16.3 Applications of change of variables

Change of variables in multiple integrals is a powerful technique for solving complex problems in physics and engineering. It allows us to transform integrals from one coordinate system to another, simplifying calculations and exploiting symmetries in the process.

This section focuses on practical applications of change of variables. We'll explore how it's used to calculate mass, moments of inertia, and surface areas in various coordinate systems. We'll also dive into its role in probability theory and statistics.

Coordinate Systems

Polar and Spherical Coordinate Systems

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• Polar coordinates $(r, \theta)$ represent points in a 2D plane using a distance $r$ from the origin and an angle $\theta$ from the positive x-axis
  • Useful for problems with circular symmetry (circular motion, gravitational fields)
• Spherical coordinates $(r, \theta, \phi)$ extend polar coordinates to 3D space by adding an azimuthal angle $\phi$ measured from the positive z-axis
  • Advantageous for problems with spherical symmetry (electric fields, angular momentum)
• Conversion formulas relate polar and spherical coordinates to Cartesian coordinates $(x, y, z)$
  • Polar: $x = r\cos\theta$, $y = r\sin\theta$
  • Spherical: $x = r\sin\phi\cos\theta$, $y = r\sin\phi\sin\theta$, $z = r\cos\phi$

Cylindrical and Elliptical Coordinate Systems

• Cylindrical coordinates $(r, \theta, z)$ combine polar coordinates in the xy-plane with a Cartesian z-coordinate
  • Suitable for problems with cylindrical symmetry (fluid flow through pipes, electromagnetic waves in waveguides)
• Elliptical coordinates $(\xi, \eta, \phi)$ are based on confocal ellipsoids and hyperboloids
  • Useful in solving partial differential equations (Laplace's equation, wave equation) in ellipsoidal domains
• Coordinate system choice depends on the geometry and symmetry of the problem
  • Simplifies expressions and calculations by aligning with natural boundaries and exploiting symmetries

Physical Applications

Mass and Moment of Inertia Calculations

• Change of variables enables calculation of mass and center of mass for objects with complex geometries
  • Mass element $dm = \rho(x, y, z) \, dV$ in Cartesian coordinates transforms to $dm = \rho(r, \theta, \phi) \, r^2\sin\phi \, dr \, d\theta \, d\phi$ in spherical coordinates
• Moment of inertia tensor $I$ quantifies an object's resistance to rotational acceleration
  • Diagonal elements $I_{xx}, I_{yy}, I_{zz}$ represent moments of inertia about principal axes
  • Off-diagonal elements $I_{xy}, I_{xz}, I_{yz}$ are products of inertia
• Appropriate coordinate system choice simplifies integration and exploits symmetry
  • Spherical coordinates for spherical objects (planets, stars)
  • Cylindrical coordinates for cylindrical objects (shafts, flywheels)

Surface Area Calculations

• Surface area of parametric surfaces $\mathbf{r}(u, v) = (x(u, v), y(u, v), z(u, v))$ given by double integral $A = \iint \left\lVert \frac{\partial \mathbf{r}}{\partial u} \times \frac{\partial \mathbf{r}}{\partial v} \right\rVert \, du \, dv$
  • Cross product of partial derivatives yields surface element $dS$
• Coordinate transformations map surface integrals to parameter space $(u, v)$
  • Sphere of radius $R$: $\mathbf{r}(\theta, \phi) = (R\sin\phi\cos\theta, R\sin\phi\sin\theta, R\cos\phi)$, $A = \int_0^{2\pi} \int_0^\pi R^2\sin\phi \, d\phi \, d\theta = 4\pi R^2$
• Gauss-Ostrogradsky theorem relates surface integrals to volume integrals
  • Flux of vector field $\mathbf{F}$ through closed surface $S$ equals volume integral of divergence $\nabla \cdot \mathbf{F}$ over enclosed volume $V$

Probability Distributions

Probability Density Functions and Random Variables

• Probability density functions (PDFs) $f(x_1, \ldots, x_n)$ describe continuous multivariate probability distributions
  • Joint PDF of random variables $X_1, \ldots, X_n$ gives probability $P((X_1, \ldots, X_n) \in A) = \int_A f(x_1, \ldots, x_n) \, dx_1 \ldots dx_n$ for any measurable set $A$
• Change of variables theorem relates PDFs under coordinate transformations
  • If $\mathbf{Y} = \mathbf{g}(\mathbf{X})$ is a one-to-one transformation with inverse $\mathbf{X} = \mathbf{h}(\mathbf{Y})$, then $f_\mathbf{Y}(\mathbf{y}) = f_\mathbf{X}(\mathbf{h}(\mathbf{y})) \, |\det J_\mathbf{h}(\mathbf{y})|$
• Marginal and conditional PDFs obtained by integration and division
  • Marginal PDF $f_{X_i}(x_i) = \int f(x_1, \ldots, x_n) \, dx_1 \ldots dx_{i-1} dx_{i+1} \ldots dx_n$
  • Conditional PDF $f_{X_i|X_j}(x_i|x_j) = \frac{f(x_i, x_j)}{f_{X_j}(x_j)}$

Applications in Statistics and Machine Learning

• Multivariate normal distribution widely used for modeling correlated random variables
  • PDF $f(\mathbf{x}) = \frac{1}{\sqrt{(2\pi)^n \det \Sigma}} \exp\left(-\frac{1}{2}(\mathbf{x} - \mathbf{\mu})^\top \Sigma^{-1} (\mathbf{x} - \mathbf{\mu})\right)$ with mean vector $\mathbf{\mu}$ and covariance matrix $\Sigma$
• Bayesian inference updates prior distributions to posterior distributions based on observed data
  • Posterior PDF $f(\mathbf{\theta}|\mathbf{x}) \propto f(\mathbf{x}|\mathbf{\theta}) \, f(\mathbf{\theta})$ by Bayes' theorem, where $f(\mathbf{x}|\mathbf{\theta})$ is likelihood and $f(\mathbf{\theta})$ is prior
• Coordinate transformations enable efficient sampling and integration in high-dimensional spaces
  • Markov chain Monte Carlo (MCMC) methods (Metropolis-Hastings, Gibbs sampling) generate samples from complex distributions
  • Variational inference approximates intractable posteriors with simpler distributions by minimizing Kullback-Leibler divergence