Chapter 29 — Key Takeaways

A one-page recap of Functions of Several Variables. Use it to review before the quiz and as a quick reference. Section numbers point back to the chapter; chapter numbers point forward and back across the book.

Functions of Several Variables

  • A function of several variables sends a point of higher-dimensional space to a single real number: $f:\mathbb{R}^2 \to \mathbb{R}$, $(x,y) \mapsto f(x,y)$, and similarly $f:\mathbb{R}^3 \to \mathbb{R}$ (Section 29.1).
  • The domain is a region of $\mathbb{R}^2$ (or $\mathbb{R}^3$), not an interval. Find it by the same rules as one variable — no division by zero, no negatives under even roots, positive arguments of $\ln$ — then sketch the region (Section 29.2).
  • The range is the set of attained output values, a subset of $\mathbb{R}$ (Section 29.2).
  • The graph of $f(x,y)$ is a surface $z = f(x,y)$ in $\mathbb{R}^3$, not a curve. Know the catalog: plane, paraboloid (bowl/dome), hyperbolic paraboloid (saddle), cone, hemisphere (Section 29.3).

Level Curves and Level Surfaces

  • A level curve (contour) is $\{(x,y): f(x,y) = c\}$ — points of equal height, like a topographic map (Section 29.4).
  • Reading a contour map: closely spaced contours = steep; widely spaced = gentle; nested closed loops = peak or pit; an "X"/cross = saddle. Spacing (not count) encodes steepness — the information the gradient of Chapter 30 will extract (Section 29.4).
  • A level surface is $\{(x,y,z): f(x,y,z) = c\}$ — the 3D analog. Examples: spheres, paraboloids, hyperboloids; physically, isotherms, isobars, equipotentials, and CT/MRI density surfaces (Section 29.5).

Multivariable Limits and Continuity

  • $\displaystyle\lim_{(x,y)\to(a,b)} f = L$ uses the planar distance $\sqrt{(x-a)^2+(y-b)^2} < \delta$ in place of $|x-a| < \delta$; the $\delta$-region is a disk (Section 29.6).
  • The new subtlety: $(x,y)$ may approach $(a,b)$ along infinitely many paths. The limit exists only if $f$ tends to the same value along every path (Section 29.6).
  • Two-path test (disproves a limit): find two paths giving different values — e.g. along $y = mx$, get $\dfrac{2xy}{x^2+y^2} = \dfrac{2m}{1+m^2}$, which depends on $m$, so the limit fails to exist (Section 29.6).
  • Polar trick (proves a limit): set $x = r\cos\theta,\ y = r\sin\theta$; if $|f| \le g(r)$ with $g(r) \to 0$ independent of $\theta$, the squeeze theorem (Chapter 3) gives the limit (Section 29.6).
  • Continuity: $f$ is continuous at $(a,b)$ if $\lim_{(x,y)\to(a,b)} f = f(a,b)$. Polynomials are continuous everywhere; rational functions wherever the denominator is nonzero (Section 29.7).

Partial Derivatives

  • The partial derivative $f_x$ differentiates in $x$ while holding $y$ constant (and symmetrically for $f_y$). The procedure adds no new rules: differentiate one variable, freeze the rest (Section 29.8).
  • Notation: $\dfrac{\partial f}{\partial x} = f_x = D_x f = \partial_x f$ (Section 29.8).
  • Geometric meaning: $f_x(a,b)$ is the slope of the slice $z = f(x, b)$ (cut by the plane $y = b$); $f_y(a,b)$ is the slope of the slice $x = a$. They are the east–west and north–south slopes — two of infinitely many directional slopes, the rest handled by the directional derivative of Chapter 30 (Section 29.8).
  • Marginal interpretation: in economics $Q_L$ is the marginal product of labor; every "marginal" quantity is a partial derivative (Section 29.8).

Higher Partials and Clairaut's Theorem

  • Differentiating twice gives four second partials: $f_{xx}, f_{yy}, f_{xy}, f_{yx}$ (the last two are mixed) (Section 29.9).
  • Subscript vs. Leibniz order run opposite: $f_{xy}$ means $x$ first then $y$ (left to right); $\dfrac{\partial^2 f}{\partial y\,\partial x}$ means the same but reads right to left (inside-out) (Section 29.9).
  • Clairaut's theorem: if $f_{xy}$ and $f_{yx}$ are both continuous near $(a,b)$, then $f_{xy}(a,b) = f_{yx}(a,b)$ — the order of mixed differentiation doesn't matter (Section 29.9).
  • Clairaut underwrites the symmetric Hessian of the second-derivative test (Chapter 31) and the existence of gradient (curl-free) fields (Chapter 34) (Section 29.9).

Tangent Plane and Linearization

  • The tangent plane at $(a,b)$ is $z = f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b)$ — the surface's best linear approximation, generalizing the Chapter 11 tangent line (Section 29.10).
  • The linearization $L(x,y)$ is that right-hand side; $f(x,y) \approx L(x,y)$ near $(a,b)$ (Section 29.10).
  • The total differential $df = f_x\,dx + f_y\,dy$ gives the first-order change in $f$ — the engine of error propagation (Section 29.10).
  • Approximation error grows like the square of the distance from the base point: trust a linearization only near $(a,b)$ (Section 29.10).

Common Errors to Avoid

  • Concluding a multivariable limit exists because the $x$- and $y$-axes agree — the most common error. Always test $y = x$, the family $y = mx$, and a parabola $y = mx^2$ (Section 29.6).
  • Believing the polar substitution proves a limit by itself: you still need a $\theta$-independent bound $g(r) \to 0$ (Section 29.6 Sidebar).
  • Reading evenly labeled contours as evenly sloped terrain — spacing carries steepness, not the labels (Section 29.4).
  • Confusing the subscript and Leibniz orderings for mixed partials (Section 29.9).
  • Trusting a linearization far from its base point (Section 29.10).

Connections

  • Back to Chapter 3 (squeeze theorem powers the polar limit proof), Chapter 4 (continuity), Chapter 7 (the single-variable rules you reuse for partials), Chapter 9 (critical points where $f' = 0$), Chapter 11 (single-variable linearization), and Chapter 28 (vector-valued functions, the prior step into higher dimensions).
  • Forward to Chapter 30 (gradient and directional derivatives bundle the partials; multivariable chain rule), Chapter 31 (optimization of several variables; second-derivative test with the symmetric Hessian), Chapter 32 (double integrals over regions), and Chapter 34 (vector fields, where Clairaut becomes the curl-free condition).
  • Threads: geometry + algebra are inseparable (every surface has a formula and a picture); approximation is the soul of calculus (the tangent plane); calculus appears in every quantitative field (thermodynamics, economics, machine learning).