Chapter 34 — Key Takeaways
A structured recap of vector fields and the two operators that read their local structure. Use it as a pre-exam checklist; each item points back to the section that develops it.
Vector Field vs. Vector-Valued Function (§34.1)
- A vector field $\mathbf{F}:\mathbb{R}^n\to\mathbb{R}^n$ assigns a vector to every point of space: $\mathbf{F}(x,y) = \langle P, Q\rangle$ in 2D, $\mathbf{F}(x,y,z) = \langle P, Q, R\rangle$ in 3D. Picture: a carpet of arrows filling the room.
- Do not confuse it with the vector-valued function $\mathbf{r}(t) = \langle x(t), y(t), z(t)\rangle$ of Chapter 28, which takes one parameter $t$ and outputs one arrow tracing a single curve.
| Vector-valued function (Ch. 28) | Vector field (Ch. 34) | |
|---|---|---|
| Input | parameter $t\in\mathbb{R}$ | point $(x,y)$ or $(x,y,z)$ |
| Output | one vector $\mathbf{r}(t)$ | a vector at every point |
| Picture | one arrow tracing a curve | a field of arrows |
- Common error: writing $\mathbf{F}(t)$ for a field. Reserve $t$ for a path's parameter; use position variables for a field.
Gradient Fields Are Your First Vector Fields (§34.1–34.2)
- The gradient $\nabla f = \langle f_x, f_y\rangle$ (from Chapter 30) attaches a vector to each point — it is a vector field. Gradient fields are the most important family and reappear as conservative fields in §34.6.
- The gallery worth memorizing: constant $\langle 1,0\rangle$ (uniform), radial $\langle x, y\rangle$ (source), rotational $\langle -y, x\rangle$ (swirl), inverse-square $-\mathbf{r}/r^3$ (decays as $1/r^2$), vortex $\langle -y, x\rangle/r^2$ (decays as $1/r$).
Streamlines / Flow Lines (§34.3)
- A streamline is a curve $\mathbf{r}(t)$ with $\mathbf{r}'(t) = \mathbf{F}(\mathbf{r}(t))$: the particle's velocity equals the field there.
- Solving for streamlines is solving a system of differential equations (Chapter 19). This is where the path $\mathbf{r}(t)$ (Ch. 28) and the field $\mathbf{F}$ (this chapter) meet.
- Examples: constant field → straight lines; radial → rays from the origin; rotational → circles.
Divergence — Measuring Outflow (§34.4)
$$\nabla\cdot\mathbf{F} = P_x + Q_y + R_z \quad (\text{a } \textbf{scalar} \text{ field}).$$
- $\nabla\cdot\mathbf{F} > 0$: source (net outflow). $< 0$: sink. $= 0$: solenoidal (divergence-free, incompressible — a co-moving blob keeps its volume).
- Worked: $\nabla\cdot\langle x,y,z\rangle = 3$; $\nabla\cdot\langle -y, x, 0\rangle = 0$; the inverse-square field has $\nabla\cdot\mathbf{F}=0$ for $r>0$ (all source concentrated at the origin).
Curl — Measuring Rotation (§34.5)
$$\nabla\times\mathbf{F} = \left\langle R_y - Q_z,\ P_z - R_x,\ Q_x - P_y\right\rangle, \qquad \text{2D scalar curl } = Q_x - P_y.$$
- Curl is a vector (3D) or a single scalar (2D). Direction = spin axis (right-hand rule); magnitude = twice the local angular speed.
- $\nabla\times\mathbf{F} = \mathbf{0}$: irrotational (no paddle-wheel spin).
- Worked: $\nabla\times\langle -y, x, 0\rangle = \langle 0,0,2\rangle$; radial and constant fields have zero curl; shear $\langle y, 0\rangle$ has curl $-1$ — shear is rotation in disguise.
- Local, not global: the vortex $\langle -y, x\rangle/r^2$ has zero curl everywhere it is defined yet circulates around the origin. Local irrotationality $\ne$ no global circulation.
Conservative Fields and Potentials (§34.6–34.8)
- $\mathbf{F}$ is conservative (a gradient field) if $\mathbf{F} = \nabla f$ for a potential function $f$. The single scalar $f$ encodes the whole field.
