2.3 Change of Variables and Coordinate Systems

Changing variables in multiple integrals is a powerful technique for simplifying complex calculations. By transforming coordinates, we can make integrals easier to evaluate, especially when dealing with symmetrical problems.

Cylindrical and spherical coordinate systems are particularly useful for problems with specific symmetries. These alternative systems, along with the Jacobian determinant, allow us to tackle a wider range of mathematical and physical scenarios efficiently.

Change of Variables and Coordinate Systems

Change of variables in multiple integrals

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• Change of variables technique simplifies evaluation of multiple integrals by transforming integral from one coordinate system to another
  • New coordinate system chosen to make integral easier to evaluate (Cartesian to polar)
  • Transformation defined by set of equations relating old coordinates to new coordinates ($x = r \cos \theta$, $y = r \sin \theta$)
• Transform multiple integral using change of variables:
  1. Define transformation equations relating old coordinates to new coordinates
  2. Calculate Jacobian matrix, matrix of partial derivatives of transformation equations
  3. Compute Jacobian determinant $|J|$, determinant of Jacobian matrix
  4. Multiply integrand by absolute value of Jacobian determinant $|J|$
  5. Replace old coordinates in integrand and differential elements with new coordinates ($dx dy$ to $r dr d\theta$)
  6. Update limits of integration based on transformation equations
• Transformed integral has form: $\int_D f(x(u,v), y(u,v)) |J| du dv$
  • $u$ and $v$ are new coordinates
  • $x(u,v)$ and $y(u,v)$ are transformation equations

Cylindrical and spherical coordinate systems

• Cylindrical coordinates $(r, \theta, z)$ useful for problems with cylindrical symmetry
  • $r$ represents distance from z-axis
  • $\theta$ represents angle in xy-plane, measured counterclockwise from positive x-axis
  • $z$ represents height along z-axis
  • Transformation equations: $x = r \cos \theta$, $y = r \sin \theta$, $z = z$
  • Volume element: $dV = r dr d\theta dz$
• Spherical coordinates $(\rho, \theta, \phi)$ useful for problems with spherical symmetry
  • $\rho$ represents distance from origin
  • $\theta$ represents angle in xy-plane, measured counterclockwise from positive x-axis
  • $\phi$ represents angle from positive z-axis
  • Transformation equations: $x = \rho \sin \phi \cos \theta$, $y = \rho \sin \phi \sin \theta$, $z = \rho \cos \phi$
  • Volume element: $dV = \rho^2 \sin \phi d\rho d\theta d\phi$
• Evaluate multiple integrals using cylindrical or spherical coordinates:
  1. Identify symmetry of problem and choose appropriate coordinate system
  2. Express integrand and region of integration in terms of new coordinates
  3. Determine limits of integration for each variable based on region of integration
  4. Multiply integrand by appropriate volume element ($r dr d\theta dz$ for cylindrical, $\rho^2 \sin \phi d\rho d\theta d\phi$ for spherical)
  5. Evaluate integral using new coordinates and limits of integration

Jacobian determinant for variable changes

• Jacobian determinant $|J|$ represents scale factor by which volume element changes under coordinate transformation
  • Calculated as determinant of Jacobian matrix, matrix of partial derivatives of transformation equations
• For transformation from $(x, y)$ to $(u, v)$, Jacobian matrix is: $J = \begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{vmatrix}$
• Jacobian determinant adjusts volume element in transformed integral
  • $|J| > 1$, volume element increases under transformation
  • $|J| < 1$, volume element decreases under transformation
  • $|J| = 1$, volume element remains unchanged under transformation
• Solving problems involving Jacobian determinant:
  1. Calculate Jacobian matrix by taking partial derivatives of transformation equations
  2. Compute Jacobian determinant by evaluating determinant of Jacobian matrix
  3. Multiply integrand by absolute value of Jacobian determinant $|J|$ to adjust for change in volume element
  4. Proceed with change of variables process as described in previous objectives