Chapter 17

Hard

Vector Calculus — Gradient, Divergence, Curl

00 · Symbol Glossary

$\vec F(x,y,z) = \langle P,Q,R\rangle$vector field

A rule assigning a directed quantity (Chapter 14) to every point in a region, rather than a single directed quantity or a single number. $P,Q,R$ are themselves ordinary functions of $x,y,z$ — the components of the arrow attached at each point.

$\nabla$del — the symbolic operator

A directed quantity built not from numbers but from the partial-derivative instructions of Chapter 15: $\nabla = \left\langle\dfrac{\partial}{\partial x},\ \dfrac{\partial}{\partial y},\ \dfrac{\partial}{\partial z}\right\rangle$ . It does nothing by itself — it only produces a result once "multiplied" against a function or a vector field, exactly as Chapter 02's $\dfrac{d}{dx}[\,\cdot\,]$ was an instruction, not a number.

$\nabla\cdot\vec F$divergence of F

The result of combining $\nabla$ and $\vec F$ the way Chapter 14's dot product combines two directed quantities — a plain number at each point, measuring how much the field spreads outward from that point.

$\nabla\times\vec F$curl of F

The result of combining $\nabla$ and $\vec F$ the way Chapter 14's cross product combines two directed quantities — a new directed quantity at each point, measuring the field's tendency to rotate around that point.

$\vec F = \nabla f$conservative vector field

A vector field that is itself the gradient (Chapter 15) of some scalar function $f$ , called a potential function for $\vec F$ .

01 · Vector Fields

Chapter 14 attached a single directed quantity to a curve's motion at one instant. A vector field goes further: it attaches a directed quantity to every point in a region simultaneously — a whole plane or space full of arrows.

Definition — Vector Field

A vector field in the plane is $\vec F(x,y) = \langle P(x,y),\ Q(x,y)\rangle$ — a directed quantity whose components are themselves functions of position. In space, $\vec F(x,y,z) = \langle P,Q,R\rangle$ with three component functions.

Example — Two contrasting vector fields

$\vec F(x,y) = \langle x,y\rangle$ : at the point $(x,y)$ , the attached arrow is $\langle x,y\rangle$ itself — pointing directly away from the origin, growing longer farther out. This is a radial field.

$\vec G(x,y) = \langle -y,x\rangle$ : at $(x,y)$ , the arrow $\langle-y,x\rangle$ is perpendicular to the radial direction $\langle x,y\rangle$ (check with the dot product, Chapter 14: $\langle x,y\rangle\cdot\langle-y,x\rangle = -xy+xy=0$ ) — this field circulates around the origin rather than away from it.

02 · The Del Operator and Divergence

Chapter 15's gradient $\nabla f = \langle f_x,f_y\rangle$ came from applying partial derivatives to a single function and bundling the results into a directed quantity. Writing $\nabla = \langle\partial_x,\partial_y,\partial_z\rangle$ as a directed quantity of operators in its own right lets the same symbol combine with a vector field two different ways — matching the two products Chapter 14 already built.

Definition — Divergence

\nabla\cdot\vec F = \frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}

Exactly Chapter 14's dot-product pattern — multiply matching components and sum — except each "multiplication" of $\partial_x$ against $P$ means "apply $\dfrac{\partial}{\partial x}$ to $P$ ," not ordinary multiplication.

Step-by-step — Compute the divergence of $\vec F(x,y) = \langle x^2y,\ 3xy^2\rangle$

Match the components to $P,Q$ :

P=x^2y

Q=3xy^2

Differentiate $P$ with respect to $x$ (Chapter 15, treating $y$ as constant):

\dfrac{\partial P}{\partial x} = 2xy

Differentiate $Q$ with respect to $y$ (treating $x$ as constant):

\dfrac{\partial Q}{\partial y} = 6xy

Sum, following the divergence formula:

\nabla\cdot\vec F = 2xy+6xy = 8xy

Interpret at a specific point, say $(1,1)$ :

\nabla\cdot\vec F(1,1) = 8

, a positive number — the field is spreading outward (a net "source") at that point.