- Master identity: $\nabla\times(\nabla f) = \mathbf{0}$ (Clairaut, Chapter 29) — every conservative field is irrotational. So $\nabla\times\mathbf{F}\ne\mathbf{0}$ instantly proves $\mathbf{F}$ is not conservative.
- Curl Test: on a simply connected domain (no holes), if $\nabla\times\mathbf{F}=\mathbf{0}$ then $\mathbf{F}$ is conservative. The topological hypothesis is essential — the vortex satisfies $\nabla\times\mathbf{F}=\mathbf{0}$ but lives on a punctured plane and is not conservative (full proof via the loop integral in Chapter 35).
- Finding a potential (2D): integrate $f_x = P$ in $x$ to get $f = \int P\,dx + g(y)$; differentiate in $y$, set equal to $Q$, solve for $g'(y)$; integrate.
- Why "conservative": for $\mathbf{F}=\nabla f$, work is path-independent, $\int_C \nabla f\cdot d\mathbf{r} = f(B)-f(A)$ (the Fundamental Theorem for Line Integrals, previewed §34.8, proved Chapter 35); loop work is $0$; energy is conserved. Physical potentials: gravity $\Phi=-GM/r$, voltage $V$, spring $U=\tfrac12 k\|\mathbf{x}\|^2$.
The Operator Zoo (§34.9)
| Operator | Input → Output | Formula (3D) |
|---|---|---|
| Gradient $\nabla f$ | scalar → vector | $\langle f_x, f_y, f_z\rangle$ |
| Divergence $\nabla\cdot\mathbf{F}$ | vector → scalar | $P_x + Q_y + R_z$ |
| Curl $\nabla\times\mathbf{F}$ | vector → vector | $\langle R_y - Q_z, P_z - R_x, Q_x - P_y\rangle$ |
| Laplacian $\Delta f = \nabla\cdot\nabla f$ | scalar → scalar | $f_{xx} + f_{yy} + f_{zz}$ |
- Three independent properties: conservative ($\mathbf{F}=\nabla f$), solenoidal ($\nabla\cdot\mathbf{F}=0$), irrotational ($\nabla\times\mathbf{F}=\mathbf{0}$). A field both solenoidal and irrotational is harmonic ($\Delta f = 0$).
- Two reusable identities: $\nabla\times(\nabla f)=\mathbf{0}$ (curl of a gradient) and $\nabla\cdot(\nabla\times\mathbf{F})=0$ (divergence of a curl). The second is why $\nabla\cdot\mathbf{B}=0$ is automatic once $\mathbf{B}=\nabla\times\mathbf{A}$.
Applications (§34.11–34.12)
- Fluid dynamics: incompressibility is $\nabla\cdot\mathbf{u}=0$; vorticity $\boldsymbol{\omega}=\nabla\times\mathbf{u}$ measures spin; CFD computes divergence and curl on a grid.
- Electromagnetism: Maxwell's equations are written entirely in divergence and curl. $\nabla\cdot\mathbf{E}=\rho/\varepsilon_0$ (charge is the source of $\mathbf{E}$); $\nabla\cdot\mathbf{B}=0$ (no monopoles); the two curl equations couple $\mathbf{E}$ and $\mathbf{B}$ into light.
Common Errors to Avoid
- Writing a field as $\mathbf{F}(t)$ (collapses it to a curve). Use $\mathbf{F}(x,y)$.
- Confusing the outputs: divergence is a scalar, curl is a vector (or 2D scalar).
- Thinking zero curl means "no circulation anywhere." It means no local spin; global circulation can still be nonzero on a non-simply-connected domain.
- Applying the curl test without checking the domain is simply connected.
- Sign slips in the curl formula — keep the cyclic pattern $R_y - Q_z,\ P_z - R_x,\ Q_x - P_y$.
Connections
- Back: gradient (Ch. 30), Clairaut's theorem (Ch. 29), systems of ODEs (Ch. 19), vector-valued functions (Ch. 28), gradient descent (Ch. 6).
- Forward: line integrals and Green's Theorem (Chapter 35), surface integrals/flux (Chapter 36), Stokes' Theorem and the Divergence Theorem give curl and divergence their deepest, integral meaning (Chapter 37), and all the integral theorems unify as one generalized Fundamental Theorem of Calculus (Chapter 38).