2: Partial Derivatives
- Page ID
- 7220
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 2.1: Introduction
- This page offers an overview of key formulas related to partial derivatives crucial for thermodynamics. It emphasizes their significance in thermodynamic equations and prepares readers for upcoming mathematical concepts, highlighting their relevance in thermodynamic analysis.
- 2.2: Partial Derivatives
- This page covers partial derivatives for a two-dimensional surface defined by \( z = z(x, y) \), demonstrating how to calculate them with respect to \( x \) and \( y \) while holding the other variable constant. It provides examples to clarify these concepts and highlights the necessity of indicating the constant variable. Finally, it illustrates how variations in \( x \) and \( y \) influence \( z \) and presents a formula for the total rate of change of \( z \) over time.
- 2.3: Implicit Differentiation
- This page covers the differentiation of implicit functions using the equation \( \ln(xy) = x^2 y^3 \). It highlights the complexity of computing \( dy/dx \) directly and presents an approach of interpreting the implicit function as an intersection of a surface and a plane.
- 2.4: Product of Three Partial Derivatives
- This page explores the relationships among three variables, x, y, and z, using algebraic manipulation to express one variable in terms of the others. It outlines equations relating changes among the variables and discusses the significance of partial derivatives, stressing the importance of specifying held constant variables during differentiation. The page concludes by presenting a key relation that shows the interdependence of derivatives in the context of multiple variables.
- 2.5: Second Derivatives and Exact Differentials
- This page covers the differentiation of functions with two variables, focusing on the equality of mixed second derivatives for well-behaved functions. It introduces exact and inexact differentials and illustrates their applications with examples, including integrating factors.
- 2.6: Euler's Theorem for Homogeneous Functions
- This page explains Euler's theorem on homogeneous functions, which are characterized by all terms being of a specific degree n. It provides an example function and illustrates how to evaluate its partial derivatives. The page emphasizes the key equation \( x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} + z \frac{\partial f}{\partial z} = nf \) for a homogeneous function of degree n, encouraging readers to verify this relationship through calculation.
- 2.7: Undetermined Multipliers
- This page explains how to determine the maxima or minima of a function ψ(x, y, z) under constraints defined by f(x, y, z) = 0. It emphasizes that the derivatives of the Lagrangian function Φ = ψ + λf must equal zero, and this methodology can be applied to multiple constraints. The discussion highlights how dependencies among the variables alter standard optimization conditions.
- 2.8: Dee and Delta
- This page explains the differential symbols \(∆\), \(δ\), and \(d\) used in calculus and thermodynamics, highlighting their meanings: \(∆\) for increments and \(δ\) for small increments. It illustrates the relationship with \(y = x^2\), showing how \(δy\) approximates \(2x \, δx\) and converges to \(dy/dx = 2x\) as increments approach zero. The author addresses criticisms of using \(dy\) or \(dx\) independently while applying these symbols to clarify concepts in infinitesimal calculus.