Geometric intuition — divergence measures outflow

Picture $\vec F$ as the velocity of a flowing fluid at each point. A small region around a point where $\nabla\cdot\vec F>0$ has more fluid leaving than entering — a source. Where $\nabla\cdot\vec F<0$ , more fluid enters than leaves — a sink. Where $\nabla\cdot\vec F=0$ everywhere, the flow is incompressible — whatever flows into any small region flows back out in equal amount.

❌ Confusing the gradient of a function with the divergence of a field

Given $\vec F(x,y)=\langle x^2,y^2\rangle$ , a common mix-up: computing something that looks like $\nabla\vec F = \langle 2x, 2y\rangle$ and calling it "the divergence."

Why it breaks: the gradient (Chapter 15) takes a single scalar-valued function $f(x,y)$ and produces a directed quantity $\langle f_x,f_y\rangle$ . The divergence takes a vector field $\vec F=\langle P,Q\rangle$ (already a directed quantity at each point) and produces a single number, $P_x+Q_y$ . These are different operations on different kinds of input — a vector field has no "gradient" of its own, and a scalar function has no "divergence."

Consequence: for $\vec F=\langle x^2,y^2\rangle$ , the correct divergence is $\nabla\cdot\vec F = \dfrac{\partial}{\partial x}(x^2)+\dfrac{\partial}{\partial y}(y^2) = 2x+2y$ — a single number-valued function, not the directed quantity $\langle2x,2y\rangle$ produced by the mistaken computation. Keep track of what kind of object is being differentiated: a scalar function gets a gradient (output: directed quantity); a vector field gets a divergence (output: number).

03 · Curl

Definition — Curl

For $\vec F(x,y,z) = \langle P,Q,R\rangle$ :

\nabla\times\vec F = \left\langle\ \frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z},\ \ \frac{\partial P}{\partial z}-\frac{\partial R}{\partial x},\ \ \frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\ \right\rangle

Exactly Chapter 14's cross-product component pattern — each component skips one matching index — but with $\partial_x,\partial_y,\partial_z$ standing in for the first directed quantity's components, and $P,Q,R$ for the second.

Step-by-step — Compute the curl of $\vec F(x,y,z) = \langle -y,\ x,\ 0\rangle$ (the rotation field from Section 01, extended into 3D)

Identify $P,Q,R$ :

P=-y

Q=x

R=0

First component — skip $x$ , use $y,z$ derivatives of $R,Q$ :

\dfrac{\partial R}{\partial y}-\dfrac{\partial Q}{\partial z} = 0 - 0 = 0

(

R=0

is constant;

Q=x

does not depend on

z

Second component — skip $y$ , use $z,x$ derivatives of $P,R$ :

\dfrac{\partial P}{\partial z}-\dfrac{\partial R}{\partial x} = 0-0=0

(

P=-y

does not depend on

z

;

R=0

is constant).

Third component — skip $z$ , use $x,y$ derivatives of $Q,P$ :

\dfrac{\partial Q}{\partial x}-\dfrac{\partial P}{\partial y} = 1-(-1) = 2

(

Q=x

differentiated w.r.t.

x

gives

1

;

P=-y

differentiated w.r.t.

y

gives

-1

Assemble:

\nabla\times\vec F = \langle0,0,2\rangle

— a directed quantity pointing straight up along the

z

-axis, with magnitude

2

, matching the visible circulation of the rotation field from Section 01.

❌ Reversing the subtraction order in a curl component

On the same field, computing the third component as $\dfrac{\partial P}{\partial y}-\dfrac{\partial Q}{\partial x} = -1-1=-2$ instead of $\dfrac{\partial Q}{\partial x}-\dfrac{\partial P}{\partial y}=1-(-1)=2$ .

