordinate transformation

  • If $|J| > 1$, the transformation expands the volume or area elements
  • If $0 < |J| < 1$, the transformation contracts the volume or area elements
  • If $|J| < 0$, the transformation reverses the orientation of the volume or area elements

• The Jacobian determinant is a scalar function of the new coordinates and is included as a multiplicative factor in the transformed integral

Change of Variables Theorem

• The Change of Variables Theorem relates the original integral to the transformed integral using the Jacobian determinant

• For a double integral over a region $R$ in the $(x, y)$ plane, the Change of Variables Theorem states:

  $\iint_R f(x, y) \, dA = \iint_S f(x(u, v), y(u, v)) \, |J| \, du \, dv$

  where $S$ is the transformed region in the $(u, v)$ plane and $|J|$ is the Jacobian determinant

• For a triple integral over a region $E$ in $(x, y, z)$ space, the Change of Variables Theorem states:

  $\iiint_E f(x, y, z) \, dV = \iiint_T f(x(u, v, w), y(u, v, w), z(u, v, w)) \, |J| \, du \, dv \, dw$

  where $T$ is the transformed region in the $(u, v, w)$ space and $|J|$ is the Jacobian determinant

• The Jacobian determinant accounts for the change in volume or area elements during the transformation and ensures that the transformed integral is equivalent to the original integral

Common Coordinate Transformations

• Polar coordinates $(r, \theta)$ transform a point $(x, y)$ in the Cartesian plane to a point $(r, \theta)$ in the polar plane
  • $x = r \cos \theta$, $y = r \sin \theta$
  • Useful for integrating over circular or radial domains
  • Jacobian determinant for polar coordinates: $|J| = r$
• Cylindrical coordinates $(r, \theta, z)$ extend polar coordinates by adding a vertical component $z$
  • $x = r \cos \theta$, $y = r \sin \theta$, $z = z$
  • Suitable for problems with cylindrical symmetry
  • Jacobian determinant for cylindrical coordinates: $|J| = r$
• Spherical coordinates $(r, \theta, \phi)$ represent a point in 3D space using a radial distance $r$, polar angle $\theta$, and azimuthal angle $\phi$
  • $x = r \sin \phi \cos \theta$, $y = r \sin \phi \sin \theta$, $z = r \cos \phi$
  • Used for integrating over spherical regions
  • Jacobian determinant for spherical coordinates: $|J| = r^2 \sin \phi$
• Other coordinate transformations include elliptic coordinates, parabolic coordinates, and hyperbolic coordinates, which are useful in specific applications

Solving Problems Step-by-Step

• Identify the region of integration in the original coordinate system
• Choose an appropriate coordinate transformation based on the geometry of the region and the integrand
• Express the original coordinates in terms of the new coordinates using the transformation equations
• Compute the Jacobian matrix and the Jacobian determinant
• Transform the integrand by substituting the original coordinates with their expressions in terms of the new coordinates
• Determine the new limits of integration in the transformed coordinate system
  • Sketch the region in the new coordinate system to visualize the new limits
  • Use the transformation equations to find the corresponding limits in the new coordinates
• Rewrite the integral in the new coordinate system, including the Jacobian determinant as a multiplicative factor
• Evaluate the transformed integral using appropriate integration techniques (iterated integrals, substitution, etc.)
• If required, convert the final result back to the original coordinate system

Common Pitfalls and Tips

• Ensure that the Jacobian determinant is included in the transformed integral to account for the change in volume or area elements
• Pay attention to the order of integration when evaluating iterated integrals in the new coordinate system
• Be careful when determining the new limits of integration, especially when the transformation involves trigonometric functions
  • Sketch the region in the new coordinate system to avoid mistakes in the limits
• Remember to adjust the differentials ($dx \, dy$, $dx \, dy \, dz$) to the corresponding differentials in the new coordinate system ($du \, dv$, $du \, dv \, dw$)
• When transforming to polar or spherical coordinates, consider the symmetry of the integrand and the region to simplify the integral
  • If the integrand is independent of $\theta$ in polar coordinates, the integral with respect to $\theta$ can be evaluated separately
• Verify that the Jacobian determinant is non-zero in the region of integration to ensure the transformation is valid
• Practice various types of problems to develop intuition for choosing suitable coordinate transformations

Practice Problems and Examples

• Example 1: Evaluate $\iint_R xy \, dA$ where $R$ is the region bounded by $y = 0$, $y = \sqrt{4 - x^2}$, and $x = 0$ using polar coordinates
  • Solution: Transform to polar coordinates, $x = r \cos \theta$, $y = r \sin \theta$, $|J| = r$
    • $\iint_R xy \, dA = \int_0^{\pi/2} \int_0^2 r^3 \cos \theta \sin \theta \, dr \, d\theta = \frac{\pi}{4}$
• Example 2: Calculate the volume of the solid bounded by the paraboloid $z = x^2 + y^2$ and the plane $z = 2y$ using cylindrical coordinates
  • Solution: Transform to cylindrical coordinates, $x = r \cos \theta$, $y = r \sin \theta$, $z = z$, $|J| = r$
    • $\iiint_E dV = \int_0^{2\pi} \int_0^1 \int_0^{2r\sin\theta} r \, dz \, dr \, d\theta = \frac{2\pi}{3}$
• Example 3: Evaluate $\iiint_E \sqrt{x^2 + y^2 + z^2} \, dV$ where $E$ is the sphere $x^2 + y^2 + z^2 \leq 1$ using spherical coordinates
  • Solution: Transform to spherical coordinates, $x = r \sin \phi \cos \theta$, $y = r \sin \phi \sin \theta$, $z = r \cos \phi$, $|J| = r^2 \sin \phi$
    • $\iiint_E \sqrt{x^2 + y^2 + z^2} \, dV = \int_0^{2\pi} \int_0^{\pi} \int_0^1 r^3 \sin \phi \, dr \, d\phi \, d\theta = \frac{4\pi}{5}$
• Practice problems:
  • Find the area of the region bounded by the cardioid $r = 1 + \cos \theta$ using polar coordinates
  • Calculate the volume of the solid enclosed by the cone $z = \sqrt{x^2 + y^2}$ and the sphere $x^2 + y^2 + z^2 = 1$ using spherical coordinates
  • Evaluate $\iint_R e^{-x^2-y^2} \, dA$ where $R$ is the disk $x^2 + y^2 \leq 4$ using polar coordinates