Why it breaks: exactly like Chapter 14's cross product, curl inherits the anti-symmetric structure of the cross-product pattern — each component is a specific difference, and the definition fixes which term comes first ( $\dfrac{\partial Q}{\partial x}$ before $\dfrac{\partial P}{\partial y}$ in the third slot, matching the index pattern from Section 03's definition). Swapping the order flips the sign, the same failure mode as Chapter 14's $\vec a\times\vec b$ vs. $\vec b\times\vec a$ .

Consequence: $-2\neq2$ — the sign of a curl component determines the direction of rotation (by the same right-hand convention as the cross product), so reversing it points the result in exactly the opposite direction, even though its magnitude happens to come out the same in this example.

04 · Conservative Vector Fields

Some vector fields are secretly gradients of an ordinary function — a fact with a direct, checkable consequence involving curl.

Definition — Conservative Field and the Curl Test

$\vec F$ is conservative if $\vec F=\nabla f$ for some scalar function $f$ (called a potential function).

If $\vec F=\langle P,Q\rangle$ is conservative in the plane, then $\nabla\times\vec F = \vec 0$ — this follows directly from Clairaut's theorem (Chapter 15, Section 02): if $P=f_x$ and $Q=f_y$ , then $\dfrac{\partial Q}{\partial x}-\dfrac{\partial P}{\partial y} = f_{yx}-f_{xy}=0$ .

Step-by-step — Find a potential function for $\vec F(x,y) = \langle 2xy,\ x^2+3y^2\rangle$

Check the curl test first (Section 03's third component, 2D case):

\dfrac{\partial Q}{\partial x}-\dfrac{\partial P}{\partial y} = 2x - 2x = 0

— passes the necessary condition, so a potential function may exist.

Set $f_x=P=2xy$ and integrate with respect to $x$ (Chapter 06, treating $y$ as constant):

f(x,y) = x^2y + g(y)

for some as-yet-unknown function

g(y)

(the "constant" of integration with respect to

x

may still depend on

y

Differentiate this candidate $f$ with respect to $y$ and match against $Q$ :

f_y = x^2+g'(y)

. This must equal

Q=x^2+3y^2

, so

g'(y) = 3y^2

Integrate to find $g(y)$ :

g(y) = y^3+C

(Chapter 06 power rule for integration).

Assemble the potential function:

f(x,y) = x^2y+y^3+C

Verify:

f_x = 2xy = P

✓;

f_y = x^2+3y^2=Q

✓.

The curl test is necessary, not always sufficient

On a region with a "hole" removed (for instance, the plane with the origin taken out), a field can satisfy $\nabla\times\vec F=\vec 0$ everywhere it's defined without being conservative — the standard example is the circulation-like field $\vec F = \left\langle \dfrac{-y}{x^2+y^2},\ \dfrac{x}{x^2+y^2}\right\rangle$ , which has zero curl away from the origin but still exhibits genuine rotation when traced all the way around it. Confirming this requires tools (line integrals around closed loops) beyond this chapter's scope — here, the curl test is used only on regions without such holes, where it is both necessary and sufficient.

05 · Exercises

EXERCISE 17.1

Match the field's components to $P$ and $Q$ , differentiate each with respect to its own matching variable, and sum.

$\vec F(x,y) = \langle3x^2y,\ -xy^2\rangle$ . $P=3x^2y$ , $Q=-xy^2$ . $\dfrac{\partial P}{\partial x}=6xy$ . $\dfrac{\partial Q}{\partial y}=-2xy$ . $\nabla\cdot\vec F = 6xy-2xy=4xy$ .

Find the divergence of $\vec F(x,y) = \langle3x^2y,\ -xy^2\rangle$ .

EXERCISE 17.2

Apply each of the three curl component formulas in turn, being careful with the subtraction order in each.

$\vec F(x,y,z)=\langle yz,\ xz,\ xy\rangle$ . First: $\partial_y(xy)-\partial_z(xz) = x-x=0$ . Second: $\partial_z(yz)-\partial_x(xy)=y-y=0$ . Third: $\partial_x(xz)-\partial_y(yz)=z-z=0$ . $\nabla\times\vec F=\langle0,0,0\rangle=\vec 0$ .

Find the curl of $\vec F(x,y,z) = \langle yz,\ xz,\ xy\rangle$ .

EXERCISE 17.3

A field with zero curl everywhere in the plane (no holes removed) passes the necessary condition for having a potential function — build it the same way as Section 04's step-block.

$\vec F(x,y)=\langle3x^2+y,\ x+2y\rangle$ . Curl test: $\partial_x(x+2y)-\partial_y(3x^2+y) = 1-1=0$ ✓. Set $f_x=3x^2+y \implies f=x^3+xy+g(y)$ . Then $f_y=x+g'(y)$ must equal $x+2y$ , so $g'(y)=2y \implies g(y)=y^2$ . $f(x,y)=x^3+xy+y^2+C$ . Verify: $f_x=3x^2+y$ ✓; $f_y=x+2y$ ✓.

Find a potential function for $\vec F(x,y) = \langle3x^2+y,\ x+2y\rangle$ , after first checking the curl test.

EXERCISE 17.4

Recall the two different objects divergence and gradient act on, and what kind of output each produces.

The gradient $\nabla f$ takes a scalar-valued function $f(x,y,z)$ and produces a directed quantity $\langle f_x,f_y,f_z\rangle$ (Chapter 15). The divergence $\nabla\cdot\vec F$ takes a vector field $\vec F=\langle P,Q,R\rangle$ (already a directed quantity at each point) and produces a single number, $P_x+Q_y+R_z$ . A scalar function has no divergence (there's no vector field to dot $\nabla$ against), and a vector field has no gradient (gradient is only defined for scalar-valued functions).

Explain the difference between the gradient and the divergence: what type of object does each take as input, and what type does each produce as output?

EXERCISE 17.5

Model the flow's velocity as a vector field with components proportional to position, then apply the divergence formula directly.

A capital flow field is modeled as $\vec F(x,y) = \langle 2x,\ 2y\rangle$ (dollars per unit time flowing outward at position $(x,y)$ in some abstract market-position space). $\nabla\cdot\vec F = \partial_x(2x)+\partial_y(2y) = 2+2=4$ , a positive constant everywhere — the field is a uniform "source": capital is flowing outward from every point at the same constant rate, consistent with $\vec F$ being a pure radial-scaling field.

A capital flow field is modeled as $\vec F(x,y) = \langle2x,\ 2y\rangle$ . Compute its divergence and interpret whether the flow acts as a source or a sink.

06 · Chapter Summary

Concept	Formula / Rule
Vector field	$\vec F(x,y,z)=\langle P,Q,R\rangle$ ; a directed quantity at every point
Del operator	$\nabla = \langle\partial_x,\partial_y,\partial_z\rangle$
Divergence	$\nabla\cdot\vec F = P_x+Q_y+R_z$ ; scalar; measures outflow
Curl	$\nabla\times\vec F$ ; directed quantity; measures rotation
Conservative field	$\vec F=\nabla f$ for a potential function $f$
Curl test	$\vec F$ conservative $\implies \nabla\times\vec F=\vec 0$ (converse needs no holes in the domain)

This closes the second batch of the Calculus sequence: sequences and series, power/Taylor series with error bounds, the full convergence-test toolkit, parametric and polar curves, directed quantities in 2D/3D, partial derivatives, multiple integrals, and vector calculus — the multivariable and infinite-series extensions of the single-variable toolkit built in Chapters 01–08.

Next: Chapter 18 — Differential Equations: First Order, where derivatives appear inside an equation to be solved, rather than computed directly.